Trio of 4-patterns There all together, 317, different equivalence classes For the equivalence class of patterns, { {[2, 1, 3, 4], [4, 2, 3, 1], [4, 2, 1, 3]}, {[1, 2, 4, 3], [2, 4, 3, 1], [4, 2, 3, 1]}, {[1, 3, 4, 2], [1, 3, 2, 4], [3, 4, 2, 1]}, {[1, 3, 2, 4], [2, 3, 1, 4], [3, 4, 2, 1]}, {[1, 4, 2, 3], [1, 3, 2, 4], [4, 3, 1, 2]}, {[1, 3, 2, 4], [3, 1, 2, 4], [4, 3, 1, 2]}, {[1, 2, 4, 3], [4, 1, 3, 2], [4, 2, 3, 1]}, {[2, 1, 3, 4], [3, 2, 4, 1], [4, 2, 3, 1]}} the member , {[1, 4, 2, 3], [1, 3, 2, 4], [4, 3, 1, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[3, 4, 1, 2, 5], {}, {1}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[4, 1, 2, 3], {}, {3}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}], [[2, 1, 3, 4], {}, {3}], [[2, 1, 3], {}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[4, 5, 2, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[1, 2, 3], {}, {2}], [[3, 4, 1, 2], {}, {}], [[3, 1, 2, 4], {}, {}], [[2, 1, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[4, 5, 1, 2, 3], {}, {4}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [ [4, 5, 1, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 5, 1, 2, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 2, 4, 5], {}, {4}], [[4, 2, 3, 5, 1], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {2}], [[2, 3, 1, 4], {}, {1}], [[3, 1, 2], {}, {}], [[4, 1, 2, 5, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [ [4, 1, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[3, 1, 2, 5, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+4*q^2+6*q^3+5*q^4+2*q^5+q^6, 1+2*q+4*q^2+8*q^3+ 12*q^4+15*q^5+15*q^6+10*q^7+5*q^8+2*q^9+q^10, 1+2*q+4*q^2+8*q^3+14*q^4+21*q^5+ 30*q^6+38*q^7+41*q^8+35*q^9+28*q^10+19*q^11+10*q^12+5*q^13+2*q^14+q^15, 1+2*q+4 *q^2+8*q^3+14*q^4+23*q^5+36*q^6+51*q^7+68*q^8+88*q^9+98*q^10+101*q^11+96*q^12+ 83*q^13+64*q^14+47*q^15+32*q^16+19*q^17+10*q^18+5*q^19+2*q^20+q^21, 1+2*q+4*q^2 +8*q^3+14*q^4+23*q^5+38*q^6+57*q^7+81*q^8+113*q^9+149*q^10+185*q^11+219*q^12+ 238*q^13+251*q^14+253*q^15+236*q^16+206*q^17+172*q^18+135*q^19+103*q^20+76*q^21 +51*q^22+32*q^23+19*q^24+10*q^25+5*q^26+2*q^27+q^28, 1+2*q+4*q^2+8*q^3+14*q^4+ 23*q^5+38*q^6+59*q^7+87*q^8+126*q^9+174*q^10+234*q^11+303*q^12+380*q^13+448*q^ 14+512*q^15+566*q^16+609*q^17+627*q^18+623*q^19+592*q^20+533*q^21+475*q^22+406* q^23+334*q^24+265*q^25+207*q^26+155*q^27+115*q^28+80*q^29+51*q^30+32*q^31+19*q^ 32+10*q^33+5*q^34+2*q^35+q^36, 1+2*q+4*q^2+8*q^3+14*q^4+23*q^5+38*q^6+59*q^7+89 *q^8+132*q^9+187*q^10+259*q^11+352*q^12+462*q^13+591*q^14+737*q^15+885*q^16+ 1025*q^17+1171*q^18+1299*q^19+1410*q^20+1492*q^21+1529*q^22+1525*q^23+1486*q^24 +1407*q^25+1295*q^26+1163*q^27+1021*q^28+877*q^29+736*q^30+610*q^31+491*q^32+ 389*q^33+300*q^34+227*q^35+167*q^36+119*q^37+80*q^38+51*q^39+32*q^40+19*q^41+10 *q^42+5*q^43+2*q^44+q^45, 1+2*q+4*q^2+8*q^3+14*q^4+23*q^5+38*q^6+59*q^7+89*q^8+ 134*q^9+193*q^10+272*q^11+377*q^12+511*q^13+673*q^14+878*q^15+1112*q^16+1378*q^ 17+1667*q^18+1964*q^19+2270*q^20+2588*q^21+2881*q^22+3152*q^23+3392*q^24+3560*q ^25+3670*q^26+3718*q^27+3695*q^28+3587*q^29+3436*q^30+3212*q^31+2952*q^32+2669* q^33+2367*q^34+2069*q^35+1790*q^36+1524*q^37+1282*q^38+1065*q^39+859*q^40+691*q ^41+546*q^42+424*q^43+320*q^44+239*q^45+171*q^46+119*q^47+80*q^48+51*q^49+32*q^ 50+19*q^51+10*q^52+5*q^53+2*q^54+q^55, 1+2*q+4*q^2+8*q^3+14*q^4+23*q^5+38*q^6+ 59*q^7+89*q^8+134*q^9+195*q^10+278*q^11+390*q^12+536*q^13+722*q^14+960*q^15+ 1253*q^16+1603*q^17+2023*q^18+2500*q^19+3030*q^20+3603*q^21+4222*q^22+4865*q^23 +5529*q^24+6193*q^25+6829*q^26+7412*q^27+7933*q^28+8362*q^29+8684*q^30+8900*q^ 31+8986*q^32+8936*q^33+8764*q^34+8472*q^35+8055*q^36+7567*q^37+7010*q^38+6412*q ^39+5795*q^40+5188*q^41+4580*q^42+4008*q^43+3477*q^44+2983*q^45+2538*q^46+2137* q^47+1773*q^48+1458*q^49+1184*q^50+940*q^51+746*q^52+581*q^53+444*q^54+332*q^55 +243*q^56+171*q^57+119*q^58+80*q^59+51*q^60+32*q^61+19*q^62+10*q^63+5*q^64+2*q^ 65+q^66, 1+2*q+4*q^2+8*q^3+14*q^4+23*q^5+38*q^6+59*q^7+89*q^8+134*q^9+195*q^10+ 280*q^11+396*q^12+549*q^13+747*q^14+1009*q^15+1335*q^16+1744*q^17+2248*q^18+ 2854*q^19+3570*q^20+4409*q^21+5347*q^22+6389*q^23+7544*q^24+8788*q^25+10109*q^ 26+11498*q^27+12908*q^28+14314*q^29+15707*q^30+17009*q^31+18208*q^32+19268*q^33 +20174*q^34+20881*q^35+21379*q^36+21618*q^37+21614*q^38+21381*q^39+20913*q^40+ 20223*q^41+19353*q^42+18313*q^43+17149*q^44+15916*q^45+14627*q^46+13330*q^47+ 12036*q^48+10799*q^49+9606*q^50+8490*q^51+7435*q^52+6476*q^53+5591*q^54+4797*q^ 55+4083*q^56+3442*q^57+2873*q^58+2374*q^59+1945*q^60+1577*q^61+1265*q^62+995*q^ 63+781*q^64+601*q^65+456*q^66+336*q^67+243*q^68+171*q^69+119*q^70+80*q^71+51*q^ 72+32*q^73+19*q^74+10*q^75+5*q^76+2*q^77+q^78, 1+2*q+4*q^2+8*q^3+14*q^4+23*q^5+ 38*q^6+59*q^7+89*q^8+134*q^9+195*q^10+280*q^11+398*q^12+555*q^13+760*q^14+1034* q^15+1384*q^16+1826*q^17+2389*q^18+3079*q^19+3924*q^20+4947*q^21+6158*q^22+7566 *q^23+9193*q^24+11014*q^25+13057*q^26+15328*q^27+17789*q^28+20431*q^29+23251*q^ 30+26176*q^31+29186*q^32+32248*q^33+35253*q^34+38174*q^35+40988*q^36+43592*q^37 +45926*q^38+47979*q^39+49634*q^40+50869*q^41+51710*q^42+52071*q^43+51942*q^44+ 51397*q^45+50368*q^46+48949*q^47+47188*q^48+45082*q^49+42711*q^50+40181*q^51+ 37498*q^52+34740*q^53+31991*q^54+29241*q^55+26561*q^56+24016*q^57+21586*q^58+ 19286*q^59+17154*q^60+15144*q^61+13312*q^62+11633*q^63+10100*q^64+8712*q^65+ 7485*q^66+6374*q^67+5393*q^68+4530*q^69+3765*q^70+3106*q^71+2546*q^72+2064*q^73 +1658*q^74+1320*q^75+1030*q^76+801*q^77+613*q^78+460*q^79+336*q^80+243*q^81+171 *q^82+119*q^83+80*q^84+51*q^85+32*q^86+19*q^87+10*q^88+5*q^89+2*q^90+q^91, 1+2* q+4*q^2+8*q^3+14*q^4+23*q^5+38*q^6+59*q^7+89*q^8+134*q^9+195*q^10+280*q^11+398* q^12+557*q^13+766*q^14+1047*q^15+1409*q^16+1875*q^17+2471*q^18+3220*q^19+4149*q ^20+5301*q^21+6696*q^22+8375*q^23+10376*q^24+12721*q^25+15423*q^26+18515*q^27+ 22023*q^28+25948*q^29+30307*q^30+35090*q^31+40270*q^32+45832*q^33+51743*q^34+ 57926*q^35+64336*q^36+70871*q^37+77466*q^38+84014*q^39+90452*q^40+96626*q^41+ 102440*q^42+107813*q^43+112601*q^44+116717*q^45+120101*q^46+122666*q^47+124359* q^48+125172*q^49+125083*q^50+124083*q^51+122204*q^52+119529*q^53+116079*q^54+ 111974*q^55+107310*q^56+102165*q^57+96652*q^58+90917*q^59+84997*q^60+79060*q^61 +73150*q^62+67334*q^63+61685*q^64+56272*q^65+51094*q^66+46197*q^67+41615*q^68+ 37308*q^69+33316*q^70+29625*q^71+26213*q^72+23085*q^73+20254*q^74+17656*q^75+ 15340*q^76+13248*q^77+11379*q^78+9715*q^79+8258*q^80+6958*q^81+5828*q^82+4849*q ^83+3998*q^84+3278*q^85+2665*q^86+2145*q^87+1713*q^88+1355*q^89+1050*q^90+813*q ^91+617*q^92+460*q^93+336*q^94+243*q^95+171*q^96+119*q^97+80*q^98+51*q^99+32*q^ 100+19*q^101+10*q^102+5*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+4*q^4+6*q^3+5*q^2+2*q+1, q^10+2*q^9+4*q^8+8 *q^7+12*q^6+15*q^5+15*q^4+10*q^3+5*q^2+2*q+1, q^15+2*q^14+4*q^13+8*q^12+14*q^11 +21*q^10+30*q^9+38*q^8+41*q^7+35*q^6+28*q^5+19*q^4+10*q^3+5*q^2+2*q+1, q^21+2*q ^20+4*q^19+8*q^18+14*q^17+23*q^16+36*q^15+51*q^14+68*q^13+88*q^12+98*q^11+101*q ^10+96*q^9+83*q^8+64*q^7+47*q^6+32*q^5+19*q^4+10*q^3+5*q^2+2*q+1, 1+8*q^25+38*q ^22+14*q^24+57*q^21+23*q^23+2*q+5*q^2+4*q^26+10*q^3+19*q^4+32*q^5+51*q^6+76*q^7 +103*q^8+135*q^9+172*q^10+206*q^11+236*q^12+253*q^13+251*q^14+238*q^15+219*q^16 +q^28+185*q^17+149*q^18+113*q^19+81*q^20+2*q^27, 1+234*q^25+448*q^22+303*q^24+ 512*q^21+380*q^23+2*q+38*q^30+59*q^29+4*q^34+8*q^33+5*q^2+174*q^26+10*q^3+19*q^ 4+32*q^5+51*q^6+14*q^32+80*q^7+115*q^8+155*q^9+207*q^10+265*q^11+334*q^12+406*q ^13+475*q^14+533*q^15+592*q^16+87*q^28+623*q^17+627*q^18+609*q^19+566*q^20+126* q^27+23*q^31+2*q^35+q^36, 1+1410*q^25+8*q^42+1525*q^22+1492*q^24+1486*q^21+1529 *q^23+2*q+2*q^44+737*q^30+q^45+89*q^37+885*q^29+38*q^39+14*q^41+259*q^34+59*q^ 38+352*q^33+5*q^2+1299*q^26+10*q^3+19*q^4+32*q^5+51*q^6+462*q^32+80*q^7+119*q^8 +167*q^9+227*q^10+300*q^11+389*q^12+491*q^13+610*q^14+736*q^15+877*q^16+4*q^43+ 1025*q^28+1021*q^17+1163*q^18+1295*q^19+1407*q^20+1171*q^27+23*q^40+591*q^31+ 187*q^35+132*q^36, 1+3436*q^25+511*q^42+2669*q^22+3212*q^24+2367*q^21+2952*q^23 +2*q+14*q^51+272*q^44+4*q^53+3560*q^30+8*q^52+q^55+2*q^54+193*q^45+1667*q^37+ 3670*q^29+134*q^46+1112*q^39+673*q^41+2588*q^34+23*q^50+1378*q^38+2881*q^33+5*q ^2+3587*q^26+10*q^3+19*q^4+32*q^5+51*q^6+3152*q^32+80*q^7+119*q^8+171*q^9+239*q ^10+320*q^11+424*q^12+546*q^13+691*q^14+859*q^15+1065*q^16+377*q^43+38*q^49+ 3718*q^28+1282*q^17+1524*q^18+1790*q^19+2069*q^20+3695*q^27+878*q^40+3392*q^31+ 2270*q^35+59*q^48+1964*q^36+89*q^47, 1+5188*q^25+5529*q^42+3477*q^22+4580*q^24+ 2983*q^21+4008*q^23+2*q+960*q^51+4222*q^44+536*q^53+134*q^57+8055*q^30+722*q^52 +89*q^58+278*q^55+38*q^60+390*q^54+3603*q^45+8362*q^37+7567*q^29+3030*q^46+2*q^ 65+7412*q^39+23*q^61+6193*q^41+8986*q^34+1253*q^50+4*q^64+7933*q^38+8936*q^33+ 195*q^56+5*q^2+5795*q^26+10*q^3+19*q^4+32*q^5+51*q^6+8764*q^32+80*q^7+119*q^8+ 171*q^9+14*q^62+243*q^10+332*q^11+444*q^12+581*q^13+746*q^14+940*q^15+1184*q^16 +q^66+4865*q^43+1603*q^49+7010*q^28+1458*q^17+1773*q^18+2137*q^19+2538*q^20+ 6412*q^27+6829*q^40+8*q^63+8472*q^31+8900*q^35+2023*q^48+59*q^59+8684*q^36+2500 *q^47, 1+6476*q^25+21379*q^42+4083*q^22+5591*q^24+3442*q^21+4797*q^23+2*q+11498 *q^51+20174*q^44+8788*q^53+4409*q^57+12036*q^30+10109*q^52+3570*q^58+6389*q^55+ 2248*q^60+7544*q^54+19268*q^45+59*q^71+14*q^74+20223*q^37+280*q^67+10799*q^29+ 18208*q^46+549*q^65+134*q^69+21381*q^39+1744*q^61+21618*q^41+17149*q^34+12908*q ^50+747*q^64+20913*q^38+15916*q^33+5347*q^56+5*q^2+195*q^68+7435*q^26+10*q^3+19 *q^4+32*q^5+51*q^6+14627*q^32+80*q^7+119*q^8+171*q^9+1335*q^62+243*q^10+336*q^ 11+456*q^12+4*q^76+601*q^13+781*q^14+995*q^15+1265*q^16+396*q^66+89*q^70+2*q^77 +q^78+20881*q^43+14314*q^49+9606*q^28+1577*q^17+1945*q^18+2374*q^19+2873*q^20+ 8490*q^27+38*q^72+21614*q^40+23*q^73+1009*q^63+13330*q^31+8*q^75+18313*q^35+ 15707*q^48+2854*q^59+19353*q^36+17009*q^47, 1+7485*q^25+45082*q^42+4530*q^22+2* q^90+6374*q^24+3765*q^21+5393*q^23+2*q+49634*q^51+48949*q^44+45926*q^53+35253*q ^57+15144*q^30+47979*q^52+32248*q^58+40988*q^55+26176*q^60+q^91+43592*q^54+ 50368*q^45+3924*q^71+1826*q^74+31991*q^37+9193*q^67+13312*q^29+51397*q^46+13057 *q^65+6158*q^69+37498*q^39+23251*q^61+42711*q^41+24016*q^34+398*q^79+50869*q^50 +15328*q^64+34740*q^38+21586*q^33+38174*q^56+5*q^2+7566*q^68+8712*q^26+10*q^3+ 19*q^4+32*q^5+51*q^6+19286*q^32+80*q^7+119*q^8+171*q^9+20431*q^62+243*q^10+336* q^11+460*q^12+1034*q^76+613*q^13+801*q^14+1030*q^15+1320*q^16+11014*q^66+4947*q ^70+760*q^77+555*q^78+47188*q^43+51710*q^49+11633*q^28+1658*q^17+2064*q^18+2546 *q^19+3106*q^20+10100*q^27+3079*q^72+40181*q^40+2389*q^73+17789*q^63+17154*q^31 +1384*q^75+26561*q^35+280*q^80+52071*q^48+89*q^83+195*q^81+38*q^85+23*q^86+14*q ^87+8*q^88+4*q^89+59*q^84+134*q^82+29186*q^59+29241*q^36+51942*q^47, 1+8258*q^ 25+67334*q^42+4849*q^22+1047*q^90+6958*q^24+3998*q^21+5828*q^23+2*q+89*q^97+ 116079*q^51+79060*q^44+122204*q^53+124359*q^57+17656*q^30+280*q^94+119529*q^52+ 122666*q^58+125083*q^55+116717*q^60+766*q^91+124083*q^54+84997*q^45+51743*q^71+ 35090*q^74+134*q^96+59*q^98+41615*q^37+77466*q^67+15340*q^29+90917*q^46+398*q^ 93+38*q^99+90452*q^65+64336*q^69+51094*q^39+112601*q^61+61685*q^41+29625*q^34+ 15423*q^79+111974*q^50+96626*q^64+23*q^100+46197*q^38+26213*q^33+125172*q^56+5* q^2+70871*q^68+9715*q^26+10*q^3+19*q^4+32*q^5+51*q^6+23085*q^32+80*q^7+119*q^8+ 171*q^9+195*q^95+107813*q^62+14*q^101+243*q^10+336*q^11+460*q^12+25948*q^76+617 *q^13+813*q^14+1050*q^15+1355*q^16+84014*q^66+57926*q^70+22023*q^77+18515*q^78+ 73150*q^43+8*q^102+107310*q^49+13248*q^28+1713*q^17+2145*q^18+2665*q^19+3278*q^ 20+11379*q^27+45832*q^72+56272*q^40+40270*q^73+4*q^103+102440*q^63+20254*q^31+ 30307*q^75+33316*q^35+2*q^104+12721*q^80+102165*q^48+q^105+6696*q^83+10376*q^81 +4149*q^85+3220*q^86+2471*q^87+1875*q^88+1409*q^89+5301*q^84+8375*q^82+120101*q ^59+37308*q^36+557*q^92+96652*q^47] The number of permutations avoiding, {[1, 4, 2, 3], [1, 3, 2, 4], [4, 3, 1, 2]}, is given by [1, 2, 6, 21, 75, 259, 853, 2684, 8120, 23782, 67845, 189493, 520359, 1409742, 3778514] The number of EVEN permutations avoiding, {[1, 4, 2, 3], [1, 3, 2, 4], [4, 3, 1, 2]}, is given by [1, 1, 3, 11, 38, 130, 425, 1344, 4061, 11891, 33923, 94746, 260178, 704873, 1889258] The number of ODD permutations avoiding, {[1, 4, 2, 3], [1, 3, 2, 4], [4, 3, 1, 2]}, is given by [0, 1, 3, 10, 37, 129, 428, 1340, 4059, 11891, 33922, 94747, 260181, 704869, 1889256] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 38, 129, 428, 1344, 4061, 11891, 33922, 94746, 260178, 704869, 1889256] ODD:, [0, 1, 3, 10, 37, 130, 425, 1340, 4059, 11891, 33923, 94747, 260181, 704873, 1889258] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.133333333, 7.725868726, 10.79484174, 14.31594635, 18.27032020, 22.64313346, 27.42247771, 32.59861314, 38.16348521, 44.11042943, 50.43400342] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.430157850, 1.975404319, 2.622693272, 3.376544377, 4.230331780, 5.174035776, 6.197294033, 7.290359942, 8.444283213, 9.650844557, 10.90244596, 12.19202899] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.04535, 3.90222, 6.87852, 11.4011, 17.8957, 26.7706, 38.4065, 53.1493, 71.3059, 93.1388, 118.863, 148.646] Skewness: , [Float(undefined), 0., 0., -0.08364977329, -0.1232282447, -0.1114064500, -0.07533950983, -0.03344308707, 0.006616036966, 0.04254746725, 0.07413594581, 0.1018489228, 0.1262632447, 0.1478977112, 0.1671589550] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.679411074, 2.911214665, 2.954831329, 2.940491297, 2.921059311, 2.911108996, 2.911971179, 2.921587192, 2.937615858, 2.958246105, 2.982245825, 3.008657821] end of this data For the equivalence class of patterns, { {[3, 4, 2, 1], [3, 2, 1, 4], [4, 1, 3, 2]}, {[2, 4, 3, 1], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 2, 4, 3], [2, 3, 4, 1], [3, 1, 2, 4]}, {[1, 4, 2, 3], [2, 3, 4, 1], [2, 1, 3, 4]}, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, {[1, 2, 4, 3], [2, 3, 1, 4], [4, 1, 2, 3]}, {[1, 3, 4, 2], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [3, 2, 4, 1], [4, 3, 1, 2]}} the member , {[1, 4, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 1, 0, 0]}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[1, 4, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 5, 1, 4, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {[1, 1, 0, 0, 0], [0, 1, 1, 0, 0], [1, 0, 1, 1, 0]}, {3}], [[1, 3, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 2, 4, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {2}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [1, 0, 1, 0, 0]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0]}, {1}], [[3, 5, 2, 4, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[1, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 1, 1, 0, 0]}, {1}], [ [2, 5, 1, 3, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0]}, {4}], [ [2, 4, 1, 3, 5], {[0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}], [ [3, 5, 1, 4, 2], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 3, 5, 1, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0]}, {1}], [ [2, 4, 5, 1, 3], {[0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {2}], [[3, 4, 5, 2, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 3, 4, 1, 5], {[1, 0, 0, 0, 0, 0]}, {1}], [[3, 4, 5, 1, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[2, 1, 3], {[1, 1, 0, 0]}, {1}], [[2, 3, 4, 1], {[1, 0, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 1, 1, 0]}, {}], [[1, 2, 3], {}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 2, 3, 4], {}, {2}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+2*q^5+q^6, 1+4*q+9*q^2+13*q^3 +14*q^4+12*q^5+10*q^6+5*q^7+3*q^8+2*q^9+q^10, 1+5*q+14*q^2+26*q^3+36*q^4+39*q^5 +38*q^6+30*q^7+23*q^8+16*q^9+10*q^10+7*q^11+3*q^12+3*q^13+2*q^14+q^15, 1+6*q+20 *q^2+45*q^3+76*q^4+101*q^5+114*q^6+112*q^7+99*q^8+77*q^9+60*q^10+44*q^11+33*q^ 12+24*q^13+16*q^14+10*q^15+6*q^16+5*q^17+3*q^18+3*q^19+2*q^20+q^21, 1+7*q+27*q^ 2+71*q^3+141*q^4+222*q^5+292*q^6+332*q^7+340*q^8+312*q^9+266*q^10+209*q^11+166* q^12+128*q^13+99*q^14+80*q^15+56*q^16+43*q^17+27*q^18+20*q^19+14*q^20+11*q^21+7 *q^22+4*q^23+5*q^24+3*q^25+3*q^26+2*q^27+q^28, 1+8*q+35*q^2+105*q^3+239*q^4+434 *q^5+656*q^6+852*q^7+984*q^8+1030*q^9+992*q^10+885*q^11+749*q^12+604*q^13+485*q ^14+391*q^15+309*q^16+244*q^17+190*q^18+147*q^19+113*q^20+88*q^21+61*q^22+44*q^ 23+33*q^24+23*q^25+16*q^26+15*q^27+11*q^28+8*q^29+5*q^30+4*q^31+5*q^32+3*q^33+3 *q^34+2*q^35+q^36, 1+9*q+44*q^2+148*q^3+379*q^4+778*q^5+1330*q^6+1949*q^7+2519* q^8+2937*q^9+3149*q^10+3132*q^11+2933*q^12+2598*q^13+2216*q^14+1832*q^15+1510*q ^16+1229*q^17+990*q^18+796*q^19+627*q^20+501*q^21+394*q^22+317*q^23+246*q^24+ 196*q^25+146*q^26+113*q^27+83*q^28+63*q^29+45*q^30+31*q^31+28*q^32+21*q^33+19*q ^34+13*q^35+12*q^36+8*q^37+6*q^38+5*q^39+4*q^40+5*q^41+3*q^42+3*q^43+2*q^44+q^ 45, 1+10*q+54*q^2+201*q^3+571*q^4+1305*q^5+2488*q^6+4065*q^7+5833*q^8+7510*q^9+ 8845*q^10+9661*q^11+9899*q^12+9599*q^13+8889*q^14+7912*q^15+6846*q^16+5805*q^17 +4862*q^18+4028*q^19+3300*q^20+2692*q^21+2159*q^22+1731*q^23+1387*q^24+1117*q^ 25+897*q^26+730*q^27+588*q^28+473*q^29+374*q^30+291*q^31+222*q^32+173*q^33+129* q^34+98*q^35+80*q^36+61*q^37+49*q^38+39*q^39+27*q^40+23*q^41+22*q^42+19*q^43+16 *q^44+10*q^45+9*q^46+6*q^47+6*q^48+5*q^49+4*q^50+5*q^51+3*q^52+3*q^53+2*q^54+q^ 55, 1+11*q+65*q^2+265*q^3+826*q^4+2077*q^5+4365*q^6+7867*q^7+12430*q^8+17556*q^ 9+22568*q^10+26803*q^11+29779*q^12+31241*q^13+31229*q^14+29930*q^15+27705*q^16+ 24900*q^17+21915*q^18+18934*q^19+16158*q^20+13628*q^21+11367*q^22+9389*q^23+ 7683*q^24+6239*q^25+5029*q^26+4084*q^27+3313*q^28+2715*q^29+2217*q^30+1805*q^31 +1458*q^32+1195*q^33+961*q^34+783*q^35+635*q^36+505*q^37+395*q^38+307*q^39+230* q^40+176*q^41+150*q^42+116*q^43+98*q^44+78*q^45+65*q^46+49*q^47+42*q^48+32*q^49 +27*q^50+25*q^51+19*q^52+20*q^53+16*q^54+13*q^55+7*q^56+7*q^57+6*q^58+6*q^59+5* q^60+4*q^61+5*q^62+3*q^63+3*q^64+2*q^65+q^66, 1+12*q+77*q^2+341*q^3+1156*q^4+ 3168*q^5+7269*q^6+14319*q^7+24716*q^8+38054*q^9+53126*q^10+68242*q^11+81691*q^ 12+92080*q^13+98595*q^14+101027*q^15+99698*q^16+95254*q^17+88587*q^18+80565*q^ 19+71952*q^20+63293*q^21+54968*q^22+47177*q^23+40025*q^24+33656*q^25+27992*q^26 +23109*q^27+18951*q^28+15518*q^29+12686*q^30+10400*q^31+8522*q^32+6995*q^33+ 5735*q^34+4710*q^35+3856*q^36+3179*q^37+2627*q^38+2175*q^39+1781*q^40+1440*q^41 +1168*q^42+950*q^43+769*q^44+615*q^45+481*q^46+379*q^47+304*q^48+240*q^49+196*q ^50+158*q^51+133*q^52+115*q^53+100*q^54+83*q^55+66*q^56+53*q^57+42*q^58+35*q^59 +31*q^60+26*q^61+27*q^62+21*q^63+17*q^64+17*q^65+13*q^66+10*q^67+5*q^68+7*q^69+ 6*q^70+6*q^71+5*q^72+4*q^73+5*q^74+3*q^75+3*q^76+2*q^77+q^78, 1+13*q+90*q^2+430 *q^3+1574*q^4+4665*q^5+11594*q^6+24767*q^7+46369*q^8+77354*q^9+116723*q^10+ 161510*q^11+207488*q^12+250134*q^13+285561*q^14+311095*q^15+325594*q^16+329177* q^17+323090*q^18+309133*q^19+289536*q^20+266254*q^21+241159*q^22+215458*q^23+ 190184*q^24+165949*q^25+143299*q^26+122534*q^27+103817*q^28+87277*q^29+72861*q^ 30+60504*q^31+50037*q^32+41328*q^33+34075*q^34+28125*q^35+23196*q^36+19127*q^37 +15753*q^38+13025*q^39+10755*q^40+8902*q^41+7379*q^42+6140*q^43+5124*q^44+4290* q^45+3549*q^46+2922*q^47+2417*q^48+1977*q^49+1611*q^50+1298*q^51+1041*q^52+839* q^53+693*q^54+560*q^55+463*q^56+378*q^57+308*q^58+251*q^59+216*q^60+180*q^61+ 152*q^62+130*q^63+111*q^64+100*q^65+86*q^66+74*q^67+53*q^68+47*q^69+35*q^70+34* q^71+30*q^72+27*q^73+26*q^74+23*q^75+19*q^76+14*q^77+14*q^78+10*q^79+8*q^80+5*q ^81+7*q^82+6*q^83+6*q^84+5*q^85+4*q^86+5*q^87+3*q^88+3*q^89+2*q^90+q^91, 1+14*q +104*q^2+533*q^3+2094*q^4+6669*q^5+17834*q^6+41038*q^7+82807*q^8+148831*q^9+ 241603*q^10+358767*q^11+493066*q^12+633780*q^13+768981*q^14+887708*q^15+981702* q^16+1046162*q^17+1079771*q^18+1084065*q^19+1062753*q^20+1020691*q^21+963182*q^ 22+895177*q^23+820908*q^24+743666*q^25+666138*q^26+590507*q^27+518270*q^28+ 450830*q^29+388863*q^30+332873*q^31+282832*q^32+238886*q^33+200639*q^34+167916* q^35+140111*q^36+116644*q^37+96878*q^38+80434*q^39+66688*q^40+55254*q^41+45712* q^42+37816*q^43+31358*q^44+26168*q^45+21891*q^46+18359*q^47+15401*q^48+12886*q^ 49+10764*q^50+9003*q^51+7514*q^52+6288*q^53+5288*q^54+4405*q^55+3661*q^56+3036* q^57+2481*q^58+2016*q^59+1635*q^60+1321*q^61+1088*q^62+905*q^63+759*q^64+642*q^ 65+554*q^66+466*q^67+387*q^68+324*q^69+270*q^70+227*q^71+195*q^72+167*q^73+148* q^74+139*q^75+115*q^76+97*q^77+89*q^78+72*q^79+60*q^80+50*q^81+39*q^82+35*q^83+ 33*q^84+31*q^85+26*q^86+27*q^87+22*q^88+21*q^89+16*q^90+11*q^91+11*q^92+8*q^93+ 8*q^94+5*q^95+7*q^96+6*q^97+6*q^98+5*q^99+4*q^100+5*q^101+3*q^102+3*q^103+2*q^ 104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+5*q^3+4*q^2+2*q+1, q^10+4*q^9+9*q^8+ 13*q^7+14*q^6+12*q^5+10*q^4+5*q^3+3*q^2+2*q+1, q^15+5*q^14+14*q^13+26*q^12+36*q ^11+39*q^10+38*q^9+30*q^8+23*q^7+16*q^6+10*q^5+7*q^4+3*q^3+3*q^2+2*q+1, q^21+6* q^20+20*q^19+45*q^18+76*q^17+101*q^16+114*q^15+112*q^14+99*q^13+77*q^12+60*q^11 +44*q^10+33*q^9+24*q^8+16*q^7+10*q^6+6*q^5+5*q^4+3*q^3+3*q^2+2*q+1, 1+71*q^25+ 292*q^22+141*q^24+332*q^21+222*q^23+2*q+3*q^2+27*q^26+3*q^3+5*q^4+4*q^5+7*q^6+ 11*q^7+14*q^8+20*q^9+27*q^10+43*q^11+56*q^12+80*q^13+99*q^14+128*q^15+166*q^16+ q^28+209*q^17+266*q^18+312*q^19+340*q^20+7*q^27, 1+885*q^25+485*q^22+749*q^24+ 391*q^21+604*q^23+2*q+656*q^30+852*q^29+35*q^34+105*q^33+3*q^2+992*q^26+3*q^3+5 *q^4+4*q^5+5*q^6+239*q^32+8*q^7+11*q^8+15*q^9+16*q^10+23*q^11+33*q^12+44*q^13+ 61*q^14+88*q^15+113*q^16+984*q^28+147*q^17+190*q^18+244*q^19+309*q^20+1030*q^27 +434*q^31+8*q^35+q^36, 1+627*q^25+148*q^42+317*q^22+501*q^24+246*q^21+394*q^23+ 2*q+9*q^44+1832*q^30+q^45+2519*q^37+1510*q^29+1330*q^39+379*q^41+3132*q^34+1949 *q^38+2933*q^33+3*q^2+796*q^26+3*q^3+5*q^4+4*q^5+5*q^6+2598*q^32+6*q^7+8*q^8+12 *q^9+13*q^10+19*q^11+21*q^12+28*q^13+31*q^14+45*q^15+63*q^16+44*q^43+1229*q^28+ 83*q^17+113*q^18+146*q^19+196*q^20+990*q^27+778*q^40+2216*q^31+3149*q^35+2937*q ^36, 1+374*q^25+9599*q^42+173*q^22+291*q^24+129*q^21+222*q^23+2*q+571*q^51+9661 *q^44+54*q^53+1117*q^30+201*q^52+q^55+10*q^54+8845*q^45+4862*q^37+897*q^29+7510 *q^46+6846*q^39+8889*q^41+2692*q^34+1305*q^50+5805*q^38+2159*q^33+3*q^2+473*q^ 26+3*q^3+5*q^4+4*q^5+5*q^6+1731*q^32+6*q^7+6*q^8+9*q^9+10*q^10+16*q^11+19*q^12+ 22*q^13+23*q^14+27*q^15+39*q^16+9899*q^43+2488*q^49+730*q^28+49*q^17+61*q^18+80 *q^19+98*q^20+588*q^27+7912*q^40+1387*q^31+3300*q^35+4065*q^48+4028*q^36+5833*q ^47, 1+176*q^25+7683*q^42+98*q^22+150*q^24+78*q^21+116*q^23+2*q+29930*q^51+ 11367*q^44+31241*q^53+17556*q^57+635*q^30+31229*q^52+12430*q^58+26803*q^55+4365 *q^60+29779*q^54+13628*q^45+2715*q^37+505*q^29+16158*q^46+11*q^65+4084*q^39+ 2077*q^61+6239*q^41+1458*q^34+27705*q^50+65*q^64+3313*q^38+1195*q^33+22568*q^56 +3*q^2+230*q^26+3*q^3+5*q^4+4*q^5+5*q^6+961*q^32+6*q^7+6*q^8+7*q^9+826*q^62+7*q ^10+13*q^11+16*q^12+20*q^13+19*q^14+25*q^15+27*q^16+q^66+9389*q^43+24900*q^49+ 395*q^28+32*q^17+42*q^18+49*q^19+65*q^20+307*q^27+5029*q^40+265*q^63+783*q^31+ 1805*q^35+21915*q^48+7867*q^59+2217*q^36+18934*q^47, 1+115*q^25+3856*q^42+66*q^ 22+100*q^24+53*q^21+83*q^23+2*q+23109*q^51+5735*q^44+33656*q^53+63293*q^57+304* q^30+27992*q^52+71952*q^58+47177*q^55+88587*q^60+40025*q^54+6995*q^45+14319*q^ 71+1156*q^74+1440*q^37+68242*q^67+240*q^29+8522*q^46+92080*q^65+38054*q^69+2175 *q^39+95254*q^61+3179*q^41+769*q^34+18951*q^50+98595*q^64+1781*q^38+615*q^33+ 54968*q^56+3*q^2+53126*q^68+133*q^26+3*q^3+5*q^4+4*q^5+5*q^6+481*q^32+6*q^7+6*q ^8+7*q^9+99698*q^62+5*q^10+10*q^11+13*q^12+77*q^76+17*q^13+17*q^14+21*q^15+27*q ^16+81691*q^66+24716*q^70+12*q^77+q^78+4710*q^43+15518*q^49+196*q^28+26*q^17+31 *q^18+35*q^19+42*q^20+158*q^27+7269*q^72+2627*q^40+3168*q^73+101027*q^63+379*q^ 31+341*q^75+950*q^35+12686*q^48+80565*q^59+1168*q^36+10400*q^47, 1+86*q^25+1977 *q^42+47*q^22+13*q^90+74*q^24+35*q^21+53*q^23+2*q+10755*q^51+2922*q^44+15753*q^ 53+34075*q^57+180*q^30+13025*q^52+41328*q^58+23196*q^55+60504*q^60+q^91+19127*q ^54+3549*q^45+289536*q^71+329177*q^74+693*q^37+190184*q^67+152*q^29+4290*q^46+ 143299*q^65+241159*q^69+1041*q^39+72861*q^61+1611*q^41+378*q^34+207488*q^79+ 8902*q^50+122534*q^64+839*q^38+308*q^33+28125*q^56+3*q^2+215458*q^68+100*q^26+3 *q^3+5*q^4+4*q^5+5*q^6+251*q^32+6*q^7+6*q^8+7*q^9+87277*q^62+5*q^10+8*q^11+10*q ^12+311095*q^76+14*q^13+14*q^14+19*q^15+23*q^16+165949*q^66+266254*q^70+285561* q^77+250134*q^78+2417*q^43+7379*q^49+130*q^28+26*q^17+27*q^18+30*q^19+34*q^20+ 111*q^27+309133*q^72+1298*q^40+323090*q^73+103817*q^63+216*q^31+325594*q^75+463 *q^35+161510*q^80+6140*q^48+46369*q^83+116723*q^81+11594*q^85+4665*q^86+1574*q^ 87+430*q^88+90*q^89+24767*q^84+77354*q^82+50037*q^59+560*q^36+5124*q^47, 1+60*q ^25+905*q^42+35*q^22+887708*q^90+50*q^24+33*q^21+39*q^23+2*q+82807*q^97+5288*q^ 51+1321*q^44+7514*q^53+15401*q^57+139*q^30+358767*q^94+6288*q^52+18359*q^58+ 10764*q^55+26168*q^60+768981*q^91+9003*q^54+1635*q^45+200639*q^71+332873*q^74+ 148831*q^96+41038*q^98+387*q^37+96878*q^67+115*q^29+2016*q^46+493066*q^93+17834 *q^99+66688*q^65+140111*q^69+554*q^39+31358*q^61+759*q^41+227*q^34+666138*q^79+ 4405*q^50+55254*q^64+6669*q^100+466*q^38+195*q^33+12886*q^56+3*q^2+116644*q^68+ 72*q^26+3*q^3+5*q^4+4*q^5+5*q^6+167*q^32+6*q^7+6*q^8+7*q^9+241603*q^95+37816*q^ 62+2094*q^101+5*q^10+8*q^11+8*q^12+450830*q^76+11*q^13+11*q^14+16*q^15+21*q^16+ 80434*q^66+167916*q^70+518270*q^77+590507*q^78+1088*q^43+533*q^102+3661*q^49+97 *q^28+22*q^17+27*q^18+26*q^19+31*q^20+89*q^27+238886*q^72+642*q^40+282832*q^73+ 104*q^103+45712*q^63+148*q^31+388863*q^75+270*q^35+14*q^104+743666*q^80+3036*q^ 48+q^105+963182*q^83+820908*q^81+1062753*q^85+1084065*q^86+1079771*q^87+1046162 *q^88+981702*q^89+1020691*q^84+895177*q^82+21891*q^59+324*q^36+633780*q^92+2481 *q^47] The number of permutations avoiding, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 74, 254, 858, 2889, 9775, 33371, 115135, 401538, 1414821, 5032193, 18049762] The number of EVEN permutations avoiding, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 11, 38, 127, 430, 1445, 4888, 16686, 57562, 200773, 707383, 2516113, 9024772] The number of ODD permutations avoiding, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 10, 36, 127, 428, 1444, 4887, 16685, 57573, 200765, 707438, 2516080, 9024990] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 38, 127, 428, 1445, 4888, 16685, 57573, 200773, 707383, 2516080, 9024990] ODD:, [0, 1, 3, 10, 36, 127, 430, 1444, 4887, 16686, 57562, 200765, 707438, 2516113, 9024772] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.857142857, 4.378378378, 5.948818898, 7.531468531, 9.123572170, 10.73626598, 12.38485511, 14.08430972, 15.84724982, 17.68313023, 19.59816406, 21.59573484] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.489190073, 2.084062449, 2.710677811, 3.331640075, 3.921737167, 4.472548764, 4.987456340, 5.475985867, 5.950033282, 6.421532797, 6.900973610, 7.396584778] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.21769, 4.34332, 7.34777, 11.0998, 15.3800, 20.0037, 24.8747, 29.9864, 35.4029, 41.2361, 47.6234, 54.7095] Skewness: , [Float(undefined), 0., 0., 0.1589038037, 0.3868984957, 0.6150592287, 0.8021174213, 0.9382608799, 1.026095044, 1.072484830, 1.085283345, 1.072231984, 1.040778014, 0.9979774702, 0.9500731465] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.435936804, 2.835421156, 3.340283794, 3.891423407, 4.407206427, 4.822615467, 5.104075033, 5.246889435, 5.265615581, 5.185284245, 5.035475872, 4.845664223] end of this data For the equivalence class of patterns, { {[3, 4, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, {[3, 4, 1, 2], [3, 2, 4, 1], [4, 2, 3, 1]}, {[1, 3, 4, 2], [1, 3, 2, 4], [2, 1, 4, 3]}, {[2, 4, 3, 1], [3, 4, 1, 2], [4, 2, 3, 1]}, {[1, 4, 2, 3], [1, 3, 2, 4], [2, 1, 4, 3]}, {[1, 3, 2, 4], [2, 3, 1, 4], [2, 1, 4, 3]}, {[1, 3, 2, 4], [2, 1, 4, 3], [3, 1, 2, 4]}, {[3, 4, 1, 2], [4, 1, 3, 2], [4, 2, 3, 1]}} the member , {[3, 4, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {}, {1}], [[2, 1], {[1, 1, 0]}, {2}], [[1, 3, 2], {[1, 0, 1, 0], [0, 1, 1, 0]}, {3}], [[2, 3, 1], {[1, 0, 1, 0], [0, 1, 0, 0]}, {3}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+3*q^4+2*q^5+q^6, 1+4*q+9*q^2+15*q^3 +16*q^4+12*q^5+9*q^6+5*q^7+3*q^8+2*q^9+q^10, 1+5*q+14*q^2+29*q^3+43*q^4+48*q^5+ 42*q^6+33*q^7+24*q^8+17*q^9+13*q^10+7*q^11+5*q^12+3*q^13+2*q^14+q^15, 1+6*q+20* q^2+49*q^3+90*q^4+129*q^5+148*q^6+139*q^7+121*q^8+97*q^9+77*q^10+59*q^11+42*q^ 12+31*q^13+23*q^14+19*q^15+10*q^16+7*q^17+5*q^18+3*q^19+2*q^20+q^21, 1+7*q+27*q ^2+76*q^3+164*q^4+282*q^5+396*q^6+460*q^7+462*q^8+427*q^9+370*q^10+311*q^11+253 *q^12+197*q^13+155*q^14+127*q^15+98*q^16+72*q^17+55*q^18+42*q^19+32*q^20+25*q^ 21+15*q^22+10*q^23+7*q^24+5*q^25+3*q^26+2*q^27+q^28, 1+8*q+35*q^2+111*q^3+273*q ^4+542*q^5+891*q^6+1229*q^7+1455*q^8+1527*q^9+1482*q^10+1357*q^11+1197*q^12+ 1022*q^13+844*q^14+705*q^15+583*q^16+476*q^17+380*q^18+305*q^19+248*q^20+203*q^ 21+162*q^22+122*q^23+95*q^24+72*q^25+57*q^26+43*q^27+34*q^28+20*q^29+15*q^30+10 *q^31+7*q^32+5*q^33+3*q^34+2*q^35+q^36, 1+9*q+44*q^2+155*q^3+426*q^4+953*q^5+ 1782*q^6+2827*q^7+3866*q^8+4647*q^9+5041*q^10+5088*q^11+4859*q^12+4464*q^13+ 3965*q^14+3441*q^15+2965*q^16+2532*q^17+2142*q^18+1781*q^19+1492*q^20+1253*q^21 +1053*q^22+876*q^23+718*q^24+583*q^25+481*q^26+397*q^27+327*q^28+261*q^29+204*q ^30+159*q^31+123*q^32+97*q^33+74*q^34+58*q^35+45*q^36+28*q^37+20*q^38+15*q^39+ 10*q^40+7*q^41+5*q^42+3*q^43+2*q^44+q^45, 1+10*q+54*q^2+209*q^3+633*q^4+1569*q^ 5+3272*q^6+5836*q^7+9027*q^8+12281*q^9+14951*q^10+16649*q^11+17335*q^12+17134*q ^13+16278*q^14+14982*q^15+13472*q^16+11964*q^17+10515*q^18+9132*q^19+7862*q^20+ 6762*q^21+5819*q^22+5005*q^23+4288*q^24+3629*q^25+3078*q^26+2622*q^27+2236*q^28 +1902*q^29+1603*q^30+1332*q^31+1104*q^32+920*q^33+758*q^34+629*q^35+520*q^36+ 420*q^37+328*q^38+263*q^39+206*q^40+162*q^41+125*q^42+99*q^43+75*q^44+60*q^45+ 38*q^46+28*q^47+20*q^48+15*q^49+10*q^50+7*q^51+5*q^52+3*q^53+2*q^54+q^55, 1+11* q+65*q^2+274*q^3+905*q^4+2455*q^5+5629*q^6+11104*q^7+19099*q^8+28987*q^9+39315* q^10+48393*q^11+55020*q^12+58778*q^13+59762*q^14+58452*q^15+55422*q^16+51337*q^ 17+46838*q^18+42175*q^19+37567*q^20+33174*q^21+29189*q^22+25644*q^23+22497*q^24 +19652*q^25+17067*q^26+14843*q^27+12915*q^28+11264*q^29+9806*q^30+8498*q^31+ 7314*q^32+6308*q^33+5416*q^34+4656*q^35+4004*q^36+3432*q^37+2907*q^38+2446*q^39 +2052*q^40+1718*q^41+1430*q^42+1190*q^43+984*q^44+817*q^45+663*q^46+527*q^47+ 423*q^48+336*q^49+267*q^50+209*q^51+164*q^52+127*q^53+100*q^54+77*q^55+52*q^56+ 38*q^57+28*q^58+20*q^59+15*q^60+10*q^61+7*q^62+5*q^63+3*q^64+2*q^65+q^66, 1+12* q+77*q^2+351*q^3+1254*q^4+3688*q^5+9198*q^6+19822*q^7+37413*q^8+62535*q^9+93539 *q^10+126668*q^11+157373*q^12+182004*q^13+198642*q^14+206860*q^15+207445*q^16+ 201793*q^17+191764*q^18+179106*q^19+164945*q^20+150171*q^21+135499*q^22+121629* q^23+108816*q^24+97078*q^25+86268*q^26+76404*q^27+67615*q^28+59880*q^29+53062*q ^30+46973*q^31+41480*q^32+36560*q^33+32185*q^34+28320*q^35+24931*q^36+21939*q^ 37+19275*q^38+16856*q^39+14677*q^40+12757*q^41+11065*q^42+9565*q^43+8269*q^44+ 7133*q^45+6129*q^46+5224*q^47+4430*q^48+3734*q^49+3153*q^50+2644*q^51+2210*q^52 +1837*q^53+1531*q^54+1264*q^55+1035*q^56+833*q^57+673*q^58+537*q^59+433*q^60+ 340*q^61+270*q^62+211*q^63+166*q^64+128*q^65+102*q^66+68*q^67+52*q^68+38*q^69+ 28*q^70+20*q^71+15*q^72+10*q^73+7*q^74+5*q^75+3*q^76+2*q^77+q^78, 1+13*q+90*q^2 +441*q^3+1693*q^4+5358*q^5+14414*q^6+33614*q^7+68901*q^8+125485*q^9+204979*q^10 +303173*q^11+410240*q^12+513762*q^13+602708*q^14+669843*q^15+712001*q^16+729667 *q^17+725860*q^18+705246*q^19+672745*q^20+632223*q^21+586959*q^22+539818*q^23+ 493215*q^24+448532*q^25+406291*q^26+366643*q^27+329793*q^28+296283*q^29+266080* q^30+238904*q^31+214346*q^32+192067*q^33+171889*q^34+153747*q^35+137526*q^36+ 122995*q^37+110062*q^38+98374*q^39+87760*q^40+78109*q^41+69458*q^42+61639*q^43+ 54662*q^44+48438*q^45+42860*q^46+37823*q^47+33287*q^48+29147*q^49+25493*q^50+ 22236*q^51+19350*q^52+16790*q^53+14548*q^54+12564*q^55+10819*q^56+9268*q^57+ 7893*q^58+6698*q^59+5680*q^60+4790*q^61+4028*q^62+3374*q^63+2815*q^64+2341*q^65 +1947*q^66+1596*q^67+1295*q^68+1055*q^69+851*q^70+685*q^71+547*q^72+437*q^73+ 343*q^74+272*q^75+213*q^76+167*q^77+130*q^78+92*q^79+68*q^80+52*q^81+38*q^82+28 *q^83+20*q^84+15*q^85+10*q^86+7*q^87+5*q^88+3*q^89+2*q^90+q^91, 1+14*q+104*q^2+ 545*q^3+2236*q^4+7569*q^5+21816*q^6+54641*q^7+120631*q^8+237298*q^9+419675*q^10 +672868*q^11+986393*q^12+1334423*q^13+1682697*q^14+1998068*q^15+2255450*q^16+ 2440322*q^17+2548894*q^18+2585932*q^19+2562289*q^20+2491071*q^21+2384316*q^22+ 2253303*q^23+2108082*q^24+1957311*q^25+1806790*q^26+1659922*q^27+1518640*q^28+ 1384787*q^29+1260318*q^30+1145646*q^31+1040626*q^32+944378*q^33+856158*q^34+ 775495*q^35+702209*q^36+635686*q^37+575601*q^38+521294*q^39+471923*q^40+426845* q^41+385700*q^42+348231*q^43+314148*q^44+283321*q^45+255383*q^46+229993*q^47+ 206899*q^48+185731*q^49+166376*q^50+148836*q^51+132902*q^52+118511*q^53+105490* q^54+93765*q^55+83174*q^56+73649*q^57+64994*q^58+57188*q^59+50202*q^60+43982*q^ 61+38415*q^62+33507*q^63+29122*q^64+25266*q^65+21864*q^66+18872*q^67+16196*q^68 +13866*q^69+11823*q^70+10065*q^71+8527*q^72+7216*q^73+6067*q^74+5098*q^75+4259* q^76+3556*q^77+2956*q^78+2441*q^79+1990*q^80+1632*q^81+1324*q^82+1075*q^83+863* q^84+695*q^85+551*q^86+440*q^87+345*q^88+274*q^89+214*q^90+169*q^91+119*q^92+92 *q^93+68*q^94+52*q^95+38*q^96+28*q^97+20*q^98+15*q^99+10*q^100+7*q^101+5*q^102+ 3*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+3*q^2+2*q+1, q^10+4*q^9+9*q^8+ 15*q^7+16*q^6+12*q^5+9*q^4+5*q^3+3*q^2+2*q+1, q^15+5*q^14+14*q^13+29*q^12+43*q^ 11+48*q^10+42*q^9+33*q^8+24*q^7+17*q^6+13*q^5+7*q^4+5*q^3+3*q^2+2*q+1, q^21+6*q ^20+20*q^19+49*q^18+90*q^17+129*q^16+148*q^15+139*q^14+121*q^13+97*q^12+77*q^11 +59*q^10+42*q^9+31*q^8+23*q^7+19*q^6+10*q^5+7*q^4+5*q^3+3*q^2+2*q+1, 1+76*q^25+ 396*q^22+164*q^24+460*q^21+282*q^23+2*q+3*q^2+27*q^26+5*q^3+7*q^4+10*q^5+15*q^6 +25*q^7+32*q^8+42*q^9+55*q^10+72*q^11+98*q^12+127*q^13+155*q^14+197*q^15+253*q^ 16+q^28+311*q^17+370*q^18+427*q^19+462*q^20+7*q^27, 1+1357*q^25+844*q^22+1197*q ^24+705*q^21+1022*q^23+2*q+891*q^30+1229*q^29+35*q^34+111*q^33+3*q^2+1482*q^26+ 5*q^3+7*q^4+10*q^5+15*q^6+273*q^32+20*q^7+34*q^8+43*q^9+57*q^10+72*q^11+95*q^12 +122*q^13+162*q^14+203*q^15+248*q^16+1455*q^28+305*q^17+380*q^18+476*q^19+583*q ^20+1527*q^27+542*q^31+8*q^35+q^36, 1+1492*q^25+155*q^42+876*q^22+1253*q^24+718 *q^21+1053*q^23+2*q+9*q^44+3441*q^30+q^45+3866*q^37+2965*q^29+1782*q^39+426*q^ 41+5088*q^34+2827*q^38+4859*q^33+3*q^2+1781*q^26+5*q^3+7*q^4+10*q^5+15*q^6+4464 *q^32+20*q^7+28*q^8+45*q^9+58*q^10+74*q^11+97*q^12+123*q^13+159*q^14+204*q^15+ 261*q^16+44*q^43+2532*q^28+327*q^17+397*q^18+481*q^19+583*q^20+2142*q^27+953*q^ 40+3965*q^31+5041*q^35+4647*q^36, 1+1603*q^25+17134*q^42+920*q^22+1332*q^24+758 *q^21+1104*q^23+2*q+633*q^51+16649*q^44+54*q^53+3629*q^30+209*q^52+q^55+10*q^54 +14951*q^45+10515*q^37+3078*q^29+12281*q^46+13472*q^39+16278*q^41+6762*q^34+ 1569*q^50+11964*q^38+5819*q^33+3*q^2+1902*q^26+5*q^3+7*q^4+10*q^5+15*q^6+5005*q ^32+20*q^7+28*q^8+38*q^9+60*q^10+75*q^11+99*q^12+125*q^13+162*q^14+206*q^15+263 *q^16+17335*q^43+3272*q^49+2622*q^28+328*q^17+420*q^18+520*q^19+629*q^20+2236*q ^27+14982*q^40+4288*q^31+7862*q^35+5836*q^48+9132*q^36+9027*q^47, 1+1718*q^25+ 22497*q^42+984*q^22+1430*q^24+817*q^21+1190*q^23+2*q+58452*q^51+29189*q^44+ 58778*q^53+28987*q^57+4004*q^30+59762*q^52+19099*q^58+48393*q^55+5629*q^60+ 55020*q^54+33174*q^45+11264*q^37+3432*q^29+37567*q^46+11*q^65+14843*q^39+2455*q ^61+19652*q^41+7314*q^34+55422*q^50+65*q^64+12915*q^38+6308*q^33+39315*q^56+3*q ^2+2052*q^26+5*q^3+7*q^4+10*q^5+15*q^6+5416*q^32+20*q^7+28*q^8+38*q^9+905*q^62+ 52*q^10+77*q^11+100*q^12+127*q^13+164*q^14+209*q^15+267*q^16+q^66+25644*q^43+ 51337*q^49+2907*q^28+336*q^17+423*q^18+527*q^19+663*q^20+2446*q^27+17067*q^40+ 274*q^63+4656*q^31+8498*q^35+46838*q^48+11104*q^59+9806*q^36+42175*q^47, 1+1837 *q^25+24931*q^42+1035*q^22+1531*q^24+833*q^21+1264*q^23+2*q+76404*q^51+32185*q^ 44+97078*q^53+150171*q^57+4430*q^30+86268*q^52+164945*q^58+121629*q^55+191764*q ^60+108816*q^54+36560*q^45+19822*q^71+1254*q^74+12757*q^37+126668*q^67+3734*q^ 29+41480*q^46+182004*q^65+62535*q^69+16856*q^39+201793*q^61+21939*q^41+8269*q^ 34+67615*q^50+198642*q^64+14677*q^38+7133*q^33+135499*q^56+3*q^2+93539*q^68+ 2210*q^26+5*q^3+7*q^4+10*q^5+15*q^6+6129*q^32+20*q^7+28*q^8+38*q^9+207445*q^62+ 52*q^10+68*q^11+102*q^12+77*q^76+128*q^13+166*q^14+211*q^15+270*q^16+157373*q^ 66+37413*q^70+12*q^77+q^78+28320*q^43+59880*q^49+3153*q^28+340*q^17+433*q^18+ 537*q^19+673*q^20+2644*q^27+9198*q^72+19275*q^40+3688*q^73+206860*q^63+5224*q^ 31+351*q^75+9565*q^35+53062*q^48+179106*q^59+11065*q^36+46973*q^47, 1+1947*q^25 +29147*q^42+1055*q^22+13*q^90+1596*q^24+851*q^21+1295*q^23+2*q+87760*q^51+37823 *q^44+110062*q^53+171889*q^57+4790*q^30+98374*q^52+192067*q^58+137526*q^55+ 238904*q^60+q^91+122995*q^54+42860*q^45+672745*q^71+729667*q^74+14548*q^37+ 493215*q^67+4028*q^29+48438*q^46+406291*q^65+586959*q^69+19350*q^39+266080*q^61 +25493*q^41+9268*q^34+410240*q^79+78109*q^50+366643*q^64+16790*q^38+7893*q^33+ 153747*q^56+3*q^2+539818*q^68+2341*q^26+5*q^3+7*q^4+10*q^5+15*q^6+6698*q^32+20* q^7+28*q^8+38*q^9+296283*q^62+52*q^10+68*q^11+92*q^12+669843*q^76+130*q^13+167* q^14+213*q^15+272*q^16+448532*q^66+632223*q^70+602708*q^77+513762*q^78+33287*q^ 43+69458*q^49+3374*q^28+343*q^17+437*q^18+547*q^19+685*q^20+2815*q^27+705246*q^ 72+22236*q^40+725860*q^73+329793*q^63+5680*q^31+712001*q^75+10819*q^35+303173*q ^80+61639*q^48+68901*q^83+204979*q^81+14414*q^85+5358*q^86+1693*q^87+441*q^88+ 90*q^89+33614*q^84+125485*q^82+214346*q^59+12564*q^36+54662*q^47, 1+1990*q^25+ 33507*q^42+1075*q^22+1998068*q^90+1632*q^24+863*q^21+1324*q^23+2*q+120631*q^97+ 105490*q^51+43982*q^44+132902*q^53+206899*q^57+5098*q^30+672868*q^94+118511*q^ 52+229993*q^58+166376*q^55+283321*q^60+1682697*q^91+148836*q^54+50202*q^45+ 856158*q^71+1145646*q^74+237298*q^96+54641*q^98+16196*q^37+575601*q^67+4259*q^ 29+57188*q^46+986393*q^93+21816*q^99+471923*q^65+702209*q^69+21864*q^39+314148* q^61+29122*q^41+10065*q^34+1806790*q^79+93765*q^50+426845*q^64+7569*q^100+18872 *q^38+8527*q^33+185731*q^56+3*q^2+635686*q^68+2441*q^26+5*q^3+7*q^4+10*q^5+15*q ^6+7216*q^32+20*q^7+28*q^8+38*q^9+419675*q^95+348231*q^62+2236*q^101+52*q^10+68 *q^11+92*q^12+1384787*q^76+119*q^13+169*q^14+214*q^15+274*q^16+521294*q^66+ 775495*q^70+1518640*q^77+1659922*q^78+38415*q^43+545*q^102+83174*q^49+3556*q^28 +345*q^17+440*q^18+551*q^19+695*q^20+2956*q^27+944378*q^72+25266*q^40+1040626*q ^73+104*q^103+385700*q^63+6067*q^31+1260318*q^75+11823*q^35+14*q^104+1957311*q^ 80+73649*q^48+q^105+2384316*q^83+2108082*q^81+2562289*q^85+2585932*q^86+2548894 *q^87+2440322*q^88+2255450*q^89+2491071*q^84+2253303*q^82+255383*q^59+13866*q^ 36+1334423*q^92+64994*q^47] The number of permutations avoiding, {[3, 4, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59280, 227240, 873886, 3370030, 13027730, 50469890] The number of EVEN permutations avoiding, {[3, 4, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 10, 39, 144, 539, 2039, 7763, 29645, 113615, 436927, 1685031, 6513915, 25234895] The number of ODD permutations avoiding, {[3, 4, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 11, 38, 143, 540, 2043, 7759, 29635, 113625, 436959, 1684999, 6513815, 25234995] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 39, 143, 540, 2039, 7763, 29635, 113625, 436927, 1685031, 6513815, 25234995] ODD:, [0, 1, 3, 11, 38, 144, 539, 2043, 7759, 29645, 113615, 436959, 1684999, 6513915, 25234895] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.809523810, 4.311688312, 5.954703833, 7.710843373, 9.564184223, 11.50579822, 13.53071862, 15.63601039, 17.81968014, 20.08012777, 22.41589970, 24.82559255] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.467717620, 2.046598320, 2.683288812, 3.364452334, 4.082073348, 4.833018830, 5.617191029, 6.435922474, 7.290902162, 8.183586687, 9.114957793, 10.08548681] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.15420, 4.18856, 7.20004, 11.3195, 16.6633, 23.3581, 31.5528, 41.4211, 53.1573, 66.9711, 83.0825, 101.717] Skewness: , [Float(undefined), 0., 0., 0.2409100532, 0.4758492292, 0.6531955621, 0.7851332947, 0.8859691431, 0.9652210118, 1.029113357, 1.081884464, 1.126451512, 1.164782193, 1.198191692, 1.227549148] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.589095783, 2.985937978, 3.337327888, 3.650753582, 3.925117587, 4.161632189, 4.363841725, 4.536513296, 4.684753097, 4.813017347, 4.924810742, 5.022728749] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2]}, {[2, 1, 4, 3], [3, 4, 1, 2], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 2, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 2]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 3, 0], [0, 0, 2, 1], [0, 0, 1, 2], [0, 0, 0, 3]}, {1}], [[2, 1], {[0, 0, 3]}, {}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 2, 0]}, {1}], [ [2, 3, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], %1, {3}], [[1, 3, 2, 4], %1, {1}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 2], [0, 0, 3, 0, 0], [0, 0, 2, 1, 0], [0, 0, 2, 0, 1]}, {2}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 2], [0, 0, 0, 2, 1], [0, 0, 0, 3, 0]}, {3}], [[2, 4, 1, 3], { [0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 2]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], %1, {2}], [[4, 1, 2, 3], %1, {1}], [[1, 4, 2, 3], %1, {2}], [[2, 3, 1], {[0, 0, 3, 0], [0, 0, 2, 1], [0, 0, 0, 2], [0, 1, 0, 0]}, {}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 2], [0, 0, 3, 0, 0], [0, 0, 2, 1, 0], [0, 0, 2, 0, 1]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 2, 0], [0, 0, 0, 1, 1], [0, 0, 0, 0, 2]}, {2}], [[3, 1, 2], {[0, 0, 0, 2]}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[2, 1, 3], {[0, 0, 0, 2], [0, 0, 1, 0]}, {2}], [[1, 3, 2], {[0, 0, 0, 2]}, {}]} %1 := {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 7*q^3+12*q^4+14*q^5+ 14*q^6+11*q^7+6*q^8+4*q^9+q^10, 11*q^6+24*q^7+36*q^8+36*q^9+33*q^10+24*q^11+16* q^12+8*q^13+5*q^14+q^15, 15*q^10+36*q^11+64*q^12+80*q^13+81*q^14+70*q^15+52*q^ 16+34*q^17+21*q^18+10*q^19+6*q^20+q^21, 19*q^15+48*q^16+92*q^17+132*q^18+159*q^ 19+156*q^20+138*q^21+104*q^22+71*q^23+44*q^24+26*q^25+12*q^26+7*q^27+q^28, 23*q ^21+60*q^22+120*q^23+184*q^24+247*q^25+280*q^26+278*q^27+246*q^28+195*q^29+138* q^30+90*q^31+54*q^32+31*q^33+14*q^34+8*q^35+q^36, 27*q^28+72*q^29+148*q^30+236* q^31+335*q^32+416*q^33+464*q^34+456*q^35+413*q^36+336*q^37+252*q^38+172*q^39+ 109*q^40+64*q^41+36*q^42+16*q^43+9*q^44+q^45, 31*q^36+84*q^37+176*q^38+288*q^39 +423*q^40+552*q^41+664*q^42+720*q^43+713*q^44+650*q^45+548*q^46+426*q^47+309*q^ 48+206*q^49+128*q^50+74*q^51+41*q^52+18*q^53+10*q^54+q^55, 35*q^45+96*q^46+204* q^47+340*q^48+511*q^49+688*q^50+864*q^51+1000*q^52+1075*q^53+1060*q^54+982*q^55 +844*q^56+683*q^57+516*q^58+366*q^59+240*q^60+147*q^61+84*q^62+46*q^63+20*q^64+ 11*q^65+q^66, 39*q^55+108*q^56+232*q^57+392*q^58+599*q^59+824*q^60+1064*q^61+ 1280*q^62+1455*q^63+1540*q^64+1526*q^65+1422*q^66+1251*q^67+1038*q^68+818*q^69+ 606*q^70+423*q^71+274*q^72+166*q^73+94*q^74+51*q^75+22*q^76+12*q^77+q^78, 43*q^ 66+120*q^67+260*q^68+444*q^69+687*q^70+960*q^71+1264*q^72+1560*q^73+1835*q^74+ 2040*q^75+2148*q^76+2124*q^77+2001*q^78+1784*q^79+1520*q^80+1232*q^81+953*q^82+ 696*q^83+480*q^84+308*q^85+185*q^86+104*q^87+56*q^88+24*q^89+13*q^90+q^91, 47*q ^78+132*q^79+288*q^80+496*q^81+775*q^82+1096*q^83+1464*q^84+1840*q^85+2215*q^86 +2540*q^87+2792*q^88+2912*q^89+2889*q^90+2734*q^91+2476*q^92+2146*q^93+1789*q^ 94+1426*q^95+1088*q^96+786*q^97+537*q^98+342*q^99+204*q^100+114*q^101+61*q^102+ 26*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+4*q^4+6*q^3+4*q^2+3*q+1, 7*q^7+12*q^6+14*q^5+14 *q^4+11*q^3+6*q^2+4*q+1, 11*q^9+24*q^8+36*q^7+36*q^6+33*q^5+24*q^4+16*q^3+8*q^2 +5*q+1, 15*q^11+36*q^10+64*q^9+80*q^8+81*q^7+70*q^6+52*q^5+34*q^4+21*q^3+10*q^2 +6*q+1, 19*q^13+48*q^12+92*q^11+132*q^10+159*q^9+156*q^8+138*q^7+104*q^6+71*q^5 +44*q^4+26*q^3+12*q^2+7*q+1, 23*q^15+60*q^14+120*q^13+184*q^12+247*q^11+280*q^ 10+278*q^9+246*q^8+195*q^7+138*q^6+90*q^5+54*q^4+31*q^3+14*q^2+8*q+1, 27*q^17+ 72*q^16+148*q^15+236*q^14+335*q^13+416*q^12+464*q^11+456*q^10+413*q^9+336*q^8+ 252*q^7+172*q^6+109*q^5+64*q^4+36*q^3+16*q^2+9*q+1, 31*q^19+84*q^18+176*q^17+ 288*q^16+423*q^15+552*q^14+664*q^13+720*q^12+713*q^11+650*q^10+548*q^9+426*q^8+ 309*q^7+206*q^6+128*q^5+74*q^4+41*q^3+18*q^2+10*q+1, 35*q^21+96*q^20+204*q^19+ 340*q^18+511*q^17+688*q^16+864*q^15+1000*q^14+1075*q^13+1060*q^12+982*q^11+844* q^10+683*q^9+516*q^8+366*q^7+240*q^6+147*q^5+84*q^4+46*q^3+20*q^2+11*q+1, 39*q^ 23+108*q^22+232*q^21+392*q^20+599*q^19+824*q^18+1064*q^17+1280*q^16+1455*q^15+ 1540*q^14+1526*q^13+1422*q^12+1251*q^11+1038*q^10+818*q^9+606*q^8+423*q^7+274*q ^6+166*q^5+94*q^4+51*q^3+22*q^2+12*q+1, 1+43*q^25+444*q^22+120*q^24+687*q^21+ 260*q^23+13*q+24*q^2+56*q^3+104*q^4+185*q^5+308*q^6+480*q^7+696*q^8+953*q^9+ 1232*q^10+1520*q^11+1784*q^12+2001*q^13+2124*q^14+2148*q^15+2040*q^16+1835*q^17 +1560*q^18+1264*q^19+960*q^20, 1+288*q^25+1096*q^22+496*q^24+1464*q^21+775*q^23 +14*q+26*q^2+132*q^26+61*q^3+114*q^4+204*q^5+342*q^6+537*q^7+786*q^8+1088*q^9+ 1426*q^10+1789*q^11+2146*q^12+2476*q^13+2734*q^14+2889*q^15+2912*q^16+2792*q^17 +2540*q^18+2215*q^19+1840*q^20+47*q^27] The number of permutations avoiding, {[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2]}, is given by [1, 2, 6, 21, 69, 194, 470, 1009, 1969, 3562, 6062, 9813, 15237, 22842, 33230] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2]}, is given by [1, 1, 3, 9, 33, 101, 239, 497, 977, 1793, 3043, 4889, 7601, 11445, 16639] The number of ODD permutations avoiding, {[1, 2, 3, 4], [2, 1, 4, 3], [3, 4, 1, 2]}, is given by [0, 1, 3, 12, 36, 93, 231, 512, 992, 1769, 3019, 4924, 7636, 11397, 16591] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 33, 93, 231, 497, 977, 1769, 3019, 4889, 7601, 11397, 16591] ODD:, [0, 1, 3, 12, 36, 101, 239, 512, 992, 1793, 3043, 4924, 7636, 11445, 16639] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.142857143, 5.710144928, 9.386597938, 14.16382979, 20.00792864, 26.89487049, 34.80993824, 43.74414385, 53.69183736, 64.64934042, 76.61417564, 89.58462233] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.389954240, 1.728712760, 1.984482527, 2.222891114, 2.467617143, 2.723339132, 2.989234295, 3.263379867, 3.543963543, 3.829532138, 4.118977044, 4.411464090] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.93197, 2.98845, 3.93817, 4.94124, 6.08913, 7.41658, 8.93552, 10.6496, 12.5597, 14.6653, 16.9660, 19.4610] Skewness: , [Float(undefined), 0., 0., 0.1693635266, 0.3169657364, 0.3876252549, 0.3815061632, 0.3407376522, 0.2917040560, 0.2457548276, 0.2063958149, 0.1739092004, 0.1474487560, 0.1259554907, 0.1084396126] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.227081582, 2.412372847, 2.623897089, 2.758408675, 2.813107364, 2.820964299, 2.807666878, 2.787178436, 2.765882005, 2.746399718, 2.729436239, 2.715070328] end of this data For the equivalence class of patterns, { {[3, 2, 4, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, {[1, 3, 4, 2], [1, 3, 2, 4], [2, 3, 1, 4]}, {[1, 4, 2, 3], [1, 3, 2, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [4, 1, 3, 2], [4, 2, 3, 1]}} the member , {[3, 2, 4, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {1}], [[3, 2, 1], {[1, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 1, 2], {[0, 1, 1, 0], [1, 0, 0, 0]}, {2}], [[2, 1], {[1, 1, 0]}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+3*q^4+2*q^5+q^6, 1+4*q+9*q^2+15*q^3 +16*q^4+12*q^5+9*q^6+6*q^7+3*q^8+2*q^9+q^10, 1+5*q+14*q^2+29*q^3+43*q^4+48*q^5+ 42*q^6+35*q^7+26*q^8+20*q^9+14*q^10+9*q^11+6*q^12+3*q^13+2*q^14+q^15, 1+6*q+20* q^2+49*q^3+90*q^4+129*q^5+148*q^6+142*q^7+127*q^8+107*q^9+87*q^10+70*q^11+54*q^ 12+40*q^13+30*q^14+22*q^15+14*q^16+9*q^17+6*q^18+3*q^19+2*q^20+q^21, 1+7*q+27*q ^2+76*q^3+164*q^4+282*q^5+396*q^6+464*q^7+474*q^8+452*q^9+405*q^10+353*q^11+300 *q^12+251*q^13+205*q^14+169*q^15+135*q^16+104*q^17+81*q^18+60*q^19+44*q^20+32*q ^21+22*q^22+14*q^23+9*q^24+6*q^25+3*q^26+2*q^27+q^28, 1+8*q+35*q^2+111*q^3+273* q^4+542*q^5+891*q^6+1234*q^7+1475*q^8+1578*q^9+1574*q^10+1486*q^11+1359*q^12+ 1210*q^13+1057*q^14+918*q^15+788*q^16+669*q^17+563*q^18+467*q^19+384*q^20+313*q ^21+250*q^22+196*q^23+152*q^24+116*q^25+87*q^26+64*q^27+46*q^28+32*q^29+22*q^30 +14*q^31+9*q^32+6*q^33+3*q^34+2*q^35+q^36, 1+9*q+44*q^2+155*q^3+426*q^4+953*q^5 +1782*q^6+2833*q^7+3896*q^8+4738*q^9+5240*q^10+5423*q^11+5335*q^12+5070*q^13+ 4687*q^14+4258*q^15+3816*q^16+3397*q^17+2993*q^18+2619*q^19+2279*q^20+1972*q^21 +1692*q^22+1441*q^23+1215*q^24+1013*q^25+841*q^26+690*q^27+559*q^28+449*q^29+ 356*q^30+277*q^31+214*q^32+164*q^33+122*q^34+91*q^35+66*q^36+46*q^37+32*q^38+22 *q^39+14*q^40+9*q^41+6*q^42+3*q^43+2*q^44+q^45, 1+10*q+54*q^2+209*q^3+633*q^4+ 1569*q^5+3272*q^6+5843*q^7+9069*q^8+12429*q^9+15329*q^10+17396*q^11+18557*q^12+ 18866*q^13+18518*q^14+17686*q^15+16562*q^16+15287*q^17+13967*q^18+12649*q^19+ 11375*q^20+10188*q^21+9080*q^22+8056*q^23+7116*q^24+6245*q^25+5453*q^26+4740*q^ 27+4092*q^28+3509*q^29+2992*q^30+2528*q^31+2119*q^32+1767*q^33+1461*q^34+1196*q ^35+973*q^36+783*q^37+622*q^38+492*q^39+384*q^40+295*q^41+226*q^42+170*q^43+126 *q^44+93*q^45+66*q^46+46*q^47+32*q^48+22*q^49+14*q^50+9*q^51+6*q^52+3*q^53+2*q^ 54+q^55, 1+11*q+65*q^2+274*q^3+905*q^4+2455*q^5+5629*q^6+11112*q^7+19155*q^8+ 29212*q^9+39970*q^10+49877*q^11+57789*q^12+63196*q^13+66030*q^14+66629*q^15+ 65410*q^16+62900*q^17+59540*q^18+55688*q^19+51567*q^20+47426*q^21+43361*q^22+ 39481*q^23+35813*q^24+32365*q^25+29140*q^26+26162*q^27+23402*q^28+20855*q^29+ 18518*q^30+16365*q^31+14392*q^32+12602*q^33+10980*q^34+9519*q^35+8211*q^36+7040 *q^37+6001*q^38+5086*q^39+4283*q^40+3585*q^41+2982*q^42+2463*q^43+2020*q^44+ 1647*q^45+1329*q^46+1064*q^47+846*q^48+666*q^49+520*q^50+402*q^51+307*q^52+232* q^53+174*q^54+128*q^55+93*q^56+66*q^57+46*q^58+32*q^59+22*q^60+14*q^61+9*q^62+6 *q^63+3*q^64+2*q^65+q^66, 1+12*q+77*q^2+351*q^3+1254*q^4+3688*q^5+9198*q^6+ 19831*q^7+37485*q^8+62860*q^9+94599*q^10+129370*q^11+163043*q^12+192136*q^13+ 214537*q^14+229401*q^15+237042*q^16+238325*q^17+234570*q^18+227021*q^19+216831* q^20+204893*q^21+192015*q^22+178793*q^23+165630*q^24+152802*q^25+140468*q^26+ 128741*q^27+117690*q^28+107333*q^29+97648*q^30+88606*q^31+80185*q^32+72347*q^33 +65078*q^34+58366*q^35+52173*q^36+46469*q^37+41243*q^38+36458*q^39+32098*q^40+ 28151*q^41+24585*q^42+21376*q^43+18510*q^44+15957*q^45+13686*q^46+11684*q^47+ 9924*q^48+8384*q^49+7051*q^50+5897*q^51+4901*q^52+4053*q^53+3331*q^54+2719*q^55 +2207*q^56+1778*q^57+1420*q^58+1128*q^59+890*q^60+694*q^61+538*q^62+414*q^63+ 313*q^64+236*q^65+176*q^66+128*q^67+93*q^68+66*q^69+46*q^70+32*q^71+22*q^72+14* q^73+9*q^74+6*q^75+3*q^76+2*q^77+q^78, 1+13*q+90*q^2+441*q^3+1693*q^4+5358*q^5+ 14414*q^6+33624*q^7+68991*q^8+125936*q^9+206606*q^10+307772*q^11+420952*q^12+ 534995*q^13+639463*q^14+726766*q^15+792641*q^16+836114*q^17+858411*q^18+862391* q^19+851361*q^20+828718*q^21+797444*q^22+760341*q^23+719609*q^24+677046*q^25+ 633871*q^26+591121*q^27+549417*q^28+509321*q^29+471077*q^30+434817*q^31+400561* q^32+368314*q^33+337982*q^34+309558*q^35+282967*q^36+258103*q^37+234886*q^38+ 213256*q^39+193109*q^40+174399*q^41+157083*q^42+141073*q^43+126322*q^44+112790* q^45+100383*q^46+89050*q^47+78740*q^48+69380*q^49+60924*q^50+53319*q^51+46489*q ^52+40383*q^53+34952*q^54+30132*q^55+25869*q^56+22124*q^57+18836*q^58+15965*q^ 59+13480*q^60+11326*q^61+9469*q^62+7883*q^63+6527*q^64+5374*q^65+4405*q^66+3588 *q^67+2904*q^68+2338*q^69+1870*q^70+1484*q^71+1172*q^72+918*q^73+712*q^74+550*q ^75+420*q^76+317*q^77+238*q^78+176*q^79+128*q^80+93*q^81+66*q^82+46*q^83+32*q^ 84+22*q^85+14*q^86+9*q^87+6*q^88+3*q^89+2*q^90+q^91, 1+14*q+104*q^2+545*q^3+ 2236*q^4+7569*q^5+21816*q^6+54652*q^7+120741*q^8+237904*q^9+422069*q^10+680288* q^11+1005365*q^12+1375729*q^13+1761101*q^14+2130523*q^15+2458456*q^16+2727604*q ^17+2929733*q^18+3063905*q^19+3134956*q^20+3150606*q^21+3120047*q^22+3052419*q^ 23+2956742*q^24+2840709*q^25+2710955*q^26+2572786*q^27+2430400*q^28+2287192*q^ 29+2145786*q^30+2007919*q^31+1874812*q^32+1747245*q^33+1625550*q^34+1509953*q^ 35+1400564*q^36+1297227*q^37+1199780*q^38+1108009*q^39+1021621*q^40+940386*q^41 +864123*q^42+792572*q^43+725548*q^44+662891*q^45+604378*q^46+549832*q^47+499094 *q^48+451963*q^49+408305*q^50+367979*q^51+330801*q^52+296620*q^53+265285*q^54+ 236623*q^55+210481*q^56+186715*q^57+165150*q^58+145648*q^59+128078*q^60+112286* q^61+98140*q^62+85511*q^63+74263*q^64+64283*q^65+55460*q^66+47680*q^67+40847*q^ 68+34868*q^69+29650*q^70+25119*q^71+21200*q^72+17817*q^73+14913*q^74+12430*q^75 +10313*q^76+8518*q^77+7003*q^78+5727*q^79+4660*q^80+3773*q^81+3036*q^82+2430*q^ 83+1934*q^84+1528*q^85+1200*q^86+936*q^87+724*q^88+556*q^89+424*q^90+319*q^91+ 238*q^92+176*q^93+128*q^94+93*q^95+66*q^96+46*q^97+32*q^98+22*q^99+14*q^100+9*q ^101+6*q^102+3*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+3*q^2+2*q+1, q^10+4*q^9+9*q^8+ 15*q^7+16*q^6+12*q^5+9*q^4+6*q^3+3*q^2+2*q+1, q^15+5*q^14+14*q^13+29*q^12+43*q^ 11+48*q^10+42*q^9+35*q^8+26*q^7+20*q^6+14*q^5+9*q^4+6*q^3+3*q^2+2*q+1, q^21+6*q ^20+20*q^19+49*q^18+90*q^17+129*q^16+148*q^15+142*q^14+127*q^13+107*q^12+87*q^ 11+70*q^10+54*q^9+40*q^8+30*q^7+22*q^6+14*q^5+9*q^4+6*q^3+3*q^2+2*q+1, 1+76*q^ 25+396*q^22+164*q^24+464*q^21+282*q^23+2*q+3*q^2+27*q^26+6*q^3+9*q^4+14*q^5+22* q^6+32*q^7+44*q^8+60*q^9+81*q^10+104*q^11+135*q^12+169*q^13+205*q^14+251*q^15+ 300*q^16+q^28+353*q^17+405*q^18+452*q^19+474*q^20+7*q^27, 1+1486*q^25+1057*q^22 +1359*q^24+918*q^21+1210*q^23+2*q+891*q^30+1234*q^29+35*q^34+111*q^33+3*q^2+ 1574*q^26+6*q^3+9*q^4+14*q^5+22*q^6+273*q^32+32*q^7+46*q^8+64*q^9+87*q^10+116*q ^11+152*q^12+196*q^13+250*q^14+313*q^15+384*q^16+1475*q^28+467*q^17+563*q^18+ 669*q^19+788*q^20+1578*q^27+542*q^31+8*q^35+q^36, 1+2279*q^25+155*q^42+1441*q^ 22+1972*q^24+1215*q^21+1692*q^23+2*q+9*q^44+4258*q^30+q^45+3896*q^37+3816*q^29+ 1782*q^39+426*q^41+5423*q^34+2833*q^38+5335*q^33+3*q^2+2619*q^26+6*q^3+9*q^4+14 *q^5+22*q^6+5070*q^32+32*q^7+46*q^8+66*q^9+91*q^10+122*q^11+164*q^12+214*q^13+ 277*q^14+356*q^15+449*q^16+44*q^43+3397*q^28+559*q^17+690*q^18+841*q^19+1013*q^ 20+2993*q^27+953*q^40+4687*q^31+5240*q^35+4738*q^36, 1+2992*q^25+18866*q^42+ 1767*q^22+2528*q^24+1461*q^21+2119*q^23+2*q+633*q^51+17396*q^44+54*q^53+6245*q^ 30+209*q^52+q^55+10*q^54+15329*q^45+13967*q^37+5453*q^29+12429*q^46+16562*q^39+ 18518*q^41+10188*q^34+1569*q^50+15287*q^38+9080*q^33+3*q^2+3509*q^26+6*q^3+9*q^ 4+14*q^5+22*q^6+8056*q^32+32*q^7+46*q^8+66*q^9+93*q^10+126*q^11+170*q^12+226*q^ 13+295*q^14+384*q^15+492*q^16+18557*q^43+3272*q^49+4740*q^28+622*q^17+783*q^18+ 973*q^19+1196*q^20+4092*q^27+17686*q^40+7116*q^31+11375*q^35+5843*q^48+12649*q^ 36+9069*q^47, 1+3585*q^25+35813*q^42+2020*q^22+2982*q^24+1647*q^21+2463*q^23+2* q+66629*q^51+43361*q^44+63196*q^53+29212*q^57+8211*q^30+66030*q^52+19155*q^58+ 49877*q^55+5629*q^60+57789*q^54+47426*q^45+20855*q^37+7040*q^29+51567*q^46+11*q ^65+26162*q^39+2455*q^61+32365*q^41+14392*q^34+65410*q^50+65*q^64+23402*q^38+ 12602*q^33+39970*q^56+3*q^2+4283*q^26+6*q^3+9*q^4+14*q^5+22*q^6+10980*q^32+32*q ^7+46*q^8+66*q^9+905*q^62+93*q^10+128*q^11+174*q^12+232*q^13+307*q^14+402*q^15+ 520*q^16+q^66+39481*q^43+62900*q^49+6001*q^28+666*q^17+846*q^18+1064*q^19+1329* q^20+5086*q^27+29140*q^40+274*q^63+9519*q^31+16365*q^35+59540*q^48+11112*q^59+ 18518*q^36+55688*q^47, 1+4053*q^25+52173*q^42+2207*q^22+3331*q^24+1778*q^21+ 2719*q^23+2*q+128741*q^51+65078*q^44+152802*q^53+204893*q^57+9924*q^30+140468*q ^52+216831*q^58+178793*q^55+234570*q^60+165630*q^54+72347*q^45+19831*q^71+1254* q^74+28151*q^37+129370*q^67+8384*q^29+80185*q^46+192136*q^65+62860*q^69+36458*q ^39+238325*q^61+46469*q^41+18510*q^34+117690*q^50+214537*q^64+32098*q^38+15957* q^33+192015*q^56+3*q^2+94599*q^68+4901*q^26+6*q^3+9*q^4+14*q^5+22*q^6+13686*q^ 32+32*q^7+46*q^8+66*q^9+237042*q^62+93*q^10+128*q^11+176*q^12+77*q^76+236*q^13+ 313*q^14+414*q^15+538*q^16+163043*q^66+37485*q^70+12*q^77+q^78+58366*q^43+ 107333*q^49+7051*q^28+694*q^17+890*q^18+1128*q^19+1420*q^20+5897*q^27+9198*q^72 +41243*q^40+3688*q^73+229401*q^63+11684*q^31+351*q^75+21376*q^35+97648*q^48+ 227021*q^59+24585*q^36+88606*q^47, 1+4405*q^25+69380*q^42+2338*q^22+13*q^90+ 3588*q^24+1870*q^21+2904*q^23+2*q+193109*q^51+89050*q^44+234886*q^53+337982*q^ 57+11326*q^30+213256*q^52+368314*q^58+282967*q^55+434817*q^60+q^91+258103*q^54+ 100383*q^45+851361*q^71+836114*q^74+34952*q^37+719609*q^67+9469*q^29+112790*q^ 46+633871*q^65+797444*q^69+46489*q^39+471077*q^61+60924*q^41+22124*q^34+420952* q^79+174399*q^50+591121*q^64+40383*q^38+18836*q^33+309558*q^56+3*q^2+760341*q^ 68+5374*q^26+6*q^3+9*q^4+14*q^5+22*q^6+15965*q^32+32*q^7+46*q^8+66*q^9+509321*q ^62+93*q^10+128*q^11+176*q^12+726766*q^76+238*q^13+317*q^14+420*q^15+550*q^16+ 677046*q^66+828718*q^70+639463*q^77+534995*q^78+78740*q^43+157083*q^49+7883*q^ 28+712*q^17+918*q^18+1172*q^19+1484*q^20+6527*q^27+862391*q^72+53319*q^40+ 858411*q^73+549417*q^63+13480*q^31+792641*q^75+25869*q^35+307772*q^80+141073*q^ 48+68991*q^83+206606*q^81+14414*q^85+5358*q^86+1693*q^87+441*q^88+90*q^89+33624 *q^84+125936*q^82+400561*q^59+30132*q^36+126322*q^47, 1+4660*q^25+85511*q^42+ 2430*q^22+2130523*q^90+3773*q^24+1934*q^21+3036*q^23+2*q+120741*q^97+265285*q^ 51+112286*q^44+330801*q^53+499094*q^57+12430*q^30+680288*q^94+296620*q^52+ 549832*q^58+408305*q^55+662891*q^60+1761101*q^91+367979*q^54+128078*q^45+ 1625550*q^71+2007919*q^74+237904*q^96+54652*q^98+40847*q^37+1199780*q^67+10313* q^29+145648*q^46+1005365*q^93+21816*q^99+1021621*q^65+1400564*q^69+55460*q^39+ 725548*q^61+74263*q^41+25119*q^34+2710955*q^79+236623*q^50+940386*q^64+7569*q^ 100+47680*q^38+21200*q^33+451963*q^56+3*q^2+1297227*q^68+5727*q^26+6*q^3+9*q^4+ 14*q^5+22*q^6+17817*q^32+32*q^7+46*q^8+66*q^9+422069*q^95+792572*q^62+2236*q^ 101+93*q^10+128*q^11+176*q^12+2287192*q^76+238*q^13+319*q^14+424*q^15+556*q^16+ 1108009*q^66+1509953*q^70+2430400*q^77+2572786*q^78+98140*q^43+545*q^102+210481 *q^49+8518*q^28+724*q^17+936*q^18+1200*q^19+1528*q^20+7003*q^27+1747245*q^72+ 64283*q^40+1874812*q^73+104*q^103+864123*q^63+14913*q^31+2145786*q^75+29650*q^ 35+14*q^104+2840709*q^80+186715*q^48+q^105+3120047*q^83+2956742*q^81+3134956*q^ 85+3063905*q^86+2929733*q^87+2727604*q^88+2458456*q^89+3150606*q^84+3052419*q^ 82+604378*q^59+34868*q^36+1375729*q^92+165150*q^47] The number of permutations avoiding, {[3, 2, 4, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, 73816489] The number of EVEN permutations avoiding, {[3, 2, 4, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 10, 39, 148, 579, 2267, 8970, 35618, 142101, 568514, 2280571, 9166589, 36908331] The number of ODD permutations avoiding, {[3, 2, 4, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 11, 39, 150, 578, 2272, 8966, 35633, 142087, 568562, 2280522, 9166748, 36908158] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 39, 150, 578, 2267, 8970, 35633, 142087, 568514, 2280571, 9166748, 36908158] ODD:, [0, 1, 3, 11, 39, 148, 579, 2272, 8966, 35618, 142101, 568562, 2280522, 9166589, 36908331] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.809523810, 4.346153846, 6.073825503, 7.968885048, 10.01321877, 12.19235058, 14.49443517, 16.90958450, 19.42939962, 22.04663970, 24.75498345, 27.54885321] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.467717620, 2.055804310, 2.719785719, 3.452214834, 4.246201648, 5.096035735, 5.996984717, 6.945066590, 7.936877836, 8.969468282, 10.04024924, 11.14692486] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.15420, 4.22633, 7.39723, 11.9178, 18.0302, 25.9696, 35.9638, 48.2339, 62.9940, 80.4514, 100.807, 124.254] Skewness: , [Float(undefined), 0., 0., 0.2409100532, 0.4418343024, 0.5807183672, 0.6775567310, 0.7483234878, 0.8026095408, 0.8460489510, 0.8820508912, 0.9127406071, 0.9395176501, 0.9633205109, 0.9848161688] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.589095783, 2.901387913, 3.127742896, 3.304406992, 3.447680671, 3.567464721, 3.670371642, 3.760827742, 3.841900816, 3.915733158, 3.983880715, 4.047527122] end of this data For the equivalence class of patterns, { {[2, 4, 1, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, {[2, 4, 1, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, {[1, 4, 3, 2], [2, 4, 1, 3], [2, 3, 4, 1]}, {[2, 3, 4, 1], [3, 2, 1, 4], [3, 1, 4, 2]}, {[3, 2, 1, 4], [3, 1, 4, 2], [4, 1, 2, 3]}, {[1, 4, 3, 2], [3, 1, 4, 2], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 3, 4, 1], [3, 1, 4, 2]}, {[1, 4, 3, 2], [2, 4, 1, 3], [4, 1, 2, 3]}} the member , {[2, 4, 1, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 1, 1, 1], [1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 1, 1, 1], [1, 0, 0, 1]}, {}], [[3, 1, 4, 2, 5], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[3, 1, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 1, 1]}, {2}], [[2, 3, 1], {[1, 0, 0, 1], [0, 0, 1, 0]}, {}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 1, 0, 0]}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {1}], [ [2, 1, 4, 3], {[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {}], [[3, 4, 1, 2], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 1, 1, 0, 0]}, {}], [[1, 4, 2, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 1, 1, 0]}, {1}], [ [2, 1, 5, 4, 3], {[1, 0, 0, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 1, 0]}, {1}], [[1, 2], {[1, 1, 1]}, {}], [[3, 4, 1, 2, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 5, 1, 2, 4], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 5, 2, 3, 1], { [1, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [3, 1, 5, 4, 2], {[1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 2, 5, 4, 1], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 5, 3, 4], {[0, 0, 1, 1, 1, 0], [0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 1, 1]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [ [4, 2, 5, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 5, 2, 4], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 1, 5, 3, 2], { [1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[4, 5, 1, 3, 2], {[1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[1, 3, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 1, 1]}, {1}], [[4, 1, 2, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 1, 1, 0]}, {2}], [ [4, 5, 1, 2, 3], {[0, 1, 1, 1, 0, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]}, {3}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {2}], [[3, 1, 2], {[1, 1, 1, 0], [1, 0, 0, 1]}, {}], [[4, 1, 5, 2, 3], { [0, 1, 1, 1, 0, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]}, {2}], [[1, 3, 2], {[1, 0, 1, 0], [1, 0, 0, 1]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+3*q^3+5*q^4+3*q^5+q^6, 1+4*q+9*q^2+9*q^3+ 9*q^4+9*q^5+9*q^6+9*q^7+9*q^8+4*q^9+q^10, 1+5*q+14*q^2+20*q^3+21*q^4+20*q^5+20* q^6+22*q^7+25*q^8+21*q^9+20*q^10+21*q^11+20*q^12+14*q^13+5*q^14+q^15, 1+6*q+20* q^2+37*q^3+47*q^4+46*q^5+48*q^6+48*q^7+54*q^8+57*q^9+54*q^10+56*q^11+61*q^12+59 *q^13+53*q^14+49*q^15+46*q^16+47*q^17+37*q^18+20*q^19+6*q^20+q^21, 1+7*q+27*q^2 +61*q^3+94*q^4+105*q^5+109*q^6+110*q^7+120*q^8+131*q^9+139*q^10+140*q^11+145*q^ 12+153*q^13+152*q^14+151*q^15+154*q^16+155*q^17+159*q^18+146*q^19+131*q^20+116* q^21+110*q^22+105*q^23+94*q^24+61*q^25+27*q^26+7*q^27+q^28, 1+8*q+35*q^2+93*q^3 +170*q^4+223*q^5+245*q^6+251*q^7+265*q^8+288*q^9+319*q^10+338*q^11+363*q^12+382 *q^13+388*q^14+385*q^15+396*q^16+421*q^17+446*q^18+437*q^19+419*q^20+417*q^21+ 420*q^22+422*q^23+420*q^24+393*q^25+358*q^26+318*q^27+283*q^28+258*q^29+246*q^ 30+223*q^31+170*q^32+93*q^33+35*q^34+8*q^35+q^36, 1+9*q+44*q^2+134*q^3+284*q^4+ 435*q^5+530*q^6+566*q^7+595*q^8+633*q^9+697*q^10+758*q^11+829*q^12+912*q^13+980 *q^14+993*q^15+997*q^16+1032*q^17+1102*q^18+1172*q^19+1185*q^20+1179*q^21+1183* q^22+1201*q^23+1236*q^24+1260*q^25+1251*q^26+1202*q^27+1157*q^28+1127*q^29+1131 *q^30+1123*q^31+1065*q^32+969*q^33+867*q^34+773*q^35+686*q^36+621*q^37+574*q^38 +531*q^39+435*q^40+284*q^41+134*q^42+44*q^43+9*q^44+q^45, 1+10*q+54*q^2+185*q^3 +446*q^4+786*q^5+1080*q^6+1243*q^7+1332*q^8+1408*q^9+1529*q^10+1667*q^11+1824*q ^12+2015*q^13+2247*q^14+2429*q^15+2519*q^16+2579*q^17+2693*q^18+2845*q^19+2980* q^20+3071*q^21+3166*q^22+3273*q^23+3390*q^24+3463*q^25+3485*q^26+3467*q^27+3483 *q^28+3552*q^29+3599*q^30+3624*q^31+3591*q^32+3515*q^33+3432*q^34+3352*q^35+ 3239*q^36+3107*q^37+3018*q^38+2957*q^39+2854*q^40+2645*q^41+2368*q^42+2108*q^43 +1873*q^44+1670*q^45+1493*q^46+1367*q^47+1252*q^48+1081*q^49+786*q^50+446*q^51+ 185*q^52+54*q^53+10*q^54+q^55, 1+11*q+65*q^2+247*q^3+667*q^4+1332*q^5+2064*q^6+ 2613*q^7+2935*q^8+3138*q^9+3373*q^10+3664*q^11+4010*q^12+4398*q^13+4904*q^14+ 5453*q^15+5928*q^16+6277*q^17+6589*q^18+6931*q^19+7275*q^20+7547*q^21+7819*q^22 +8179*q^23+8638*q^24+9113*q^25+9450*q^26+9539*q^27+9546*q^28+9654*q^29+9901*q^ 30+10227*q^31+10523*q^32+10670*q^33+10660*q^34+10535*q^35+10397*q^36+10311*q^37 +10347*q^38+10443*q^39+10473*q^40+10266*q^41+9882*q^42+9469*q^43+9129*q^44+8836 *q^45+8540*q^46+8200*q^47+7870*q^48+7543*q^49+7096*q^50+6477*q^51+5786*q^52+ 5120*q^53+4552*q^54+4047*q^55+3618*q^56+3265*q^57+2980*q^58+2623*q^59+2065*q^60 +1332*q^61+667*q^62+247*q^63+65*q^64+11*q^65+q^66, 1+12*q+77*q^2+321*q^3+959*q^ 4+2141*q^5+3716*q^6+5212*q^7+6276*q^8+6929*q^9+7466*q^10+8068*q^11+8807*q^12+ 9635*q^13+10660*q^14+11880*q^15+13173*q^16+14359*q^17+15476*q^18+16551*q^19+ 17551*q^20+18380*q^21+19070*q^22+19839*q^23+20869*q^24+22131*q^25+23490*q^26+ 24617*q^27+25361*q^28+25842*q^29+26392*q^30+27130*q^31+28033*q^32+28812*q^33+ 29387*q^34+29876*q^35+30334*q^36+30713*q^37+30996*q^38+31270*q^39+31564*q^40+ 31774*q^41+31737*q^42+31530*q^43+31346*q^44+31146*q^45+30792*q^46+30301*q^47+ 29881*q^48+29586*q^49+29243*q^50+28545*q^51+27504*q^52+26266*q^53+25031*q^54+ 23936*q^55+23001*q^56+22119*q^57+21179*q^58+20121*q^59+18886*q^60+17427*q^61+ 15786*q^62+14073*q^63+12466*q^64+11045*q^65+9827*q^66+8756*q^67+7865*q^68+7109* q^69+6332*q^70+5223*q^71+3717*q^72+2141*q^73+959*q^74+321*q^75+77*q^76+12*q^77+ q^78, 1+13*q+90*q^2+408*q^3+1335*q^4+3294*q^5+6348*q^6+9856*q^7+12888*q^8+14983 *q^9+16455*q^10+17798*q^11+19359*q^12+21140*q^13+23253*q^14+25793*q^15+28721*q^ 16+31714*q^17+34734*q^18+37866*q^19+41020*q^20+43778*q^21+45963*q^22+47901*q^23 +50138*q^24+52915*q^25+56125*q^26+59385*q^27+62381*q^28+65065*q^29+67650*q^30+ 70275*q^31+72926*q^32+75389*q^33+77439*q^34+79101*q^35+80732*q^36+82590*q^37+ 84799*q^38+87304*q^39+89739*q^40+91502*q^41+92382*q^42+92692*q^43+92977*q^44+ 93497*q^45+94106*q^46+94746*q^47+95557*q^48+96370*q^49+96707*q^50+96083*q^51+ 94701*q^52+93136*q^53+91775*q^54+90624*q^55+89617*q^56+88548*q^57+87183*q^58+ 85372*q^59+83189*q^60+80721*q^61+78058*q^62+75211*q^63+72048*q^64+68645*q^65+ 65195*q^66+61942*q^67+59071*q^68+56476*q^69+53774*q^70+50542*q^71+46714*q^72+ 42544*q^73+38311*q^74+34187*q^75+30302*q^76+26849*q^77+23836*q^78+21235*q^79+ 18978*q^80+17070*q^81+15228*q^82+12956*q^83+9868*q^84+6349*q^85+3294*q^86+1335* q^87+408*q^88+90*q^89+13*q^90+q^91, 1+14*q+104*q^2+509*q^3+1809*q^4+4886*q^5+ 10364*q^6+17730*q^7+25299*q^8+31412*q^9+35773*q^10+39169*q^11+42590*q^12+46427* q^13+50887*q^14+56143*q^15+62388*q^16+69216*q^17+76346*q^18+83920*q^19+92197*q^ 20+100560*q^21+107947*q^22+113971*q^23+119569*q^24+125861*q^25+133256*q^26+ 141247*q^27+149088*q^28+156494*q^29+164077*q^30+172360*q^31+181346*q^32+190322* q^33+198304*q^34+204667*q^35+209884*q^36+214892*q^37+220628*q^38+227507*q^39+ 235455*q^40+243639*q^41+250953*q^42+256636*q^43+260940*q^44+264685*q^45+268256* q^46+271795*q^47+275374*q^48+279218*q^49+283328*q^50+287195*q^51+289983*q^52+ 291305*q^53+291392*q^54+290795*q^55+290277*q^56+290029*q^57+289781*q^58+289272* q^59+288410*q^60+287020*q^61+284687*q^62+281019*q^63+275973*q^64+270106*q^65+ 264181*q^66+258707*q^67+253874*q^68+249455*q^69+244828*q^70+239034*q^71+231438* q^72+222405*q^73+212927*q^74+203803*q^75+194998*q^76+186159*q^77+176903*q^78+ 167431*q^79+158313*q^80+150073*q^81+142512*q^82+134592*q^83+125259*q^84+114557* q^85+103477*q^86+92854*q^87+82885*q^88+73637*q^89+65237*q^90+57889*q^91+51475*q ^92+45927*q^93+41100*q^94+36679*q^95+31735*q^96+25380*q^97+17743*q^98+10365*q^ 99+4886*q^100+1809*q^101+509*q^102+104*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+3*q^3+5*q^4+3*q^5+q^6, 1+4*q+9*q^2+9*q^3+ 9*q^4+9*q^5+9*q^6+9*q^7+9*q^8+4*q^9+q^10, q^15+5*q^14+14*q^13+20*q^12+21*q^11+ 20*q^10+20*q^9+22*q^8+25*q^7+21*q^6+20*q^5+21*q^4+20*q^3+14*q^2+5*q+1, q^21+6*q ^20+20*q^19+37*q^18+47*q^17+46*q^16+48*q^15+48*q^14+54*q^13+57*q^12+54*q^11+56* q^10+61*q^9+59*q^8+53*q^7+49*q^6+46*q^5+47*q^4+37*q^3+20*q^2+6*q+1, 1+61*q^25+ 109*q^22+94*q^24+110*q^21+105*q^23+7*q+27*q^2+27*q^26+61*q^3+94*q^4+105*q^5+110 *q^6+116*q^7+131*q^8+146*q^9+159*q^10+155*q^11+154*q^12+151*q^13+152*q^14+153*q ^15+145*q^16+q^28+140*q^17+139*q^18+131*q^19+120*q^20+7*q^27, 1+338*q^25+388*q^ 22+363*q^24+385*q^21+382*q^23+8*q+245*q^30+251*q^29+35*q^34+93*q^33+35*q^2+319* q^26+93*q^3+170*q^4+223*q^5+246*q^6+170*q^32+258*q^7+283*q^8+318*q^9+358*q^10+ 393*q^11+420*q^12+422*q^13+420*q^14+417*q^15+419*q^16+265*q^28+437*q^17+446*q^ 18+421*q^19+396*q^20+288*q^27+223*q^31+8*q^35+q^36, 1+1185*q^25+134*q^42+1201*q ^22+1179*q^24+1236*q^21+1183*q^23+9*q+9*q^44+993*q^30+q^45+595*q^37+997*q^29+ 530*q^39+284*q^41+758*q^34+566*q^38+829*q^33+44*q^2+1172*q^26+134*q^3+284*q^4+ 435*q^5+531*q^6+912*q^32+574*q^7+621*q^8+686*q^9+773*q^10+867*q^11+969*q^12+ 1065*q^13+1123*q^14+1131*q^15+1127*q^16+44*q^43+1032*q^28+1157*q^17+1202*q^18+ 1251*q^19+1260*q^20+1102*q^27+435*q^40+980*q^31+697*q^35+633*q^36, 1+3599*q^25+ 2015*q^42+3515*q^22+3624*q^24+3432*q^21+3591*q^23+10*q+446*q^51+1667*q^44+54*q^ 53+3463*q^30+185*q^52+q^55+10*q^54+1529*q^45+2693*q^37+3485*q^29+1408*q^46+2519 *q^39+2247*q^41+3071*q^34+786*q^50+2579*q^38+3166*q^33+54*q^2+3552*q^26+185*q^3 +446*q^4+786*q^5+1081*q^6+3273*q^32+1252*q^7+1367*q^8+1493*q^9+1670*q^10+1873*q ^11+2108*q^12+2368*q^13+2645*q^14+2854*q^15+2957*q^16+1824*q^43+1080*q^49+3467* q^28+3018*q^17+3107*q^18+3239*q^19+3352*q^20+3483*q^27+2429*q^40+3390*q^31+2980 *q^35+1243*q^48+2845*q^36+1332*q^47, 1+10266*q^25+8638*q^42+9129*q^22+9882*q^24 +8836*q^21+9469*q^23+11*q+5453*q^51+7819*q^44+4398*q^53+3138*q^57+10397*q^30+ 4904*q^52+2935*q^58+3664*q^55+2064*q^60+4010*q^54+7547*q^45+9654*q^37+10311*q^ 29+7275*q^46+11*q^65+9539*q^39+1332*q^61+9113*q^41+10523*q^34+5928*q^50+65*q^64 +9546*q^38+10670*q^33+3373*q^56+65*q^2+10473*q^26+247*q^3+667*q^4+1332*q^5+2065 *q^6+10660*q^32+2623*q^7+2980*q^8+3265*q^9+667*q^62+3618*q^10+4047*q^11+4552*q^ 12+5120*q^13+5786*q^14+6477*q^15+7096*q^16+q^66+8179*q^43+6277*q^49+10347*q^28+ 7543*q^17+7870*q^18+8200*q^19+8540*q^20+10443*q^27+9450*q^40+247*q^63+10535*q^ 31+10227*q^35+6589*q^48+2613*q^59+9901*q^36+6931*q^47, 1+26266*q^25+30334*q^42+ 23001*q^22+25031*q^24+22119*q^21+23936*q^23+12*q+24617*q^51+29387*q^44+22131*q^ 53+18380*q^57+29881*q^30+23490*q^52+17551*q^58+19839*q^55+15476*q^60+20869*q^54 +28812*q^45+5212*q^71+959*q^74+31774*q^37+8068*q^67+29586*q^29+28033*q^46+9635* q^65+6929*q^69+31270*q^39+14359*q^61+30713*q^41+31346*q^34+25361*q^50+10660*q^ 64+31564*q^38+31146*q^33+19070*q^56+77*q^2+7466*q^68+27504*q^26+321*q^3+959*q^4 +2141*q^5+3717*q^6+30792*q^32+5223*q^7+6332*q^8+7109*q^9+13173*q^62+7865*q^10+ 8756*q^11+9827*q^12+77*q^76+11045*q^13+12466*q^14+14073*q^15+15786*q^16+8807*q^ 66+6276*q^70+12*q^77+q^78+29876*q^43+25842*q^49+29243*q^28+17427*q^17+18886*q^ 18+20121*q^19+21179*q^20+28545*q^27+3716*q^72+30996*q^40+2141*q^73+11880*q^63+ 30301*q^31+321*q^75+31530*q^35+26392*q^48+16551*q^59+31737*q^36+27130*q^47, 1+ 65195*q^25+96370*q^42+56476*q^22+13*q^90+61942*q^24+53774*q^21+59071*q^23+13*q+ 89739*q^51+94746*q^44+84799*q^53+77439*q^57+80721*q^30+87304*q^52+75389*q^58+ 80732*q^55+70275*q^60+q^91+82590*q^54+94106*q^45+41020*q^71+31714*q^74+91775*q^ 37+50138*q^67+78058*q^29+93497*q^46+56125*q^65+45963*q^69+94701*q^39+67650*q^61 +96707*q^41+88548*q^34+19359*q^79+91502*q^50+59385*q^64+93136*q^38+87183*q^33+ 79101*q^56+90*q^2+47901*q^68+68645*q^26+408*q^3+1335*q^4+3294*q^5+6349*q^6+ 85372*q^32+9868*q^7+12956*q^8+15228*q^9+65065*q^62+17070*q^10+18978*q^11+21235* q^12+25793*q^76+23836*q^13+26849*q^14+30302*q^15+34187*q^16+52915*q^66+43778*q^ 70+23253*q^77+21140*q^78+95557*q^43+92382*q^49+75211*q^28+38311*q^17+42544*q^18 +46714*q^19+50542*q^20+72048*q^27+37866*q^72+96083*q^40+34734*q^73+62381*q^63+ 83189*q^31+28721*q^75+89617*q^35+17798*q^80+92692*q^48+12888*q^83+16455*q^81+ 6348*q^85+3294*q^86+1335*q^87+408*q^88+90*q^89+9856*q^84+14983*q^82+72926*q^59+ 90624*q^36+92977*q^47, 1+158313*q^25+281019*q^42+134592*q^22+56143*q^90+150073* q^24+125259*q^21+142512*q^23+14*q+25299*q^97+291392*q^51+287020*q^44+289983*q^ 53+275374*q^57+203803*q^30+39169*q^94+291305*q^52+271795*q^58+283328*q^55+ 264685*q^60+50887*q^91+287195*q^54+288410*q^45+198304*q^71+172360*q^74+31412*q^ 96+17730*q^98+253874*q^37+220628*q^67+194998*q^29+289272*q^46+42590*q^93+10364* q^99+235455*q^65+209884*q^69+264181*q^39+260940*q^61+275973*q^41+239034*q^34+ 133256*q^79+290795*q^50+243639*q^64+4886*q^100+258707*q^38+231438*q^33+279218*q ^56+104*q^2+214892*q^68+167431*q^26+509*q^3+1809*q^4+4886*q^5+10365*q^6+222405* q^32+17743*q^7+25380*q^8+31735*q^9+35773*q^95+256636*q^62+1809*q^101+36679*q^10 +41100*q^11+45927*q^12+156494*q^76+51475*q^13+57889*q^14+65237*q^15+73637*q^16+ 227507*q^66+204667*q^70+149088*q^77+141247*q^78+284687*q^43+509*q^102+290277*q^ 49+186159*q^28+82885*q^17+92854*q^18+103477*q^19+114557*q^20+176903*q^27+190322 *q^72+270106*q^40+181346*q^73+104*q^103+250953*q^63+212927*q^31+164077*q^75+ 244828*q^35+14*q^104+125861*q^80+290029*q^48+q^105+107947*q^83+119569*q^81+ 92197*q^85+83920*q^86+76346*q^87+69216*q^88+62388*q^89+100560*q^84+113971*q^82+ 268256*q^59+249455*q^36+46427*q^92+289781*q^47] The number of permutations avoiding, {[2, 4, 1, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, is given by [1, 2, 6, 21, 73, 250, 853, 2911, 9938, 33931, 115849, 395534, 1350437, 4610679, 15741842] The number of EVEN permutations avoiding, {[2, 4, 1, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, is given by [1, 1, 3, 12, 38, 126, 427, 1463, 4980, 16972, 57929, 197816, 675288, 2305388, 7870960] The number of ODD permutations avoiding, {[2, 4, 1, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, is given by [0, 1, 3, 9, 35, 124, 426, 1448, 4958, 16959, 57920, 197718, 675149, 2305291, 7870882] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 12, 38, 124, 426, 1463, 4980, 16959, 57920, 197816, 675288, 2305291, 7870882] ODD:, [0, 1, 3, 9, 35, 126, 427, 1448, 4958, 16972, 57929, 197718, 675149, 2305388, 7870960] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.512000000, 10.54865182, 14.11405015, 18.20768766, 22.82865226, 27.97662474, 33.65159506, 39.85359924, 46.58265236, 53.83875585] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.573591585, 2.427016885, 3.498836378, 4.757819619, 6.176220600, 7.734431207, 9.419281949, 11.22122296, 13.13271147, 15.14751280, 17.26035911, 19.46672999] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.47619, 5.89041, 12.2419, 22.6368, 38.1457, 59.8214, 88.7229, 125.916, 172.468, 229.447, 297.920, 378.954] Skewness: , [Float(undefined), 0., 0., 0., 0., -0.009936384252, -0.02635427343, -0.04289966318, -0.05661742622, -0.06705945706, -0.07474243025, -0.08032108990, -0.08434002569, -0.08720575977, -0.08921106959] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.081360971, 1.998531284, 1.982622717, 2.006694782, 2.051227702, 2.104166494, 2.158439173, 2.210479707, 2.258807307, 2.302989227, 2.343134382, 2.379522310] end of this data For the equivalence class of patterns, { {[4, 3, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, {[2, 4, 3, 1], [3, 4, 2, 1], [4, 2, 3, 1]}, {[3, 4, 2, 1], [3, 2, 4, 1], [4, 2, 3, 1]}, {[1, 3, 4, 2], [1, 3, 2, 4], [1, 2, 4, 3]}, {[4, 1, 3, 2], [4, 3, 1, 2], [4, 2, 3, 1]}, {[1, 4, 2, 3], [1, 3, 2, 4], [1, 2, 4, 3]}, {[1, 3, 2, 4], [2, 3, 1, 4], [2, 1, 3, 4]}, {[1, 3, 2, 4], [2, 1, 3, 4], [3, 1, 2, 4]}} the member , {[4, 3, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3, 4], {[1, 1, 0, 0, 0]}, {3}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 1, 0]}, {4}], [[1, 2], {}, {1}], [[2, 1, 4, 3], {[0, 0, 1, 1, 0], [1, 1, 0, 0, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {1, 2}], [[3, 1, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {1, 2}] , [[3, 1, 2], {[0, 1, 1, 0], [1, 0, 0, 0]}, {2}], [[2, 1, 3], {[1, 1, 0, 0]}, {}], [[2, 1], {[1, 1, 0]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+q^5+q^6, 1+4*q+9*q^2+15*q^3+ 18*q^4+14*q^5+9*q^6+5*q^7+2*q^8+q^9+q^10, 1+5*q+14*q^2+29*q^3+46*q^4+55*q^5+52* q^6+40*q^7+27*q^8+16*q^9+10*q^10+7*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+20*q^2+ 49*q^3+94*q^4+143*q^5+178*q^6+183*q^7+160*q^8+125*q^9+90*q^10+62*q^11+44*q^12+ 29*q^13+18*q^14+14*q^15+9*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+27*q^2+76* q^3+169*q^4+305*q^5+459*q^6+584*q^7+636*q^8+608*q^9+527*q^10+422*q^11+324*q^12+ 245*q^13+178*q^14+130*q^15+98*q^16+70*q^17+50*q^18+35*q^19+25*q^20+18*q^21+13*q ^22+7*q^23+5*q^24+3*q^25+2*q^26+q^27+q^28, 1+8*q+35*q^2+111*q^3+279*q^4+576*q^5 +1003*q^6+1497*q^7+1937*q^8+2203*q^9+2245*q^10+2091*q^11+1818*q^12+1508*q^13+ 1214*q^14+953*q^15+743*q^16+581*q^17+448*q^18+343*q^19+261*q^20+200*q^21+154*q^ 22+117*q^23+86*q^24+62*q^25+47*q^26+33*q^27+25*q^28+17*q^29+11*q^30+7*q^31+5*q^ 32+3*q^33+2*q^34+q^35+q^36, 1+9*q+44*q^2+155*q^3+433*q^4+1000*q^5+1962*q^6+3328 *q^7+4939*q^8+6484*q^9+7623*q^10+8138*q^11+8019*q^12+7413*q^13+6532*q^14+5563*q ^15+4624*q^16+3787*q^17+3085*q^18+2495*q^19+2000*q^20+1604*q^21+1284*q^22+1028* q^23+823*q^24+651*q^25+510*q^26+405*q^27+317*q^28+249*q^29+192*q^30+146*q^31+ 108*q^32+83*q^33+61*q^34+45*q^35+33*q^36+24*q^37+15*q^38+11*q^39+7*q^40+5*q^41+ 3*q^42+2*q^43+q^44+q^45, 1+10*q+54*q^2+209*q^3+641*q^4+1631*q^5+3542*q^6+6688*q ^7+11120*q^8+16449*q^9+21860*q^10+26364*q^11+29193*q^12+30062*q^13+29163*q^14+ 26990*q^15+24111*q^16+20976*q^17+17925*q^18+15155*q^19+12697*q^20+10571*q^21+ 8774*q^22+7262*q^23+6011*q^24+4971*q^25+4088*q^26+3356*q^27+2759*q^28+2263*q^29 +1855*q^30+1515*q^31+1222*q^32+986*q^33+793*q^34+634*q^35+505*q^36+401*q^37+312 *q^38+242*q^39+185*q^40+142*q^41+107*q^42+82*q^43+59*q^44+45*q^45+32*q^46+22*q^ 47+15*q^48+11*q^49+7*q^50+5*q^51+3*q^52+2*q^53+q^54+q^55, 1+11*q+65*q^2+274*q^3 +914*q^4+2534*q^5+6014*q^6+12458*q^7+22828*q^8+37383*q^9+55196*q^10+74094*q^11+ 91225*q^12+104015*q^13+110973*q^14+111956*q^15+107922*q^16+100357*q^17+90773*q^ 18+80456*q^19+70297*q^20+60770*q^21+52134*q^22+44501*q^23+37855*q^24+32149*q^25 +27262*q^26+23060*q^27+19476*q^28+16446*q^29+13885*q^30+11715*q^31+9870*q^32+ 8280*q^33+6940*q^34+5807*q^35+4849*q^36+4038*q^37+3356*q^38+2768*q^39+2276*q^40 +1859*q^41+1516*q^42+1228*q^43+994*q^44+796*q^45+638*q^46+503*q^47+395*q^48+306 *q^49+240*q^50+183*q^51+141*q^52+106*q^53+80*q^54+59*q^55+44*q^56+30*q^57+22*q^ 58+15*q^59+11*q^60+7*q^61+5*q^62+3*q^63+2*q^64+q^65+q^66, 1+12*q+77*q^2+351*q^3 +1264*q^4+3786*q^5+9726*q^6+21866*q^7+43627*q^8+78061*q^9+126332*q^10+186320*q^ 11+252257*q^12+315913*q^13+368943*q^14+405265*q^15+422393*q^16+421337*q^17+ 405510*q^18+379325*q^19+347110*q^20+312408*q^21+277682*q^22+244517*q^23+213880* q^24+186206*q^25+161600*q^26+139938*q^27+120931*q^28+104338*q^29+89971*q^30+ 77556*q^31+66821*q^32+57538*q^33+49468*q^34+42490*q^35+36482*q^36+31291*q^37+ 26811*q^38+22944*q^39+19586*q^40+16669*q^41+14161*q^42+11995*q^43+10136*q^44+ 8545*q^45+7183*q^46+6008*q^47+5013*q^48+4154*q^49+3434*q^50+2828*q^51+2321*q^52 +1892*q^53+1541*q^54+1244*q^55+1002*q^56+800*q^57+635*q^58+498*q^59+394*q^60+ 306*q^61+238*q^62+182*q^63+140*q^64+104*q^65+80*q^66+58*q^67+42*q^68+30*q^69+22 *q^70+15*q^71+11*q^72+7*q^73+5*q^74+3*q^75+2*q^76+q^77+q^78, 1+13*q+90*q^2+441* q^3+1704*q^4+5477*q^5+15116*q^6+36577*q^7+78733*q^8+152387*q^9+267471*q^10+ 428806*q^11+632037*q^12+862002*q^13+1095030*q^14+1304694*q^15+1468607*q^16+ 1573329*q^17+1615894*q^18+1602160*q^19+1543399*q^20+1452848*q^21+1343025*q^22+ 1224068*q^23+1103549*q^24+986764*q^25+876950*q^26+775905*q^27+684299*q^28+ 601963*q^29+528549*q^30+463572*q^31+406275*q^32+355885*q^33+311591*q^34+272597* q^35+238363*q^36+208381*q^37+182079*q^38+159059*q^39+138867*q^40+121113*q^41+ 105502*q^42+91807*q^43+79765*q^44+69230*q^45+60000*q^46+51910*q^47+44811*q^48+ 38603*q^49+33147*q^50+28411*q^51+24284*q^52+20707*q^53+17604*q^54+14933*q^55+ 12614*q^56+10634*q^57+8924*q^58+7462*q^59+6212*q^60+5166*q^61+4269*q^62+3524*q^ 63+2890*q^64+2363*q^65+1921*q^66+1560*q^67+1252*q^68+1005*q^69+798*q^70+635*q^ 71+499*q^72+394*q^73+304*q^74+237*q^75+181*q^76+138*q^77+104*q^78+79*q^79+56*q^ 80+42*q^81+30*q^82+22*q^83+15*q^84+11*q^85+7*q^86+5*q^87+3*q^88+2*q^89+q^90+q^ 91, 1+14*q+104*q^2+545*q^3+2248*q^4+7711*q^5+22726*q^6+58797*q^7+135554*q^8+ 281572*q^9+531524*q^10+918293*q^11+1460980*q^12+2152819*q^13+2954789*q^14+ 3799359*q^15+4604343*q^16+5291699*q^17+5804023*q^18+6113361*q^19+6220961*q^20+ 6150348*q^21+5937831*q^22+5623377*q^23+5244017*q^24+4830750*q^25+4407605*q^26+ 3991789*q^27+3594797*q^28+3223391*q^29+2880771*q^30+2568174*q^31+2285482*q^32+ 2031422*q^33+1804085*q^34+1601141*q^35+1420170*q^36+1259112*q^37+1116034*q^38+ 989003*q^39+876310*q^40+776307*q^41+687439*q^42+608461*q^43+538280*q^44+475900* q^45+420516*q^46+371348*q^47+327639*q^48+288781*q^49+254205*q^50+223448*q^51+ 196129*q^52+171907*q^53+150424*q^54+131419*q^55+114601*q^56+99741*q^57+86623*q^ 58+75075*q^59+64888*q^60+55969*q^61+48159*q^62+41347*q^63+35404*q^64+30245*q^65 +25753*q^66+21883*q^67+18529*q^68+15648*q^69+13161*q^70+11046*q^71+9232*q^72+ 7702*q^73+6396*q^74+5298*q^75+4366*q^76+3591*q^77+2934*q^78+2395*q^79+1940*q^80 +1567*q^81+1256*q^82+1008*q^83+800*q^84+636*q^85+499*q^86+392*q^87+303*q^88+236 *q^89+179*q^90+138*q^91+103*q^92+77*q^93+56*q^94+42*q^95+30*q^96+22*q^97+15*q^ 98+11*q^99+7*q^100+5*q^101+3*q^102+2*q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+4*q^2+q+1, q^10+4*q^9+9*q^8+15* q^7+18*q^6+14*q^5+9*q^4+5*q^3+2*q^2+q+1, q^15+5*q^14+14*q^13+29*q^12+46*q^11+55 *q^10+52*q^9+40*q^8+27*q^7+16*q^6+10*q^5+7*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+20* q^19+49*q^18+94*q^17+143*q^16+178*q^15+183*q^14+160*q^13+125*q^12+90*q^11+62*q^ 10+44*q^9+29*q^8+18*q^7+14*q^6+9*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+76*q^25+459*q^22+ 169*q^24+584*q^21+305*q^23+q+2*q^2+27*q^26+3*q^3+5*q^4+7*q^5+13*q^6+18*q^7+25*q ^8+35*q^9+50*q^10+70*q^11+98*q^12+130*q^13+178*q^14+245*q^15+324*q^16+q^28+422* q^17+527*q^18+608*q^19+636*q^20+7*q^27, 1+2091*q^25+1214*q^22+1818*q^24+953*q^ 21+1508*q^23+q+1003*q^30+1497*q^29+35*q^34+111*q^33+2*q^2+2245*q^26+3*q^3+5*q^4 +7*q^5+11*q^6+279*q^32+17*q^7+25*q^8+33*q^9+47*q^10+62*q^11+86*q^12+117*q^13+ 154*q^14+200*q^15+261*q^16+1937*q^28+343*q^17+448*q^18+581*q^19+743*q^20+2203*q ^27+576*q^31+8*q^35+q^36, 1+2000*q^25+155*q^42+1028*q^22+1604*q^24+823*q^21+ 1284*q^23+q+9*q^44+5563*q^30+q^45+4939*q^37+4624*q^29+1962*q^39+433*q^41+8138*q ^34+3328*q^38+8019*q^33+2*q^2+2495*q^26+3*q^3+5*q^4+7*q^5+11*q^6+7413*q^32+15*q ^7+24*q^8+33*q^9+45*q^10+61*q^11+83*q^12+108*q^13+146*q^14+192*q^15+249*q^16+44 *q^43+3787*q^28+317*q^17+405*q^18+510*q^19+651*q^20+3085*q^27+1000*q^40+6532*q^ 31+7623*q^35+6484*q^36, 1+1855*q^25+30062*q^42+986*q^22+1515*q^24+793*q^21+1222 *q^23+q+641*q^51+26364*q^44+54*q^53+4971*q^30+209*q^52+q^55+10*q^54+21860*q^45+ 17925*q^37+4088*q^29+16449*q^46+24111*q^39+29163*q^41+10571*q^34+1631*q^50+ 20976*q^38+8774*q^33+2*q^2+2263*q^26+3*q^3+5*q^4+7*q^5+11*q^6+7262*q^32+15*q^7+ 22*q^8+32*q^9+45*q^10+59*q^11+82*q^12+107*q^13+142*q^14+185*q^15+242*q^16+29193 *q^43+3542*q^49+3356*q^28+312*q^17+401*q^18+505*q^19+634*q^20+2759*q^27+26990*q ^40+6011*q^31+12697*q^35+6688*q^48+15155*q^36+11120*q^47, 1+1859*q^25+37855*q^ 42+994*q^22+1516*q^24+796*q^21+1228*q^23+q+111956*q^51+52134*q^44+104015*q^53+ 37383*q^57+4849*q^30+110973*q^52+22828*q^58+74094*q^55+6014*q^60+91225*q^54+ 60770*q^45+16446*q^37+4038*q^29+70297*q^46+11*q^65+23060*q^39+2534*q^61+32149*q ^41+9870*q^34+107922*q^50+65*q^64+19476*q^38+8280*q^33+55196*q^56+2*q^2+2276*q^ 26+3*q^3+5*q^4+7*q^5+11*q^6+6940*q^32+15*q^7+22*q^8+30*q^9+914*q^62+44*q^10+59* q^11+80*q^12+106*q^13+141*q^14+183*q^15+240*q^16+q^66+44501*q^43+100357*q^49+ 3356*q^28+306*q^17+395*q^18+503*q^19+638*q^20+2768*q^27+27262*q^40+274*q^63+ 5807*q^31+11715*q^35+90773*q^48+12458*q^59+13885*q^36+80456*q^47, 1+1892*q^25+ 36482*q^42+1002*q^22+1541*q^24+800*q^21+1244*q^23+q+139938*q^51+49468*q^44+ 186206*q^53+312408*q^57+5013*q^30+161600*q^52+347110*q^58+244517*q^55+405510*q^ 60+213880*q^54+57538*q^45+21866*q^71+1264*q^74+16669*q^37+186320*q^67+4154*q^29 +66821*q^46+315913*q^65+78061*q^69+22944*q^39+421337*q^61+31291*q^41+10136*q^34 +120931*q^50+368943*q^64+19586*q^38+8545*q^33+277682*q^56+2*q^2+126332*q^68+ 2321*q^26+3*q^3+5*q^4+7*q^5+11*q^6+7183*q^32+15*q^7+22*q^8+30*q^9+422393*q^62+ 42*q^10+58*q^11+80*q^12+77*q^76+104*q^13+140*q^14+182*q^15+238*q^16+252257*q^66 +43627*q^70+12*q^77+q^78+42490*q^43+104338*q^49+3434*q^28+306*q^17+394*q^18+498 *q^19+635*q^20+2828*q^27+9726*q^72+26811*q^40+3786*q^73+405265*q^63+6008*q^31+ 351*q^75+11995*q^35+89971*q^48+379325*q^59+14161*q^36+77556*q^47, 1+1921*q^25+ 38603*q^42+1005*q^22+13*q^90+1560*q^24+798*q^21+1252*q^23+q+138867*q^51+51910*q ^44+182079*q^53+311591*q^57+5166*q^30+159059*q^52+355885*q^58+238363*q^55+ 463572*q^60+q^91+208381*q^54+60000*q^45+1543399*q^71+1573329*q^74+17604*q^37+ 1103549*q^67+4269*q^29+69230*q^46+876950*q^65+1343025*q^69+24284*q^39+528549*q^ 61+33147*q^41+10634*q^34+632037*q^79+121113*q^50+775905*q^64+20707*q^38+8924*q^ 33+272597*q^56+2*q^2+1224068*q^68+2363*q^26+3*q^3+5*q^4+7*q^5+11*q^6+7462*q^32+ 15*q^7+22*q^8+30*q^9+601963*q^62+42*q^10+56*q^11+79*q^12+1304694*q^76+104*q^13+ 138*q^14+181*q^15+237*q^16+986764*q^66+1452848*q^70+1095030*q^77+862002*q^78+ 44811*q^43+105502*q^49+3524*q^28+304*q^17+394*q^18+499*q^19+635*q^20+2890*q^27+ 1602160*q^72+28411*q^40+1615894*q^73+684299*q^63+6212*q^31+1468607*q^75+12614*q ^35+428806*q^80+91807*q^48+78733*q^83+267471*q^81+15116*q^85+5477*q^86+1704*q^ 87+441*q^88+90*q^89+36577*q^84+152387*q^82+406275*q^59+14933*q^36+79765*q^47, 1 +1940*q^25+41347*q^42+1008*q^22+3799359*q^90+1567*q^24+800*q^21+1256*q^23+q+ 135554*q^97+150424*q^51+55969*q^44+196129*q^53+327639*q^57+5298*q^30+918293*q^ 94+171907*q^52+371348*q^58+254205*q^55+475900*q^60+2954789*q^91+223448*q^54+ 64888*q^45+1804085*q^71+2568174*q^74+281572*q^96+58797*q^98+18529*q^37+1116034* q^67+4366*q^29+75075*q^46+1460980*q^93+22726*q^99+876310*q^65+1420170*q^69+ 25753*q^39+538280*q^61+35404*q^41+11046*q^34+4407605*q^79+131419*q^50+776307*q^ 64+7711*q^100+21883*q^38+9232*q^33+288781*q^56+2*q^2+1259112*q^68+2395*q^26+3*q ^3+5*q^4+7*q^5+11*q^6+7702*q^32+15*q^7+22*q^8+30*q^9+531524*q^95+608461*q^62+ 2248*q^101+42*q^10+56*q^11+77*q^12+3223391*q^76+103*q^13+138*q^14+179*q^15+236* q^16+989003*q^66+1601141*q^70+3594797*q^77+3991789*q^78+48159*q^43+545*q^102+ 114601*q^49+3591*q^28+303*q^17+392*q^18+499*q^19+636*q^20+2934*q^27+2031422*q^ 72+30245*q^40+2285482*q^73+104*q^103+687439*q^63+6396*q^31+2880771*q^75+13161*q ^35+14*q^104+4830750*q^80+99741*q^48+q^105+5937831*q^83+5244017*q^81+6220961*q^ 85+6113361*q^86+5804023*q^87+5291699*q^88+4604343*q^89+6150348*q^84+5623377*q^ 82+420516*q^59+15648*q^36+2152819*q^92+86623*q^47] The number of permutations avoiding, {[4, 3, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550] The number of EVEN permutations avoiding, {[4, 3, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 11, 40, 154, 618, 2515, 10315, 42616, 177035, 738200, 3086833, 12936822, 54314225] The number of ODD permutations avoiding, {[4, 3, 1, 2], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 10, 39, 155, 619, 2511, 10311, 42626, 177045, 738168, 3086801, 12936922, 54314325] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 40, 155, 619, 2515, 10315, 42626, 177045, 738200, 3086833, 12936922, 54314325] ODD:, [0, 1, 3, 10, 39, 154, 618, 2511, 10311, 42616, 177035, 738168, 3086801, 12936822, 54314225] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.761904762, 4.215189873, 5.822006472, 7.557801132, 9.404894548, 11.34970426, 13.38133784, 15.49079587, 17.67048053, 19.91387228, 22.21530583, 24.56980852] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.411003085, 1.907154326, 2.448968052, 3.034311630, 3.660020495, 4.322815562, 5.019560334, 5.747332827, 6.503436602, 7.285390676, 8.090912675, 8.917900767] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.99093, 3.63724, 5.99744, 9.20705, 13.3958, 18.6867, 25.1960, 33.0318, 42.2947, 53.0769, 65.4629, 79.5290] Skewness: , [Float(undefined), 0., 0., 0.2228634801, 0.4465515529, 0.6227485406, 0.7583402646, 0.8641478553, 0.9486824674, 1.017925761, 1.076006098, 1.125805066, 1.169326351, 1.207996370, 1.242854357] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.801956116, 3.295623505, 3.675802683, 3.984698805, 4.241893478, 4.460774083, 4.650994530, 4.819797370, 4.972407709, 5.112707029, 5.243511865, 5.366947271] end of this data For the equivalence class of patterns, { {[1, 2, 4, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, {[3, 4, 2, 1], [3, 2, 1, 4], [4, 1, 2, 3]}, {[2, 3, 4, 1], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 2, 4, 3], [2, 3, 4, 1], [3, 2, 1, 4]}, {[1, 4, 3, 2], [2, 3, 4, 1], [2, 1, 3, 4]}, {[1, 4, 3, 2], [2, 3, 4, 1], [4, 3, 1, 2]}, {[1, 4, 3, 2], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 1, 2, 3]}} the member , {[1, 2, 4, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {4}], [[2, 1, 3], {}, {}], [[3, 4, 1, 2], %2, {1}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 1, 0]}, {}], [[3, 2, 4, 1], %3, {1}], [[3, 4, 2, 1], %3, {1}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {2}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[3, 1, 2, 4], %2, {4}], [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {5}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 2, 0]}, {3}], [[3, 2, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[4, 3, 2, 1], %3, {1}], [[4, 3, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[4, 2, 3, 1], %3, {1}], [[3, 5, 1, 4, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 2, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 1]}, {}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[3, 2, 5, 4, 1], %1, {1}], [[3, 5, 2, 4, 1], %1, {1}], [[1, 3, 2, 4], %2, {4}], [[4, 2, 1, 3], %3, {4}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0]}, {}], [[2, 1, 5, 4, 3], { [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[1, 4, 2, 3], %2, {3}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {}], [[2, 4, 1, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {5}], [[2, 5, 1, 4, 3], %1, {4}], [[2, 5, 1, 3, 4], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {4}], [ [3, 1, 5, 4, 2], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}]} %1 := {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} %2 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} %3 := {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+4*q^3+5*q^4+3*q^5+q^6, 1+2*q+5*q^2+6*q^3+ 9*q^4+11*q^5+12*q^6+11*q^7+9*q^8+4*q^9+q^10, 1+2*q+5*q^2+6*q^3+8*q^4+9*q^5+16*q ^6+17*q^7+24*q^8+30*q^9+29*q^10+29*q^11+23*q^12+14*q^13+5*q^14+q^15, 1+2*q+5*q^ 2+6*q^3+8*q^4+8*q^5+10*q^6+14*q^7+19*q^8+26*q^9+31*q^10+37*q^11+51*q^12+65*q^13 +73*q^14+77*q^15+72*q^16+62*q^17+41*q^18+20*q^19+6*q^20+q^21, 1+2*q+5*q^2+6*q^3 +8*q^4+8*q^5+10*q^6+10*q^7+16*q^8+17*q^9+23*q^10+25*q^11+33*q^12+43*q^13+61*q^ 14+70*q^15+82*q^16+104*q^17+137*q^18+166*q^19+188*q^20+193*q^21+188*q^22+161*q^ 23+118*q^24+66*q^25+27*q^26+7*q^27+q^28, 1+2*q+5*q^2+6*q^3+8*q^4+8*q^5+10*q^6+ 10*q^7+12*q^8+16*q^9+19*q^10+22*q^11+24*q^12+28*q^13+33*q^14+42*q^15+54*q^16+65 *q^17+81*q^18+103*q^19+129*q^20+153*q^21+186*q^22+224*q^23+280*q^24+357*q^25+ 427*q^26+472*q^27+497*q^28+480*q^29+426*q^30+325*q^31+205*q^32+99*q^33+35*q^34+ 8*q^35+q^36, 1+2*q+5*q^2+6*q^3+8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+18*q^10+ 19*q^11+24*q^12+23*q^13+28*q^14+29*q^15+36*q^16+42*q^17+55*q^18+62*q^19+73*q^20 +82*q^21+98*q^22+121*q^23+154*q^24+194*q^25+243*q^26+285*q^27+332*q^28+404*q^29 +490*q^30+596*q^31+745*q^32+922*q^33+1079*q^34+1205*q^35+1261*q^36+1242*q^37+ 1113*q^38+890*q^39+602*q^40+332*q^41+141*q^42+44*q^43+9*q^44+q^45, 1+2*q+5*q^2+ 6*q^3+8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+14*q^10+18*q^11+21*q^12+24*q^13+ 25*q^14+27*q^15+29*q^16+33*q^17+37*q^18+45*q^19+55*q^20+63*q^21+71*q^22+78*q^23 +89*q^24+101*q^25+122*q^26+148*q^27+176*q^28+203*q^29+237*q^30+283*q^31+348*q^ 32+448*q^33+549*q^34+641*q^35+742*q^36+878*q^37+1054*q^38+1282*q^39+1577*q^40+ 1949*q^41+2354*q^42+2747*q^43+3050*q^44+3228*q^45+3190*q^46+2920*q^47+2400*q^48 +1725*q^49+1040*q^50+509*q^51+193*q^52+54*q^53+10*q^54+q^55, 1+2*q+5*q^2+6*q^3+ 8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+14*q^10+14*q^11+20*q^12+21*q^13+26*q^14 +25*q^15+29*q^16+28*q^17+33*q^18+34*q^19+41*q^20+46*q^21+58*q^22+63*q^23+72*q^ 24+76*q^25+86*q^26+94*q^27+108*q^28+125*q^29+150*q^30+174*q^31+196*q^32+218*q^ 33+248*q^34+286*q^35+338*q^36+406*q^37+482*q^38+568*q^39+672*q^40+811*q^41+1004 *q^42+1207*q^43+1424*q^44+1664*q^45+1947*q^46+2290*q^47+2752*q^48+3351*q^49+ 4125*q^50+5034*q^51+6019*q^52+6961*q^53+7766*q^54+8226*q^55+8225*q^56+7620*q^57 +6443*q^58+4833*q^59+3133*q^60+1698*q^61+747*q^62+256*q^63+65*q^64+11*q^65+q^66 , 1+2*q+5*q^2+6*q^3+8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+14*q^10+14*q^11+16* q^12+20*q^13+23*q^14+26*q^15+27*q^16+29*q^17+30*q^18+32*q^19+34*q^20+38*q^21+42 *q^22+50*q^23+59*q^24+66*q^25+72*q^26+77*q^27+84*q^28+91*q^29+102*q^30+113*q^31 +132*q^32+153*q^33+176*q^34+195*q^35+214*q^36+234*q^37+262*q^38+298*q^39+344*q^ 40+407*q^41+469*q^42+535*q^43+600*q^44+687*q^45+788*q^46+926*q^47+1103*q^48+ 1323*q^49+1580*q^50+1899*q^51+2268*q^52+2652*q^53+3124*q^54+3683*q^55+4323*q^56 +5057*q^57+5978*q^58+7165*q^59+8732*q^60+10690*q^61+12946*q^62+15330*q^63+17713 *q^64+19727*q^65+21027*q^66+21151*q^67+19888*q^68+17171*q^69+13369*q^70+9155*q^ 71+5385*q^72+2647*q^73+1058*q^74+331*q^75+77*q^76+12*q^77+q^78, 1+2*q+5*q^2+6*q ^3+8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+14*q^10+14*q^11+16*q^12+16*q^13+22*q ^14+23*q^15+28*q^16+27*q^17+31*q^18+30*q^19+34*q^20+33*q^21+38*q^22+39*q^23+46* q^24+51*q^25+63*q^26+67*q^27+75*q^28+77*q^29+85*q^30+89*q^31+99*q^32+107*q^33+ 121*q^34+137*q^35+160*q^36+179*q^37+197*q^38+213*q^39+231*q^40+251*q^41+278*q^ 42+312*q^43+356*q^44+414*q^45+473*q^46+528*q^47+580*q^48+640*q^49+717*q^50+813* q^51+937*q^52+1084*q^53+1260*q^54+1438*q^55+1641*q^56+1863*q^57+2150*q^58+2517* q^59+3010*q^60+3633*q^61+4376*q^62+5135*q^63+5925*q^64+6889*q^65+8081*q^66+9517 *q^67+11174*q^68+13136*q^69+15535*q^70+18638*q^71+22648*q^72+27574*q^73+33130*q ^74+39108*q^75+45023*q^76+50243*q^77+53680*q^78+54446*q^79+51767*q^80+45595*q^ 81+36562*q^82+26211*q^83+16440*q^84+8834*q^85+3971*q^86+1455*q^87+419*q^88+90*q ^89+13*q^90+q^91, 1+2*q+5*q^2+6*q^3+8*q^4+8*q^5+10*q^6+10*q^7+12*q^8+12*q^9+14* q^10+14*q^11+16*q^12+16*q^13+18*q^14+22*q^15+25*q^16+28*q^17+29*q^18+31*q^19+32 *q^20+34*q^21+35*q^22+37*q^23+39*q^24+43*q^25+47*q^26+55*q^27+64*q^28+71*q^29+ 76*q^30+80*q^31+85*q^32+90*q^33+97*q^34+104*q^35+115*q^36+126*q^37+145*q^38+165 *q^39+186*q^40+200*q^41+215*q^42+230*q^43+248*q^44+268*q^45+295*q^46+328*q^47+ 370*q^48+426*q^49+480*q^50+533*q^51+576*q^52+626*q^53+680*q^54+756*q^55+843*q^ 56+965*q^57+1103*q^58+1258*q^59+1406*q^60+1560*q^61+1731*q^62+1938*q^63+2192*q^ 64+2502*q^65+2897*q^66+3350*q^67+3860*q^68+4398*q^69+5037*q^70+5821*q^71+6863*q ^72+8232*q^73+9859*q^74+11544*q^75+13318*q^76+15377*q^77+17841*q^78+20890*q^79+ 24548*q^80+28882*q^81+34006*q^82+40319*q^83+48346*q^84+58561*q^85+70816*q^86+ 84743*q^87+99647*q^88+114668*q^89+127921*q^90+137218*q^91+140005*q^92+134637*q^ 93+120555*q^94+99213*q^95+73828*q^96+48838*q^97+28187*q^98+13929*q^99+5768*q^ 100+1952*q^101+521*q^102+104*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+5*q^4+4*q^3+5*q^2+3*q+1, q^10+2*q^9+5*q^8+6 *q^7+9*q^6+11*q^5+12*q^4+11*q^3+9*q^2+4*q+1, q^15+2*q^14+5*q^13+6*q^12+8*q^11+9 *q^10+16*q^9+17*q^8+24*q^7+30*q^6+29*q^5+29*q^4+23*q^3+14*q^2+5*q+1, q^21+2*q^ 20+5*q^19+6*q^18+8*q^17+8*q^16+10*q^15+14*q^14+19*q^13+26*q^12+31*q^11+37*q^10+ 51*q^9+65*q^8+73*q^7+77*q^6+72*q^5+62*q^4+41*q^3+20*q^2+6*q+1, 1+6*q^25+10*q^22 +8*q^24+10*q^21+8*q^23+7*q+27*q^2+5*q^26+66*q^3+118*q^4+161*q^5+188*q^6+193*q^7 +188*q^8+166*q^9+137*q^10+104*q^11+82*q^12+70*q^13+61*q^14+43*q^15+33*q^16+q^28 +25*q^17+23*q^18+17*q^19+16*q^20+2*q^27, 1+22*q^25+33*q^22+24*q^24+42*q^21+28*q ^23+8*q+10*q^30+10*q^29+5*q^34+6*q^33+35*q^2+19*q^26+99*q^3+205*q^4+325*q^5+426 *q^6+8*q^32+480*q^7+497*q^8+472*q^9+427*q^10+357*q^11+280*q^12+224*q^13+186*q^ 14+153*q^15+129*q^16+12*q^28+103*q^17+81*q^18+65*q^19+54*q^20+16*q^27+8*q^31+2* q^35+q^36, 1+73*q^25+6*q^42+121*q^22+82*q^24+154*q^21+98*q^23+9*q+2*q^44+29*q^ 30+q^45+12*q^37+36*q^29+10*q^39+8*q^41+19*q^34+10*q^38+24*q^33+44*q^2+62*q^26+ 141*q^3+332*q^4+602*q^5+890*q^6+23*q^32+1113*q^7+1242*q^8+1261*q^9+1205*q^10+ 1079*q^11+922*q^12+745*q^13+596*q^14+490*q^15+404*q^16+5*q^43+42*q^28+332*q^17+ 285*q^18+243*q^19+194*q^20+55*q^27+8*q^40+28*q^31+18*q^35+12*q^36, 1+237*q^25+ 24*q^42+448*q^22+283*q^24+549*q^21+348*q^23+10*q+8*q^51+18*q^44+5*q^53+101*q^30 +6*q^52+q^55+2*q^54+14*q^45+37*q^37+122*q^29+12*q^46+29*q^39+25*q^41+63*q^34+8* q^50+33*q^38+71*q^33+54*q^2+203*q^26+193*q^3+509*q^4+1040*q^5+1725*q^6+78*q^32+ 2400*q^7+2920*q^8+3190*q^9+3228*q^10+3050*q^11+2747*q^12+2354*q^13+1949*q^14+ 1577*q^15+1282*q^16+21*q^43+10*q^49+148*q^28+1054*q^17+878*q^18+742*q^19+641*q^ 20+176*q^27+27*q^40+89*q^31+55*q^35+10*q^48+45*q^36+12*q^47, 1+811*q^25+72*q^42 +1424*q^22+1004*q^24+1664*q^21+1207*q^23+11*q+25*q^51+58*q^44+21*q^53+12*q^57+ 338*q^30+26*q^52+12*q^58+14*q^55+10*q^60+20*q^54+46*q^45+125*q^37+406*q^29+41*q ^46+2*q^65+94*q^39+8*q^61+76*q^41+196*q^34+29*q^50+5*q^64+108*q^38+218*q^33+14* q^56+65*q^2+672*q^26+256*q^3+747*q^4+1698*q^5+3133*q^6+248*q^32+4833*q^7+6443*q ^8+7620*q^9+8*q^62+8225*q^10+8226*q^11+7766*q^12+6961*q^13+6019*q^14+5034*q^15+ 4125*q^16+q^66+63*q^43+28*q^49+482*q^28+3351*q^17+2752*q^18+2290*q^19+1947*q^20 +568*q^27+86*q^40+6*q^63+286*q^31+174*q^35+33*q^48+10*q^59+150*q^36+34*q^47, 1+ 2652*q^25+214*q^42+4323*q^22+3124*q^24+5057*q^21+3683*q^23+12*q+77*q^51+176*q^ 44+66*q^53+38*q^57+1103*q^30+72*q^52+34*q^58+50*q^55+30*q^60+59*q^54+153*q^45+ 10*q^71+8*q^74+407*q^37+14*q^67+1323*q^29+132*q^46+20*q^65+12*q^69+298*q^39+29* q^61+234*q^41+600*q^34+84*q^50+23*q^64+344*q^38+687*q^33+42*q^56+77*q^2+14*q^68 +2268*q^26+331*q^3+1058*q^4+2647*q^5+5385*q^6+788*q^32+9155*q^7+13369*q^8+17171 *q^9+27*q^62+19888*q^10+21151*q^11+21027*q^12+5*q^76+19727*q^13+17713*q^14+ 15330*q^15+12946*q^16+16*q^66+12*q^70+2*q^77+q^78+195*q^43+91*q^49+1580*q^28+ 10690*q^17+8732*q^18+7165*q^19+5978*q^20+1899*q^27+10*q^72+262*q^40+8*q^73+26*q ^63+926*q^31+6*q^75+535*q^35+102*q^48+32*q^59+469*q^36+113*q^47, 1+8081*q^25+ 640*q^42+13136*q^22+2*q^90+9517*q^24+15535*q^21+11174*q^23+13*q+231*q^51+528*q^ 44+197*q^53+121*q^57+3633*q^30+213*q^52+107*q^58+160*q^55+89*q^60+q^91+179*q^54 +473*q^45+34*q^71+27*q^74+1260*q^37+46*q^67+4376*q^29+414*q^46+63*q^65+38*q^69+ 937*q^39+85*q^61+717*q^41+1863*q^34+16*q^79+251*q^50+67*q^64+1084*q^38+2150*q^ 33+137*q^56+90*q^2+39*q^68+6889*q^26+419*q^3+1455*q^4+3971*q^5+8834*q^6+2517*q^ 32+16440*q^7+26211*q^8+36562*q^9+77*q^62+45595*q^10+51767*q^11+54446*q^12+23*q^ 76+53680*q^13+50243*q^14+45023*q^15+39108*q^16+51*q^66+33*q^70+22*q^77+16*q^78+ 580*q^43+278*q^49+5135*q^28+33130*q^17+27574*q^18+22648*q^19+18638*q^20+5925*q^ 27+30*q^72+813*q^40+31*q^73+75*q^63+3010*q^31+28*q^75+1641*q^35+14*q^80+312*q^ 48+12*q^83+14*q^81+10*q^85+8*q^86+8*q^87+6*q^88+5*q^89+10*q^84+12*q^82+99*q^59+ 1438*q^36+356*q^47, 1+24548*q^25+1938*q^42+40319*q^22+22*q^90+28882*q^24+48346* q^21+34006*q^23+14*q+12*q^97+680*q^51+1560*q^44+576*q^53+370*q^57+11544*q^30+14 *q^94+626*q^52+328*q^58+480*q^55+268*q^60+18*q^91+533*q^54+1406*q^45+97*q^71+80 *q^74+12*q^96+10*q^98+3860*q^37+145*q^67+13318*q^29+1258*q^46+16*q^93+10*q^99+ 186*q^65+115*q^69+2897*q^39+248*q^61+2192*q^41+5821*q^34+47*q^79+756*q^50+200*q ^64+8*q^100+3350*q^38+6863*q^33+426*q^56+104*q^2+126*q^68+20890*q^26+521*q^3+ 1952*q^4+5768*q^5+13929*q^6+8232*q^32+28187*q^7+48838*q^8+73828*q^9+14*q^95+230 *q^62+8*q^101+99213*q^10+120555*q^11+134637*q^12+71*q^76+140005*q^13+137218*q^ 14+127921*q^15+114668*q^16+165*q^66+104*q^70+64*q^77+55*q^78+1731*q^43+6*q^102+ 843*q^49+15377*q^28+99647*q^17+84743*q^18+70816*q^19+58561*q^20+17841*q^27+90*q ^72+2502*q^40+85*q^73+5*q^103+215*q^63+9859*q^31+76*q^75+5037*q^35+2*q^104+43*q ^80+965*q^48+q^105+35*q^83+39*q^81+32*q^85+31*q^86+29*q^87+28*q^88+25*q^89+34*q ^84+37*q^82+295*q^59+4398*q^36+16*q^92+1103*q^47] The number of permutations avoiding, {[1, 2, 4, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, is given by [1, 2, 6, 21, 71, 219, 635, 1776, 4853, 13068, 34862, 92438, 244118, 642947, 1690256] The number of EVEN permutations avoiding, {[1, 2, 4, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, is given by [1, 1, 3, 12, 37, 111, 317, 898, 2433, 6537, 17419, 46260, 122096, 321442, 845035] The number of ODD permutations avoiding, {[1, 2, 4, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, is given by [0, 1, 3, 9, 34, 108, 318, 878, 2420, 6531, 17443, 46178, 122022, 321505, 845221] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 12, 37, 108, 318, 898, 2433, 6531, 17443, 46260, 122096, 321505, 845221] ODD:, [0, 1, 3, 9, 34, 111, 317, 878, 2420, 6537, 17419, 46178, 122022, 321442, 845035] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.095238095, 5.464788732, 8.840182648, 13.32598425, 18.93581081, 25.67154338, 33.51645240, 42.44762205, 52.44410307, 63.48853423, 75.56763932, 88.67171659] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.508855192, 2.206462888, 3.007909204, 3.814915043, 4.569698130, 5.231828129, 5.784463155, 6.230033467, 6.581129437, 6.854633787, 7.067532337, 7.234650882] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.27664, 4.86848, 9.04752, 14.5536, 20.8821, 27.3720, 33.4600, 38.8133, 43.3113, 46.9860, 49.9500, 52.3402] Skewness: , [Float(undefined), 0., 0., -0.07871347439, -0.2754856040, -0.5641225533, -0.8610449959, -1.137576399, -1.390190702, -1.615216090, -1.808417464, -1.967162992, -2.090884650, -2.181355262, -2.242333205] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.311169730, 2.467435634, 2.863753005, 3.561940954, 4.496194460, 5.631849185, 6.920004044, 8.288370358, 9.654673551, 10.93634680, 12.06491777, 12.99662165] end of this data For the equivalence class of patterns, { {[4, 3, 2, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, {[1, 3, 2, 4], [1, 2, 3, 4], [2, 3, 1, 4]}, {[1, 3, 2, 4], [1, 2, 3, 4], [3, 1, 2, 4]}, {[4, 1, 3, 2], [4, 3, 2, 1], [4, 2, 3, 1]}, {[1, 3, 4, 2], [1, 3, 2, 4], [1, 2, 3, 4]}, {[3, 2, 4, 1], [4, 3, 2, 1], [4, 2, 3, 1]}, {[1, 4, 2, 3], [1, 3, 2, 4], [1, 2, 3, 4]}, {[2, 4, 3, 1], [4, 3, 2, 1], [4, 2, 3, 1]}} the member , {[4, 3, 2, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3, 4], {[1, 1, 0, 0, 0]}, {3}], [[1], {}, {}], [[1, 2], {}, {1}], [[3, 2, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}], [ [2, 1, 4, 3], {[0, 0, 1, 1, 0], [1, 1, 0, 0, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {1, 2}], [[3, 1, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {1, 2}] , [[3, 1, 2], {[0, 1, 1, 0], [1, 0, 0, 0]}, {2}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {4}], [[2, 1, 3], {[1, 1, 0, 0]}, {}], [[2, 1], {[1, 1, 0]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+2*q^5, 1+4*q+9*q^2+15*q^3+18* q^4+16*q^5+10*q^6+5*q^7+q^8, 1+5*q+14*q^2+29*q^3+46*q^4+58*q^5+57*q^6+46*q^7+30 *q^8+16*q^9+7*q^10+2*q^11, 1+6*q+20*q^2+49*q^3+94*q^4+147*q^5+189*q^6+203*q^7+ 185*q^8+146*q^9+102*q^10+64*q^11+35*q^12+17*q^13+6*q^14+q^15, 1+7*q+27*q^2+76*q ^3+169*q^4+310*q^5+478*q^6+629*q^7+714*q^8+710*q^9+631*q^10+510*q^11+382*q^12+ 267*q^13+174*q^14+102*q^15+53*q^16+24*q^17+9*q^18+2*q^19, 1+8*q+35*q^2+111*q^3+ 279*q^4+582*q^5+1032*q^6+1581*q^7+2117*q^8+2504*q^9+2653*q^10+2556*q^11+2277*q^ 12+1905*q^13+1514*q^14+1146*q^15+823*q^16+556*q^17+351*q^18+205*q^19+110*q^20+ 54*q^21+23*q^22+7*q^23+q^24, 1+9*q+44*q^2+155*q^3+433*q^4+1007*q^5+2003*q^6+ 3468*q^7+5292*q^8+7188*q^9+8779*q^10+9750*q^11+9975*q^12+9529*q^13+8615*q^14+ 7446*q^15+6194*q^16+4965*q^17+3831*q^18+2836*q^19+2009*q^20+1359*q^21+878*q^22+ 538*q^23+309*q^24+163*q^25+78*q^26+33*q^27+11*q^28+2*q^29, 1+10*q+54*q^2+209*q^ 3+641*q^4+1639*q^5+3597*q^6+6904*q^7+11743*q^8+17878*q^9+24580*q^10+30788*q^11+ 35477*q^12+38012*q^13+38303*q^14+36697*q^15+33744*q^16+29983*q^17+25843*q^18+ 21637*q^19+17600*q^20+13896*q^21+10644*q^22+7906*q^23+5690*q^24+3959*q^25+2657* q^26+1711*q^27+1049*q^28+606*q^29+328*q^30+166*q^31+77*q^32+30*q^33+8*q^34+q^35 , 1+11*q+65*q^2+274*q^3+914*q^4+2543*q^5+6085*q^6+12773*q^7+23848*q^8+40014*q^9 +60851*q^10+84522*q^11+108069*q^12+128259*q^13+142580*q^14+149864*q^15+150314*q ^16+145048*q^17+135538*q^18+123198*q^19+109233*q^20+94616*q^21+80119*q^22+66342 *q^23+53722*q^24+42535*q^25+32920*q^26+24894*q^27+18373*q^28+13206*q^29+9218*q^ 30+6232*q^31+4072*q^32+2563*q^33+1546*q^34+888*q^35+481*q^36+242*q^37+111*q^38+ 44*q^39+13*q^40+2*q^41, 1+12*q+77*q^2+351*q^3+1264*q^4+3796*q^5+9815*q^6+22306* q^7+45205*q^8+82568*q^9+137081*q^10+208371*q^11+291975*q^12+379735*q^13+461732* q^14+528948*q^15+575433*q^16+599116*q^17+601227*q^18+585009*q^19+554466*q^20+ 513576*q^21+465938*q^22+414662*q^23+362354*q^24+311105*q^25+262517*q^26+217740* q^27+177526*q^28+142255*q^29+111986*q^30+86534*q^31+65569*q^32+48663*q^33+35330 *q^34+25056*q^35+17332*q^36+11673*q^37+7640*q^38+4842*q^39+2952*q^40+1718*q^41+ 949*q^42+496*q^43+243*q^44+107*q^45+38*q^46+9*q^47+q^48, 1+13*q+90*q^2+441*q^3+ 1704*q^4+5488*q^5+15225*q^6+37171*q^7+81068*q^8+159688*q^9+286550*q^10+471760*q ^11+717094*q^12+1012454*q^13+1335832*q^14+1657492*q^15+1946912*q^16+2179591*q^ 17+2341099*q^18+2427472*q^19+2442905*q^20+2396525*q^21+2299643*q^22+2164007*q^ 23+2000812*q^24+1820207*q^25+1630989*q^26+1440487*q^27+1254573*q^28+1077797*q^ 29+913479*q^30+763821*q^31+630027*q^32+512490*q^33+410950*q^34+324665*q^35+ 252555*q^36+193319*q^37+145517*q^38+107642*q^39+78175*q^40+55666*q^41+38799*q^ 42+26424*q^43+17549*q^44+11334*q^45+7091*q^46+4280*q^47+2484*q^48+1379*q^49+724 *q^50+353*q^51+155*q^52+57*q^53+15*q^54+2*q^55, 1+14*q+104*q^2+545*q^3+2248*q^4 +7723*q^5+22857*q^6+59577*q^7+138887*q^8+292881*q^9+563597*q^10+996731*q^11+ 1629911*q^12+2478182*q^13+3522121*q^14+4704490*q^15+5938397*q^16+7124738*q^17+ 8172311*q^18+9013310*q^19+9609960*q^20+9952722*q^21+10053534*q^22+9938086*q^23+ 9639407*q^24+9193360*q^25+8635626*q^26+7999680*q^27+7315474*q^28+6608826*q^29+ 5901303*q^30+5210407*q^31+4549847*q^32+3929885*q^33+3357665*q^34+2837566*q^35+ 2371591*q^36+1959832*q^37+1600897*q^38+1292252*q^39+1030472*q^40+811468*q^41+ 630744*q^42+483674*q^43+365697*q^44+272431*q^45+199779*q^46+144044*q^47+101992* q^48+70835*q^49+48189*q^50+32057*q^51+20811*q^52+13148*q^53+8049*q^54+4748*q^55 +2684*q^56+1447*q^57+739*q^58+350*q^59+145*q^60+47*q^61+10*q^62+q^63] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+6*q^2+4*q+2), q^2*(q^8+4*q^7+9*q^6 +15*q^5+18*q^4+16*q^3+10*q^2+5*q+1), q^4*(q^11+5*q^10+14*q^9+29*q^8+46*q^7+58*q ^6+57*q^5+46*q^4+30*q^3+16*q^2+7*q+2), q^6*(q^15+6*q^14+20*q^13+49*q^12+94*q^11 +147*q^10+189*q^9+203*q^8+185*q^7+146*q^6+102*q^5+64*q^4+35*q^3+17*q^2+6*q+1), q^9*(q^19+7*q^18+27*q^17+76*q^16+169*q^15+310*q^14+478*q^13+629*q^12+714*q^11+ 710*q^10+631*q^9+510*q^8+382*q^7+267*q^6+174*q^5+102*q^4+53*q^3+24*q^2+9*q+2), q^12*(1+35*q^22+q^24+111*q^21+8*q^23+7*q+23*q^2+54*q^3+110*q^4+205*q^5+351*q^6+ 556*q^7+823*q^8+1146*q^9+1514*q^10+1905*q^11+2277*q^12+2556*q^13+2653*q^14+2504 *q^15+2117*q^16+1581*q^17+1032*q^18+582*q^19+279*q^20), q^16*(2+433*q^25+3468*q ^22+1007*q^24+5292*q^21+2003*q^23+11*q+q^29+33*q^2+155*q^26+78*q^3+163*q^4+309* q^5+538*q^6+878*q^7+1359*q^8+2009*q^9+2836*q^10+3831*q^11+4965*q^12+6194*q^13+ 7446*q^14+8615*q^15+9529*q^16+9*q^28+9975*q^17+9750*q^18+8779*q^19+7188*q^20+44 *q^27), q^20*(1+24580*q^25+38012*q^22+30788*q^24+38303*q^21+35477*q^23+8*q+1639 *q^30+3597*q^29+10*q^34+54*q^33+30*q^2+17878*q^26+77*q^3+166*q^4+328*q^5+606*q^ 6+209*q^32+1049*q^7+1711*q^8+2657*q^9+3959*q^10+5690*q^11+7906*q^12+10644*q^13+ 13896*q^14+17600*q^15+21637*q^16+6904*q^28+25843*q^17+29983*q^18+33744*q^19+ 36697*q^20+11743*q^27+641*q^31+q^35), q^25*(2+150314*q^25+123198*q^22+145048*q^ 24+109233*q^21+135538*q^23+13*q+84522*q^30+914*q^37+108069*q^29+65*q^39+q^41+ 12773*q^34+274*q^38+23848*q^33+44*q^2+149864*q^26+111*q^3+242*q^4+481*q^5+888*q ^6+40014*q^32+1546*q^7+2563*q^8+4072*q^9+6232*q^10+9218*q^11+13206*q^12+18373*q ^13+24894*q^14+32920*q^15+42535*q^16+128259*q^28+53722*q^17+66342*q^18+80119*q^ 19+94616*q^20+142580*q^27+11*q^40+60851*q^31+6085*q^35+2543*q^36), q^30*(1+ 414662*q^25+9815*q^42+262517*q^22+362354*q^24+217740*q^21+311105*q^23+9*q+1264* q^44+601227*q^30+351*q^45+208371*q^37+585009*q^29+77*q^46+82568*q^39+22306*q^41 +461732*q^34+137081*q^38+528948*q^33+38*q^2+465938*q^26+107*q^3+243*q^4+496*q^5 +949*q^6+575433*q^32+1718*q^7+2952*q^8+4842*q^9+7640*q^10+11673*q^11+17332*q^12 +25056*q^13+35330*q^14+48663*q^15+65569*q^16+3796*q^43+554466*q^28+86534*q^17+ 111986*q^18+142255*q^19+177526*q^20+513576*q^27+45205*q^40+599116*q^31+379735*q ^35+q^48+291975*q^36+12*q^47), q^36*(2+913479*q^25+1012454*q^42+512490*q^22+ 763821*q^24+410950*q^21+630027*q^23+15*q+1704*q^51+471760*q^44+90*q^53+1820207* q^30+441*q^52+q^55+13*q^54+286550*q^45+2341099*q^37+1630989*q^29+159688*q^46+ 1946912*q^39+1335832*q^41+2396525*q^34+5488*q^50+2179591*q^38+2299643*q^33+57*q ^2+1077797*q^26+155*q^3+353*q^4+724*q^5+1379*q^6+2164007*q^32+2484*q^7+4280*q^8 +7091*q^9+11334*q^10+17549*q^11+26424*q^12+38799*q^13+55666*q^14+78175*q^15+ 107642*q^16+717094*q^43+15225*q^49+1440487*q^28+145517*q^17+193319*q^18+252555* q^19+324665*q^20+1254573*q^27+1657492*q^40+2000812*q^31+2442905*q^35+37171*q^48 +2427472*q^36+81068*q^47), q^42*(1+1600897*q^25+9952722*q^42+811468*q^22+ 1292252*q^24+630744*q^21+1030472*q^23+10*q+1629911*q^51+9013310*q^44+563597*q^ 53+22857*q^57+3929885*q^30+996731*q^52+7723*q^58+138887*q^55+545*q^60+292881*q^ 54+8172311*q^45+8635626*q^37+3357665*q^29+7124738*q^46+9639407*q^39+104*q^61+ 10053534*q^41+6608826*q^34+2478182*q^50+9193360*q^38+5901303*q^33+59577*q^56+47 *q^2+1959832*q^26+145*q^3+350*q^4+739*q^5+1447*q^6+5210407*q^32+2684*q^7+4748*q ^8+8049*q^9+14*q^62+13148*q^10+20811*q^11+32057*q^12+48189*q^13+70835*q^14+ 101992*q^15+144044*q^16+9609960*q^43+3522121*q^49+2837566*q^28+199779*q^17+ 272431*q^18+365697*q^19+483674*q^20+2371591*q^27+9938086*q^40+q^63+4549847*q^31 +7315474*q^35+4704490*q^48+2248*q^59+7999680*q^36+5938397*q^47)] The number of permutations avoiding, {[4, 3, 2, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 79, 311, 1265, 5275, 22431, 96900, 424068, 1876143, 8377299, 37704042, 170870106] The number of EVEN permutations avoiding, {[4, 3, 2, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 10, 39, 155, 632, 2638, 11216, 48452, 212036, 938073, 4188651, 18852017, 85435049] The number of ODD permutations avoiding, {[4, 3, 2, 1], [4, 2, 3, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 11, 40, 156, 633, 2637, 11215, 48448, 212032, 938070, 4188648, 18852025, 85435057] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 39, 156, 633, 2638, 11216, 48448, 212032, 938073, 4188651, 18852025, 85435057] ODD:, [0, 1, 3, 11, 40, 155, 632, 2637, 11215, 48452, 212036, 938070, 4188648, 18852017, 85435049] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.714285714, 4.075949367, 5.575562701, 7.207114625, 8.963981043, 10.83977531, 12.82873065, 14.92573361, 17.12625424, 19.42626090, 21.82214334, 24.31064820] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.314492737, 1.667045836, 2.047643360, 2.462964563, 2.913057917, 3.396469651, 3.911489273, 4.456470429, 5.029913564, 5.630477682, 6.256969469, 6.908326031] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.72789, 2.77904, 4.19284, 6.06619, 8.48591, 11.5360, 15.2997, 19.8601, 25.3000, 31.7023, 39.1497, 47.7250] Skewness: , [Float(undefined), 0., 0., -0.09242291912, -0.03841101610, 0.05772565413, 0.1514643902, 0.2312383083, 0.2964192674, 0.3491631434, 0.3919569893, 0.4269384983, 0.4558072077, 0.4798688803, 0.5001304694] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.345354691, 2.577630037, 2.716880087, 2.805618380, 2.871255110, 2.925331075, 2.972251207, 3.013765110, 3.050789733, 3.083890069, 3.113566309, 3.140232051] end of this data For the equivalence class of patterns, { {[3, 2, 4, 1], [3, 2, 1, 4], [4, 1, 2, 3]}, {[2, 3, 4, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, {[1, 4, 3, 2], [3, 1, 2, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 4, 3, 1], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 3, 4, 1], [4, 1, 3, 2]}, {[1, 4, 3, 2], [2, 3, 4, 1], [2, 3, 1, 4]}, {[1, 4, 2, 3], [3, 2, 1, 4], [4, 1, 2, 3]}, {[1, 3, 4, 2], [2, 3, 4, 1], [3, 2, 1, 4]}} the member , {[2, 3, 4, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[3, 1, 2], {[1, 0, 1, 0], [1, 0, 0, 1]}, {}], [[2, 1, 3, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 2, 5], {[0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {}], [[3, 5, 1, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {3}], [ [2, 1, 4, 3], {[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {}], [[3, 2, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 1, 0, 0]}, {2}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[3, 4, 2, 1], %2, {1}], [[4, 2, 3, 1], %2, {1}], [[3, 5, 1, 4, 2], { [1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[2, 5, 1, 3, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 5, 2, 4, 1], %1, {1}], [[2, 4, 1, 3, 5], {[0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [2, 5, 1, 4, 3], {[0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[2, 1, 5, 3, 4], { [0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 4, 3], {[0, 1, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [ [3, 1, 5, 4, 2], {[1, 0, 1, 0, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 1, 4, 3, 5], {[0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[4, 5, 2, 3, 1], %1, {1}], [[4, 5, 1, 3, 2], {[1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [4, 1, 5, 3, 2], {[1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[4, 5, 1, 2, 3], { [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {3}], [[1, 3, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {}], [[1, 4, 2, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 1, 0, 0]}, {3}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[3, 1, 4, 2, 5], {[0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [3, 4, 1, 2], {[1, 0, 0, 0, 1], [0, 1, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {}], [[3, 2, 5, 4, 1], %1, {1}], [[3, 2, 4, 1], %2, {1}], [[4, 2, 5, 3, 1], %1, {1}], [[3, 1, 5, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[2, 1, 3], {[1, 0, 0, 1]}, {}], [[3, 1, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1]}, {2}], [[4, 1, 5, 2, 3], {[0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 1, 0, 0, 1], [0, 1, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[1, 3, 2], {[1, 0, 1, 0], [1, 0, 0, 1]}, {}], [[4, 1, 2, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {2}], [[2, 3, 1], {[1, 0, 0, 1]}, {}]} %1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} %2 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+4*q^3+4*q^4+3*q^5+q^6, 1+4*q+9*q^2+11*q^3 +11*q^4+9*q^5+8*q^6+8*q^7+7*q^8+4*q^9+q^10, 1+5*q+14*q^2+23*q^3+28*q^4+26*q^5+ 24*q^6+21*q^7+20*q^8+19*q^9+16*q^10+16*q^11+15*q^12+11*q^13+5*q^14+q^15, 1+6*q+ 20*q^2+41*q^3+61*q^4+68*q^5+67*q^6+60*q^7+56*q^8+52*q^9+48*q^10+45*q^11+44*q^12 +43*q^13+38*q^14+33*q^15+32*q^16+31*q^17+26*q^18+16*q^19+6*q^20+q^21, 1+7*q+27* q^2+66*q^3+117*q^4+156*q^5+173*q^6+167*q^7+157*q^8+144*q^9+135*q^10+125*q^11+ 119*q^12+118*q^13+113*q^14+103*q^15+97*q^16+97*q^17+96*q^18+88*q^19+77*q^20+68* q^21+65*q^22+63*q^23+57*q^24+42*q^25+22*q^26+7*q^27+q^28, 1+8*q+35*q^2+99*q^3+ 204*q^4+319*q^5+405*q^6+435*q^7+429*q^8+401*q^9+376*q^10+348*q^11+327*q^12+314* q^13+306*q^14+292*q^15+274*q^16+265*q^17+266*q^18+262*q^19+245*q^20+224*q^21+ 212*q^22+211*q^23+212*q^24+202*q^25+180*q^26+157*q^27+141*q^28+133*q^29+128*q^ 30+120*q^31+99*q^32+64*q^33+29*q^34+8*q^35+q^36, 1+9*q+44*q^2+141*q^3+331*q^4+ 595*q^5+862*q^6+1042*q^7+1113*q^8+1092*q^9+1039*q^10+968*q^11+907*q^12+854*q^13 +822*q^14+793*q^15+755*q^16+720*q^17+706*q^18+702*q^19+686*q^20+649*q^21+612*q^ 22+596*q^23+601*q^24+599*q^25+573*q^26+531*q^27+491*q^28+469*q^29+463*q^30+464* q^31+453*q^32+419*q^33+369*q^34+324*q^35+293*q^36+274*q^37+261*q^38+248*q^39+ 219*q^40+163*q^41+93*q^42+37*q^43+9*q^44+q^45, 1+10*q+54*q^2+193*q^3+508*q^4+ 1032*q^5+1689*q^6+2295*q^7+2698*q^8+2842*q^9+2809*q^10+2670*q^11+2513*q^12+2353 *q^13+2235*q^14+2142*q^15+2056*q^16+1962*q^17+1890*q^18+1856*q^19+1833*q^20+ 1777*q^21+1695*q^22+1627*q^23+1605*q^24+1612*q^25+1601*q^26+1541*q^27+1452*q^28 +1381*q^29+1351*q^30+1356*q^31+1360*q^32+1331*q^33+1255*q^34+1159*q^35+1084*q^ 36+1042*q^37+1025*q^38+1020*q^39+1005*q^40+954*q^41+864*q^42+761*q^43+674*q^44+ 611*q^45+567*q^46+535*q^47+509*q^48+467*q^49+382*q^50+256*q^51+130*q^52+46*q^53 +10*q^54+q^55, 1+11*q+65*q^2+256*q^3+746*q^4+1689*q^5+3088*q^6+4685*q^7+6087*q^ 8+6973*q^9+7307*q^10+7208*q^11+6900*q^12+6497*q^13+6133*q^14+5823*q^15+5579*q^ 16+5338*q^17+5115*q^18+4953*q^19+4860*q^20+4762*q^21+4607*q^22+4419*q^23+4288*q ^24+4247*q^25+4243*q^26+4182*q^27+4036*q^28+3855*q^29+3722*q^30+3682*q^31+3703* q^32+3712*q^33+3632*q^34+3464*q^35+3278*q^36+3145*q^37+3083*q^38+3077*q^39+3085 *q^40+3053*q^41+2937*q^42+2753*q^43+2563*q^44+2416*q^45+2325*q^46+2276*q^47+ 2254*q^48+2225*q^49+2146*q^50+1990*q^51+1783*q^52+1578*q^53+1409*q^54+1278*q^55 +1178*q^56+1102*q^57+1044*q^58+976*q^59+849*q^60+638*q^61+386*q^62+176*q^63+56* q^64+11*q^65+q^66, 1+12*q+77*q^2+331*q^3+1057*q^4+2637*q^5+5330*q^6+8954*q^7+ 12828*q^8+16056*q^9+18072*q^10+18790*q^11+18573*q^12+17796*q^13+16860*q^14+ 15940*q^15+15182*q^16+14505*q^17+13890*q^18+13350*q^19+12966*q^20+12685*q^21+ 12374*q^22+11957*q^23+11541*q^24+11265*q^25+11161*q^26+11082*q^27+10870*q^28+ 10502*q^29+10114*q^30+9863*q^31+9799*q^32+9840*q^33+9810*q^34+9596*q^35+9223*q^ 36+8844*q^37+8594*q^38+8515*q^39+8549*q^40+8585*q^41+8496*q^42+8227*q^43+7840*q ^44+7463*q^45+7202*q^46+7071*q^47+7033*q^48+7028*q^49+6982*q^50+6808*q^51+6484* q^52+6081*q^53+5707*q^54+5421*q^55+5219*q^56+5081*q^57+4994*q^58+4924*q^59+4793 *q^60+4529*q^61+4134*q^62+3691*q^63+3287*q^64+2953*q^65+2679*q^66+2456*q^67+ 2280*q^68+2146*q^69+2020*q^70+1825*q^71+1487*q^72+1024*q^73+562*q^74+232*q^75+ 67*q^76+12*q^77+q^78, 1+13*q+90*q^2+419*q^3+1454*q^4+3960*q^5+8768*q^6+16175*q^ 7+25423*q^8+34750*q^9+42271*q^10+46831*q^11+48462*q^12+47852*q^13+46021*q^14+ 43696*q^15+41485*q^16+39494*q^17+37754*q^18+36166*q^19+34868*q^20+33886*q^21+ 33072*q^22+32156*q^23+31111*q^24+30158*q^25+29552*q^26+29226*q^27+28876*q^28+ 28226*q^29+27325*q^30+26473*q^31+25964*q^32+25837*q^33+25842*q^34+25641*q^35+ 25045*q^36+24192*q^37+23396*q^38+22917*q^39+22804*q^40+22912*q^41+22966*q^42+ 22711*q^43+22074*q^44+21235*q^45+20477*q^46+19993*q^47+19812*q^48+19833*q^49+ 19900*q^50+19814*q^51+19412*q^52+18692*q^53+17853*q^54+17129*q^55+16622*q^56+ 16305*q^57+16137*q^58+16069*q^59+15990*q^60+15726*q^61+15165*q^62+14367*q^63+ 13522*q^64+12786*q^65+12202*q^66+11748*q^67+11394*q^68+11126*q^69+10910*q^70+ 10652*q^71+10202*q^72+9484*q^73+8581*q^74+7669*q^75+6869*q^76+6197*q^77+5623*q^ 78+5135*q^79+4736*q^80+4426*q^81+4166*q^82+3845*q^83+3312*q^84+2511*q^85+1586*q ^86+794*q^87+299*q^88+79*q^89+13*q^90+q^91, 1+14*q+104*q^2+521*q^3+1951*q^4+ 5756*q^5+13851*q^6+27846*q^7+47729*q^8+71018*q^9+93490*q^10+111024*q^11+121508* q^12+125073*q^13+123545*q^14+118996*q^15+113406*q^16+107815*q^17+102793*q^18+ 98232*q^19+94283*q^20+91030*q^21+88476*q^22+86148*q^23+83654*q^24+81051*q^25+ 78865*q^26+77397*q^27+76393*q^28+75173*q^29+73355*q^30+71169*q^31+69281*q^32+ 68190*q^33+67823*q^34+67579*q^35+66764*q^36+65168*q^37+63169*q^38+61452*q^39+ 60485*q^40+60299*q^41+60487*q^42+60430*q^43+59635*q^44+58084*q^45+56219*q^46+ 54634*q^47+53707*q^48+53459*q^49+53650*q^50+53853*q^51+53603*q^52+52612*q^53+ 51010*q^54+49250*q^55+47789*q^56+46826*q^57+46350*q^58+46248*q^59+46320*q^60+ 46234*q^61+45629*q^62+44366*q^63+42678*q^64+40985*q^65+39577*q^66+38520*q^67+ 37764*q^68+37257*q^69+36937*q^70+36678*q^71+36219*q^72+35282*q^73+33798*q^74+ 32003*q^75+30250*q^76+28753*q^77+27520*q^78+26483*q^79+25603*q^80+24874*q^81+ 24265*q^82+23652*q^83+22807*q^84+21514*q^85+19772*q^86+17829*q^87+15985*q^88+ 14385*q^89+13013*q^90+11810*q^91+10758*q^92+9871*q^93+9162*q^94+8592*q^95+8011* q^96+7157*q^97+5823*q^98+4097*q^99+2380*q^100+1093*q^101+378*q^102+92*q^103+14* q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+4*q^3+4*q^2+3*q+1, q^10+4*q^9+9*q^8+ 11*q^7+11*q^6+9*q^5+8*q^4+8*q^3+7*q^2+4*q+1, q^15+5*q^14+14*q^13+23*q^12+28*q^ 11+26*q^10+24*q^9+21*q^8+20*q^7+19*q^6+16*q^5+16*q^4+15*q^3+11*q^2+5*q+1, q^21+ 6*q^20+20*q^19+41*q^18+61*q^17+68*q^16+67*q^15+60*q^14+56*q^13+52*q^12+48*q^11+ 45*q^10+44*q^9+43*q^8+38*q^7+33*q^6+32*q^5+31*q^4+26*q^3+16*q^2+6*q+1, 1+66*q^ 25+173*q^22+117*q^24+167*q^21+156*q^23+7*q+22*q^2+27*q^26+42*q^3+57*q^4+63*q^5+ 65*q^6+68*q^7+77*q^8+88*q^9+96*q^10+97*q^11+97*q^12+103*q^13+113*q^14+118*q^15+ 119*q^16+q^28+125*q^17+135*q^18+144*q^19+157*q^20+7*q^27, 1+348*q^25+306*q^22+ 327*q^24+292*q^21+314*q^23+8*q+405*q^30+435*q^29+35*q^34+99*q^33+29*q^2+376*q^ 26+64*q^3+99*q^4+120*q^5+128*q^6+204*q^32+133*q^7+141*q^8+157*q^9+180*q^10+202* q^11+212*q^12+211*q^13+212*q^14+224*q^15+245*q^16+429*q^28+262*q^17+266*q^18+ 265*q^19+274*q^20+401*q^27+319*q^31+8*q^35+q^36, 1+686*q^25+141*q^42+596*q^22+ 649*q^24+601*q^21+612*q^23+9*q+9*q^44+793*q^30+q^45+1113*q^37+755*q^29+862*q^39 +331*q^41+968*q^34+1042*q^38+907*q^33+37*q^2+702*q^26+93*q^3+163*q^4+219*q^5+ 248*q^6+854*q^32+261*q^7+274*q^8+293*q^9+324*q^10+369*q^11+419*q^12+453*q^13+ 464*q^14+463*q^15+469*q^16+44*q^43+720*q^28+491*q^17+531*q^18+573*q^19+599*q^20 +706*q^27+595*q^40+822*q^31+1039*q^35+1092*q^36, 1+1351*q^25+2353*q^42+1331*q^ 22+1356*q^24+1255*q^21+1360*q^23+10*q+508*q^51+2670*q^44+54*q^53+1612*q^30+193* q^52+q^55+10*q^54+2809*q^45+1890*q^37+1601*q^29+2842*q^46+2056*q^39+2235*q^41+ 1777*q^34+1032*q^50+1962*q^38+1695*q^33+46*q^2+1381*q^26+130*q^3+256*q^4+382*q^ 5+467*q^6+1627*q^32+509*q^7+535*q^8+567*q^9+611*q^10+674*q^11+761*q^12+864*q^13 +954*q^14+1005*q^15+1020*q^16+2513*q^43+1689*q^49+1541*q^28+1025*q^17+1042*q^18 +1084*q^19+1159*q^20+1452*q^27+2142*q^40+1605*q^31+1833*q^35+2295*q^48+1856*q^ 36+2698*q^47, 1+3053*q^25+4288*q^42+2563*q^22+2937*q^24+2416*q^21+2753*q^23+11* q+5823*q^51+4607*q^44+6497*q^53+6973*q^57+3278*q^30+6133*q^52+6087*q^58+7208*q^ 55+3088*q^60+6900*q^54+4762*q^45+3855*q^37+3145*q^29+4860*q^46+11*q^65+4182*q^ 39+1689*q^61+4247*q^41+3703*q^34+5579*q^50+65*q^64+4036*q^38+3712*q^33+7307*q^ 56+56*q^2+3085*q^26+176*q^3+386*q^4+638*q^5+849*q^6+3632*q^32+976*q^7+1044*q^8+ 1102*q^9+746*q^62+1178*q^10+1278*q^11+1409*q^12+1578*q^13+1783*q^14+1990*q^15+ 2146*q^16+q^66+4419*q^43+5338*q^49+3083*q^28+2225*q^17+2254*q^18+2276*q^19+2325 *q^20+3077*q^27+4243*q^40+256*q^63+3464*q^31+3682*q^35+5115*q^48+4685*q^59+3722 *q^36+4953*q^47, 1+6081*q^25+9223*q^42+5219*q^22+5707*q^24+5081*q^21+5421*q^23+ 12*q+11082*q^51+9810*q^44+11265*q^53+12685*q^57+7033*q^30+11161*q^52+12966*q^58 +11957*q^55+13890*q^60+11541*q^54+9840*q^45+8954*q^71+1057*q^74+8585*q^37+18790 *q^67+7028*q^29+9799*q^46+17796*q^65+16056*q^69+8515*q^39+14505*q^61+8844*q^41+ 7840*q^34+10870*q^50+16860*q^64+8549*q^38+7463*q^33+12374*q^56+67*q^2+18072*q^ 68+6484*q^26+232*q^3+562*q^4+1024*q^5+1487*q^6+7202*q^32+1825*q^7+2020*q^8+2146 *q^9+15182*q^62+2280*q^10+2456*q^11+2679*q^12+77*q^76+2953*q^13+3287*q^14+3691* q^15+4134*q^16+18573*q^66+12828*q^70+12*q^77+q^78+9596*q^43+10502*q^49+6982*q^ 28+4529*q^17+4793*q^18+4924*q^19+4994*q^20+6808*q^27+5330*q^72+8594*q^40+2637*q ^73+15940*q^63+7071*q^31+331*q^75+8227*q^35+10114*q^48+13350*q^59+8496*q^36+ 9863*q^47, 1+12202*q^25+19833*q^42+11126*q^22+13*q^90+11748*q^24+10910*q^21+ 11394*q^23+13*q+22804*q^51+19993*q^44+23396*q^53+25842*q^57+15726*q^30+22917*q^ 52+25837*q^58+25045*q^55+26473*q^60+q^91+24192*q^54+20477*q^45+34868*q^71+39494 *q^74+17853*q^37+31111*q^67+15165*q^29+21235*q^46+29552*q^65+33072*q^69+19412*q ^39+27325*q^61+19900*q^41+16305*q^34+48462*q^79+22912*q^50+29226*q^64+18692*q^ 38+16137*q^33+25641*q^56+79*q^2+32156*q^68+12786*q^26+299*q^3+794*q^4+1586*q^5+ 2511*q^6+16069*q^32+3312*q^7+3845*q^8+4166*q^9+28226*q^62+4426*q^10+4736*q^11+ 5135*q^12+43696*q^76+5623*q^13+6197*q^14+6869*q^15+7669*q^16+30158*q^66+33886*q ^70+46021*q^77+47852*q^78+19812*q^43+22966*q^49+14367*q^28+8581*q^17+9484*q^18+ 10202*q^19+10652*q^20+13522*q^27+36166*q^72+19814*q^40+37754*q^73+28876*q^63+ 15990*q^31+41485*q^75+16622*q^35+46831*q^80+22711*q^48+25423*q^83+42271*q^81+ 8768*q^85+3960*q^86+1454*q^87+419*q^88+90*q^89+16175*q^84+34750*q^82+25964*q^59 +17129*q^36+22074*q^47, 1+25603*q^25+44366*q^42+23652*q^22+118996*q^90+24874*q^ 24+22807*q^21+24265*q^23+14*q+47729*q^97+51010*q^51+46234*q^44+53603*q^53+53707 *q^57+32003*q^30+111024*q^94+52612*q^52+54634*q^58+53650*q^55+58084*q^60+123545 *q^91+53853*q^54+46320*q^45+67823*q^71+71169*q^74+71018*q^96+27846*q^98+37764*q ^37+63169*q^67+30250*q^29+46248*q^46+121508*q^93+13851*q^99+60485*q^65+66764*q^ 69+39577*q^39+59635*q^61+42678*q^41+36678*q^34+78865*q^79+49250*q^50+60299*q^64 +5756*q^100+38520*q^38+36219*q^33+53459*q^56+92*q^2+65168*q^68+26483*q^26+378*q ^3+1093*q^4+2380*q^5+4097*q^6+35282*q^32+5823*q^7+7157*q^8+8011*q^9+93490*q^95+ 60430*q^62+1951*q^101+8592*q^10+9162*q^11+9871*q^12+75173*q^76+10758*q^13+11810 *q^14+13013*q^15+14385*q^16+61452*q^66+67579*q^70+76393*q^77+77397*q^78+45629*q ^43+521*q^102+47789*q^49+28753*q^28+15985*q^17+17829*q^18+19772*q^19+21514*q^20 +27520*q^27+68190*q^72+40985*q^40+69281*q^73+104*q^103+60487*q^63+33798*q^31+ 73355*q^75+36937*q^35+14*q^104+81051*q^80+46826*q^48+q^105+88476*q^83+83654*q^ 81+94283*q^85+98232*q^86+102793*q^87+107815*q^88+113406*q^89+91030*q^84+86148*q ^82+56219*q^59+37257*q^36+125073*q^92+46350*q^47] The number of permutations avoiding, {[2, 3, 4, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 73, 245, 795, 2508, 7732, 23393, 69687, 204939, 596215, 1718714, 4915914] The number of EVEN permutations avoiding, {[2, 3, 4, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, is given by [1, 1, 3, 11, 37, 123, 399, 1257, 3870, 11703, 34855, 102489, 298138, 859407, 2458039] The number of ODD permutations avoiding, {[2, 3, 4, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, is given by [0, 1, 3, 10, 36, 122, 396, 1251, 3862, 11690, 34832, 102450, 298077, 859307, 2457875] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 37, 122, 396, 1257, 3870, 11690, 34832, 102489, 298138, 859307, 2457875] ODD:, [0, 1, 3, 10, 36, 123, 399, 1251, 3862, 11703, 34855, 102450, 298077, 859407, 2458039] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.794520548, 6.979591837, 9.474213836, 12.25478469, 15.30522504, 18.61471380, 22.17592951, 25.98386837, 30.03508634, 34.32721267, 38.85863077] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.557659737, 2.381198639, 3.442174182, 4.736081413, 6.255211308, 7.993447173, 9.946891025, 12.11336613, 14.49183175, 17.08194323, 19.88376084, 22.89756662] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.42630, 5.67011, 11.8486, 22.4305, 39.1277, 63.8952, 98.9406, 146.734, 210.013, 291.793, 395.364, 524.299] Skewness: , [Float(undefined), 0., 0., 0.07913801369, 0.1773337259, 0.2652788461, 0.3368593899, 0.3938490751, 0.4392229309, 0.4756078316, 0.5050677095, 0.5291687119, 0.5490758520, 0.5656655461, 0.5795991493] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.163809718, 2.111523756, 2.089938151, 2.102668522, 2.133004087, 2.169099939, 2.204675222, 2.236900192, 2.264844299, 2.288381260, 2.307861212, 2.323741489] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 4, 1], [2, 3, 1, 4]}, {[2, 3, 1, 4], [3, 2, 1, 4], [4, 2, 1, 3]}, {[3, 1, 2, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, {[1, 4, 3, 2], [1, 3, 4, 2], [4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 3, 4, 1], [3, 2, 4, 1]}, {[3, 2, 4, 1], [3, 2, 1, 4], [3, 1, 2, 4]}, {[1, 4, 2, 3], [4, 1, 2, 3], [4, 2, 1, 3]}, {[1, 4, 3, 2], [1, 4, 2, 3], [2, 4, 3, 1]}} the member , {[1, 4, 3, 2], [1, 3, 4, 2], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[1, 3, 2], {[0, 1, 0, 0]}, {3}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[2, 1, 3], {}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[3, 2, 1], {}, {2}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+9*q^3+ 11*q^4+13*q^5+12*q^6+10*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+16*q^3+20*q^4+26*q^5 +31*q^6+35*q^7+37*q^8+36*q^9+31*q^10+25*q^11+18*q^12+11*q^13+5*q^14+q^15, 1+6*q +16*q^2+27*q^3+36*q^4+46*q^5+59*q^6+73*q^7+87*q^8+98*q^9+105*q^10+113*q^11+116* q^12+110*q^13+98*q^14+82*q^15+64*q^16+47*q^17+30*q^18+16*q^19+6*q^20+q^21, 1+7* q+22*q^2+43*q^3+63*q^4+82*q^5+105*q^6+134*q^7+168*q^8+205*q^9+237*q^10+268*q^11 +300*q^12+329*q^13+351*q^14+368*q^15+376*q^16+370*q^17+348*q^18+313*q^19+267*q^ 20+218*q^21+169*q^22+123*q^23+82*q^24+47*q^25+22*q^26+7*q^27+q^28, 1+8*q+29*q^2 +65*q^3+106*q^4+145*q^5+187*q^6+239*q^7+304*q^8+382*q^9+468*q^10+555*q^11+643*q ^12+732*q^13+822*q^14+915*q^15+1009*q^16+1095*q^17+1159*q^18+1209*q^19+1249*q^ 20+1270*q^21+1260*q^22+1211*q^23+1124*q^24+1008*q^25+873*q^26+729*q^27+585*q^28 +451*q^29+328*q^30+222*q^31+135*q^32+70*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^2+ 94*q^3+171*q^4+251*q^5+332*q^6+426*q^7+543*q^8+688*q^9+860*q^10+1056*q^11+1270* q^12+1493*q^13+1718*q^14+1956*q^15+2216*q^16+2498*q^17+2778*q^18+3045*q^19+3303 *q^20+3558*q^21+3796*q^22+4005*q^23+4174*q^24+4304*q^25+4389*q^26+4412*q^27+ 4362*q^28+4230*q^29+3996*q^30+3676*q^31+3291*q^32+2865*q^33+2424*q^34+1992*q^35 +1584*q^36+1212*q^37+884*q^38+604*q^39+380*q^40+212*q^41+100*q^42+37*q^43+9*q^ 44+q^45, 1+10*q+46*q^2+131*q^3+265*q^4+422*q^5+583*q^6+758*q^7+969*q^8+1231*q^9 +1550*q^10+1927*q^11+2367*q^12+2864*q^13+3394*q^14+3944*q^15+4534*q^16+5187*q^ 17+5895*q^18+6638*q^19+7415*q^20+8229*q^21+9061*q^22+9881*q^23+10675*q^24+11462 *q^25+12253*q^26+13025*q^27+13734*q^28+14354*q^29+14862*q^30+15256*q^31+15528*q ^32+15666*q^33+15643*q^34+15418*q^35+14950*q^36+14221*q^37+13258*q^38+12109*q^ 39+10819*q^40+9452*q^41+8068*q^42+6724*q^43+5461*q^44+4310*q^45+3283*q^46+2395* q^47+1653*q^48+1062*q^49+622*q^50+320*q^51+138*q^52+46*q^53+10*q^54+q^55, 1+11* q+56*q^2+177*q^3+396*q^4+687*q^5+1005*q^6+1341*q^7+1727*q^8+2200*q^9+2781*q^10+ 3479*q^11+4306*q^12+5281*q^13+6396*q^14+7615*q^15+8919*q^16+10334*q^17+11885*q^ 18+13571*q^19+15391*q^20+17371*q^21+19510*q^22+21770*q^23+24084*q^24+26431*q^25 +28840*q^26+31354*q^27+33955*q^28+36581*q^29+39152*q^30+41656*q^31+44095*q^32+ 46458*q^33+48703*q^34+50782*q^35+52632*q^36+54196*q^37+55413*q^38+56251*q^39+ 56687*q^40+56688*q^41+56187*q^42+55112*q^43+53377*q^44+50948*q^45+47846*q^46+ 44178*q^47+40088*q^48+35745*q^49+31296*q^50+26879*q^51+22630*q^52+18654*q^53+ 15022*q^54+11777*q^55+8942*q^56+6525*q^57+4536*q^58+2965*q^59+1793*q^60+980*q^ 61+467*q^62+185*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+233*q^3+573*q^4+1083*q ^5+1692*q^6+2346*q^7+3068*q^8+3927*q^9+4981*q^10+6260*q^11+7787*q^12+9600*q^13+ 11737*q^14+14195*q^15+16942*q^16+19959*q^17+23254*q^18+26842*q^19+30756*q^20+ 35061*q^21+39812*q^22+45001*q^23+50568*q^24+56460*q^25+62666*q^26+69199*q^27+ 76067*q^28+83250*q^29+90685*q^30+98330*q^31+106206*q^32+114333*q^33+122652*q^34 +131036*q^35+139335*q^36+147474*q^37+155452*q^38+163261*q^39+170834*q^40+178088 *q^41+184894*q^42+191128*q^43+196620*q^44+201184*q^45+204677*q^46+207050*q^47+ 208263*q^48+208233*q^49+206793*q^50+203745*q^51+198929*q^52+192214*q^53+183539* q^54+173010*q^55+160866*q^56+147465*q^57+133225*q^58+118577*q^59+103925*q^60+ 89652*q^61+76053*q^62+63377*q^63+51800*q^64+41429*q^65+32309*q^66+24457*q^67+ 17859*q^68+12482*q^69+8270*q^70+5125*q^71+2921*q^72+1494*q^73+662*q^74+242*q^75 +67*q^76+12*q^77+q^78, 1+13*q+79*q^2+300*q^3+806*q^4+1656*q^5+2775*q^6+4038*q^7 +5414*q^8+6995*q^9+8908*q^10+11241*q^11+14047*q^12+17389*q^13+21351*q^14+26003* q^15+31377*q^16+37486*q^17+44315*q^18+51818*q^19+59989*q^20+68952*q^21+78897*q^ 22+89935*q^23+102036*q^24+115144*q^25+129282*q^26+144541*q^27+160938*q^28+ 178396*q^29+196782*q^30+216031*q^31+236198*q^32+257445*q^33+279878*q^34+303452* q^35+327912*q^36+352978*q^37+378485*q^38+404440*q^39+430869*q^40+457789*q^41+ 485079*q^42+512513*q^43+539788*q^44+566641*q^45+592822*q^46+618177*q^47+642586* q^48+665937*q^49+688002*q^50+708488*q^51+727006*q^52+743095*q^53+756254*q^54+ 766109*q^55+772410*q^56+774989*q^57+773639*q^58+768112*q^59+758057*q^60+743065* q^61+722667*q^62+696579*q^63+664786*q^64+627677*q^65+585958*q^66+540621*q^67+ 492812*q^68+443703*q^69+394415*q^70+346011*q^71+299428*q^72+255441*q^73+214644* q^74+177423*q^75+144015*q^76+114516*q^77+88920*q^78+67142*q^79+49051*q^80+34443 *q^81+23060*q^82+14570*q^83+8570*q^84+4611*q^85+2213*q^86+915*q^87+310*q^88+79* q^89+13*q^90+q^91, 1+14*q+92*q^2+379*q^3+1106*q^4+2462*q^5+4431*q^6+6813*q^7+ 9452*q^8+12409*q^9+15903*q^10+20149*q^11+25288*q^12+31436*q^13+38742*q^14+47369 *q^15+57463*q^16+69170*q^17+82619*q^18+97808*q^19+114612*q^20+133024*q^21+ 153330*q^22+175924*q^23+201018*q^24+228618*q^25+258758*q^26+291653*q^27+327578* q^28+366639*q^29+408705*q^30+453551*q^31+501087*q^32+551474*q^33+605056*q^34+ 662135*q^35+722724*q^36+786542*q^37+853283*q^38+922854*q^39+995343*q^40+1070844 *q^41+1149266*q^42+1230283*q^43+1313434*q^44+1398343*q^45+1484842*q^46+1572923* q^47+1662519*q^48+1753340*q^49+1844825*q^50+1936366*q^51+2027442*q^52+2117548*q ^53+2206025*q^54+2292213*q^55+2375548*q^56+2455615*q^57+2532029*q^58+2604306*q^ 59+2671660*q^60+2733063*q^61+2787224*q^62+2832886*q^63+2868936*q^64+2894466*q^ 65+2908729*q^66+2911127*q^67+2901082*q^68+2877953*q^69+2840917*q^70+2788977*q^ 71+2721137*q^72+2636661*q^73+2535255*q^74+2417287*q^75+2283926*q^76+2137298*q^ 77+1980269*q^78+1816163*q^79+1648354*q^80+1480108*q^81+1314440*q^82+1154060*q^ 83+1001274*q^84+857997*q^85+725616*q^86+605054*q^87+496773*q^88+400871*q^89+ 317189*q^90+245373*q^91+184906*q^92+135132*q^93+95257*q^94+64328*q^95+41271*q^ 96+24887*q^97+13911*q^98+7078*q^99+3196*q^100+1237*q^101+390*q^102+92*q^103+14* q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+9 *q^7+11*q^6+13*q^5+12*q^4+10*q^3+7*q^2+4*q+1, q^15+5*q^14+11*q^13+16*q^12+20*q^ 11+26*q^10+31*q^9+35*q^8+37*q^7+36*q^6+31*q^5+25*q^4+18*q^3+11*q^2+5*q+1, q^21+ 6*q^20+16*q^19+27*q^18+36*q^17+46*q^16+59*q^15+73*q^14+87*q^13+98*q^12+105*q^11 +113*q^10+116*q^9+110*q^8+98*q^7+82*q^6+64*q^5+47*q^4+30*q^3+16*q^2+6*q+1, 1+43 *q^25+105*q^22+63*q^24+134*q^21+82*q^23+7*q+22*q^2+22*q^26+47*q^3+82*q^4+123*q^ 5+169*q^6+218*q^7+267*q^8+313*q^9+348*q^10+370*q^11+376*q^12+368*q^13+351*q^14+ 329*q^15+300*q^16+q^28+268*q^17+237*q^18+205*q^19+168*q^20+7*q^27, 1+555*q^25+ 822*q^22+643*q^24+915*q^21+732*q^23+8*q+187*q^30+239*q^29+29*q^34+65*q^33+29*q^ 2+468*q^26+70*q^3+135*q^4+222*q^5+328*q^6+106*q^32+451*q^7+585*q^8+729*q^9+873* q^10+1008*q^11+1124*q^12+1211*q^13+1260*q^14+1270*q^15+1249*q^16+304*q^28+1209* q^17+1159*q^18+1095*q^19+1009*q^20+382*q^27+145*q^31+8*q^35+q^36, 1+3303*q^25+ 94*q^42+4005*q^22+3558*q^24+4174*q^21+3796*q^23+9*q+9*q^44+1956*q^30+q^45+543*q ^37+2216*q^29+332*q^39+171*q^41+1056*q^34+426*q^38+1270*q^33+37*q^2+3045*q^26+ 100*q^3+212*q^4+380*q^5+604*q^6+1493*q^32+884*q^7+1212*q^8+1584*q^9+1992*q^10+ 2424*q^11+2865*q^12+3291*q^13+3676*q^14+3996*q^15+4230*q^16+37*q^43+2498*q^28+ 4362*q^17+4412*q^18+4389*q^19+4304*q^20+2778*q^27+251*q^40+1718*q^31+860*q^35+ 688*q^36, 1+14862*q^25+2864*q^42+15666*q^22+15256*q^24+15643*q^21+15528*q^23+10 *q+265*q^51+1927*q^44+46*q^53+11462*q^30+131*q^52+q^55+10*q^54+1550*q^45+5895*q ^37+12253*q^29+1231*q^46+4534*q^39+3394*q^41+8229*q^34+422*q^50+5187*q^38+9061* q^33+46*q^2+14354*q^26+138*q^3+320*q^4+622*q^5+1062*q^6+9881*q^32+1653*q^7+2395 *q^8+3283*q^9+4310*q^10+5461*q^11+6724*q^12+8068*q^13+9452*q^14+10819*q^15+ 12109*q^16+2367*q^43+583*q^49+13025*q^28+13258*q^17+14221*q^18+14950*q^19+15418 *q^20+13734*q^27+3944*q^40+10675*q^31+7415*q^35+758*q^48+6638*q^36+969*q^47, 1+ 56688*q^25+24084*q^42+53377*q^22+56187*q^24+50948*q^21+55112*q^23+11*q+7615*q^ 51+19510*q^44+5281*q^53+2200*q^57+52632*q^30+6396*q^52+1727*q^58+3479*q^55+1005 *q^60+4306*q^54+17371*q^45+36581*q^37+54196*q^29+15391*q^46+11*q^65+31354*q^39+ 687*q^61+26431*q^41+44095*q^34+8919*q^50+56*q^64+33955*q^38+46458*q^33+2781*q^ 56+56*q^2+56687*q^26+185*q^3+467*q^4+980*q^5+1793*q^6+48703*q^32+2965*q^7+4536* q^8+6525*q^9+396*q^62+8942*q^10+11777*q^11+15022*q^12+18654*q^13+22630*q^14+ 26879*q^15+31296*q^16+q^66+21770*q^43+10334*q^49+55413*q^28+35745*q^17+40088*q^ 18+44178*q^19+47846*q^20+56251*q^27+28840*q^40+177*q^63+50782*q^31+41656*q^35+ 11885*q^48+1341*q^59+39152*q^36+13571*q^47, 1+192214*q^25+139335*q^42+160866*q^ 22+183539*q^24+147465*q^21+173010*q^23+12*q+69199*q^51+122652*q^44+56460*q^53+ 35061*q^57+208263*q^30+62666*q^52+30756*q^58+45001*q^55+23254*q^60+50568*q^54+ 114333*q^45+2346*q^71+573*q^74+178088*q^37+6260*q^67+208233*q^29+106206*q^46+ 9600*q^65+3927*q^69+163261*q^39+19959*q^61+147474*q^41+196620*q^34+76067*q^50+ 11737*q^64+170834*q^38+201184*q^33+39812*q^56+67*q^2+4981*q^68+198929*q^26+242* q^3+662*q^4+1494*q^5+2921*q^6+204677*q^32+5125*q^7+8270*q^8+12482*q^9+16942*q^ 62+17859*q^10+24457*q^11+32309*q^12+67*q^76+41429*q^13+51800*q^14+63377*q^15+ 76053*q^16+7787*q^66+3068*q^70+12*q^77+q^78+131036*q^43+83250*q^49+206793*q^28+ 89652*q^17+103925*q^18+118577*q^19+133225*q^20+203745*q^27+1692*q^72+155452*q^ 40+1083*q^73+14195*q^63+207050*q^31+233*q^75+191128*q^35+90685*q^48+26842*q^59+ 184894*q^36+98330*q^47, 1+585958*q^25+665937*q^42+443703*q^22+13*q^90+540621*q^ 24+394415*q^21+492812*q^23+13*q+430869*q^51+618177*q^44+378485*q^53+279878*q^57 +743065*q^30+404440*q^52+257445*q^58+327912*q^55+216031*q^60+q^91+352978*q^54+ 592822*q^45+59989*q^71+37486*q^74+756254*q^37+102036*q^67+722667*q^29+566641*q^ 46+129282*q^65+78897*q^69+727006*q^39+196782*q^61+688002*q^41+774989*q^34+14047 *q^79+457789*q^50+144541*q^64+743095*q^38+773639*q^33+303452*q^56+79*q^2+89935* q^68+627677*q^26+310*q^3+915*q^4+2213*q^5+4611*q^6+768112*q^32+8570*q^7+14570*q ^8+23060*q^9+178396*q^62+34443*q^10+49051*q^11+67142*q^12+26003*q^76+88920*q^13 +114516*q^14+144015*q^15+177423*q^16+115144*q^66+68952*q^70+21351*q^77+17389*q^ 78+642586*q^43+485079*q^49+696579*q^28+214644*q^17+255441*q^18+299428*q^19+ 346011*q^20+664786*q^27+51818*q^72+708488*q^40+44315*q^73+160938*q^63+758057*q^ 31+31377*q^75+772410*q^35+11241*q^80+512513*q^48+5414*q^83+8908*q^81+2775*q^85+ 1656*q^86+806*q^87+300*q^88+79*q^89+4038*q^84+6995*q^82+236198*q^59+766109*q^36 +539788*q^47, 1+1648354*q^25+2832886*q^42+1154060*q^22+47369*q^90+1480108*q^24+ 1001274*q^21+1314440*q^23+14*q+9452*q^97+2206025*q^51+2733063*q^44+2027442*q^53 +1662519*q^57+2417287*q^30+20149*q^94+2117548*q^52+1572923*q^58+1844825*q^55+ 1398343*q^60+38742*q^91+1936366*q^54+2671660*q^45+605056*q^71+453551*q^74+12409 *q^96+6813*q^98+2901082*q^37+853283*q^67+2283926*q^29+2604306*q^46+25288*q^93+ 4431*q^99+995343*q^65+722724*q^69+2908729*q^39+1313434*q^61+2868936*q^41+ 2788977*q^34+258758*q^79+2292213*q^50+1070844*q^64+2462*q^100+2911127*q^38+ 2721137*q^33+1753340*q^56+92*q^2+786542*q^68+1816163*q^26+390*q^3+1237*q^4+3196 *q^5+7078*q^6+2636661*q^32+13911*q^7+24887*q^8+41271*q^9+15903*q^95+1230283*q^ 62+1106*q^101+64328*q^10+95257*q^11+135132*q^12+366639*q^76+184906*q^13+245373* q^14+317189*q^15+400871*q^16+922854*q^66+662135*q^70+327578*q^77+291653*q^78+ 2787224*q^43+379*q^102+2375548*q^49+2137298*q^28+496773*q^17+605054*q^18+725616 *q^19+857997*q^20+1980269*q^27+551474*q^72+2894466*q^40+501087*q^73+92*q^103+ 1149266*q^63+2535255*q^31+408705*q^75+2840917*q^35+14*q^104+228618*q^80+2455615 *q^48+q^105+153330*q^83+201018*q^81+114612*q^85+97808*q^86+82619*q^87+69170*q^ 88+57463*q^89+133024*q^84+175924*q^82+1484842*q^59+2877953*q^36+31436*q^92+ 2532029*q^47] The number of permutations avoiding, {[1, 4, 3, 2], [1, 3, 4, 2], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550] The number of EVEN permutations avoiding, {[1, 4, 3, 2], [1, 3, 4, 2], [4, 1, 3, 2]}, is given by [1, 1, 3, 10, 39, 154, 618, 2512, 10312, 42618, 177037, 738175, 3086808, 12936844, 54314247] The number of ODD permutations avoiding, {[1, 4, 3, 2], [1, 3, 4, 2], [4, 1, 3, 2]}, is given by [0, 1, 3, 11, 40, 155, 619, 2514, 10314, 42624, 177043, 738193, 3086826, 12936900, 54314303] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 39, 155, 619, 2512, 10312, 42624, 177043, 738175, 3086808, 12936900, 54314303] ODD:, [0, 1, 3, 11, 40, 154, 618, 2514, 10314, 42618, 177037, 738193, 3086826, 12936844, 54314247] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.037974684, 7.653721683, 10.88197251, 14.75169121, 19.28682246, 24.50723822, 30.42956676, 37.06785368, 44.43407368, 52.53852477, 61.39013306] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.543033500, 2.258278935, 3.093969613, 4.040295017, 5.088670785, 6.231930763, 7.464142822, 8.780373253, 10.17648078, 11.64895295, 13.19477995, 14.81135771] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.38095, 5.09982, 9.57265, 16.3240, 25.8946, 38.8370, 55.7134, 77.0950, 103.561, 135.698, 174.102, 219.376] -5 Skewness: , [Float(undefined), 0., 0., -0.2721915177 10 , -0.04055809944, -0.09542671810, -0.1512285677, -0.2030004439, -0.2493579128, -0.2902927136, -0.3262748870, -0.3578977011, -0.3857410980, -0.4103165467, -0.4320714770] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.234375069, 2.317237769, 2.390674945, 2.462422306, 2.531414280, 2.596325245, 2.656503003, 2.711773029, 2.762268483, 2.808322379, 2.850214663, 2.888303470] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 4, 1, 3], [2, 3, 1, 4]}, {[2, 4, 1, 3], [2, 3, 1, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2]}, {[3, 1, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [3, 2, 4, 1], [3, 1, 4, 2]}, {[1, 4, 2, 3], [2, 4, 3, 1], [2, 4, 1, 3]}, {[1, 4, 2, 3], [2, 4, 1, 3], [4, 2, 1, 3]}, {[3, 2, 4, 1], [3, 1, 4, 2], [3, 1, 2, 4]}} the member , {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 4, 3, 1], {}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 3, 2], {}, {}], [[3, 2, 1], {}, {2}], [[1, 4, 3, 2], {}, {3}], [[2, 3, 1], {}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+10*q^3 +11*q^4+11*q^5+12*q^6+10*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+18*q^3+23*q^4+26*q^ 5+30*q^6+32*q^7+32*q^8+31*q^9+29*q^10+25*q^11+18*q^12+11*q^13+5*q^14+q^15, 1+6* q+16*q^2+30*q^3+44*q^4+54*q^5+65*q^6+76*q^7+84*q^8+89*q^9+93*q^10+94*q^11+94*q^ 12+90*q^13+84*q^14+75*q^15+62*q^16+47*q^17+30*q^18+16*q^19+6*q^20+q^21, 1+7*q+ 22*q^2+47*q^3+78*q^4+106*q^5+133*q^6+161*q^7+189*q^8+217*q^9+241*q^10+258*q^11+ 269*q^12+278*q^13+287*q^14+292*q^15+289*q^16+281*q^17+269*q^18+251*q^19+226*q^ 20+197*q^21+160*q^22+121*q^23+82*q^24+47*q^25+22*q^26+7*q^27+q^28, 1+8*q+29*q^2 +70*q^3+130*q^4+196*q^5+262*q^6+329*q^7+398*q^8+473*q^9+551*q^10+628*q^11+695*q ^12+745*q^13+788*q^14+828*q^15+865*q^16+893*q^17+914*q^18+922*q^19+920*q^20+913 *q^21+897*q^22+866*q^23+821*q^24+764*q^25+695*q^26+613*q^27+521*q^28+421*q^29+ 317*q^30+220*q^31+135*q^32+70*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^2+100*q^3+ 206*q^4+343*q^5+494*q^6+652*q^7+815*q^8+991*q^9+1183*q^10+1390*q^11+1606*q^12+ 1814*q^13+1998*q^14+2167*q^15+2320*q^16+2456*q^17+2586*q^18+2704*q^19+2806*q^20 +2885*q^21+2941*q^22+2981*q^23+3008*q^24+3021*q^25+3009*q^26+2970*q^27+2902*q^ 28+2810*q^29+2694*q^30+2549*q^31+2369*q^32+2160*q^33+1924*q^34+1668*q^35+1395*q ^36+1116*q^37+843*q^38+591*q^39+378*q^40+212*q^41+100*q^42+37*q^43+9*q^44+q^45, 1+10*q+46*q^2+138*q^3+313*q^4+572*q^5+891*q^6+1247*q^7+1625*q^8+2031*q^9+2474*q ^10+2960*q^11+3497*q^12+4071*q^13+4651*q^14+5225*q^15+5780*q^16+6298*q^17+6786* q^18+7248*q^19+7695*q^20+8120*q^21+8513*q^22+8855*q^23+9142*q^24+9390*q^25+9605 *q^26+9786*q^27+9937*q^28+10045*q^29+10104*q^30+10102*q^31+10048*q^32+9944*q^33 +9789*q^34+9573*q^35+9288*q^36+8925*q^37+8492*q^38+7993*q^39+7426*q^40+6788*q^ 41+6086*q^42+5335*q^43+4554*q^44+3765*q^45+2987*q^46+2256*q^47+1599*q^48+1047*q ^49+620*q^50+320*q^51+138*q^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+185*q^3+459* q^4+915*q^5+1541*q^6+2298*q^7+3144*q^8+4067*q^9+5077*q^10+6186*q^11+7419*q^12+ 8785*q^13+10263*q^14+11830*q^15+13453*q^16+15083*q^17+16680*q^18+18236*q^19+ 19760*q^20+21259*q^21+22728*q^22+24158*q^23+25529*q^24+26802*q^25+27972*q^26+ 29031*q^27+29977*q^28+30837*q^29+31625*q^30+32354*q^31+32994*q^32+33517*q^33+ 33906*q^34+34166*q^35+34330*q^36+34394*q^37+34347*q^38+34180*q^39+33863*q^40+ 33383*q^41+32739*q^42+31939*q^43+30977*q^44+29851*q^45+28540*q^46+27039*q^47+ 25352*q^48+23502*q^49+21500*q^50+19364*q^51+17118*q^52+14807*q^53+12485*q^54+ 10216*q^55+8058*q^56+6077*q^57+4341*q^58+2896*q^59+1776*q^60+978*q^61+467*q^62+ 185*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+242*q^3+653*q^4+1412*q^5+2565*q^6+ 4083*q^7+5892*q^8+7936*q^9+10206*q^10+12714*q^11+15501*q^12+18610*q^13+22052*q^ 14+25829*q^15+29936*q^16+34307*q^17+38838*q^18+43431*q^19+48035*q^20+52663*q^21 +57305*q^22+61944*q^23+66585*q^24+71182*q^25+75705*q^26+80111*q^27+84321*q^28+ 88275*q^29+91963*q^30+95420*q^31+98678*q^32+101754*q^33+104637*q^34+107293*q^35 +109712*q^36+111847*q^37+113702*q^38+115264*q^39+116554*q^40+117585*q^41+118376 *q^42+118906*q^43+119121*q^44+118990*q^45+118503*q^46+117660*q^47+116484*q^48+ 114936*q^49+112996*q^50+110622*q^51+107788*q^52+104480*q^53+100710*q^54+96482*q ^55+91796*q^56+86650*q^57+81048*q^58+75023*q^59+68635*q^60+61964*q^61+55087*q^ 62+48105*q^63+41138*q^64+34338*q^65+27859*q^66+21858*q^67+16471*q^68+11824*q^69 +8004*q^70+5039*q^71+2902*q^72+1492*q^73+662*q^74+242*q^75+67*q^76+12*q^77+q^78 , 1+13*q+79*q^2+310*q^3+905*q^4+2112*q^5+4125*q^6+7008*q^7+10693*q^8+15064*q^9+ 20054*q^10+25650*q^11+31902*q^12+38897*q^13+46700*q^14+55368*q^15+64970*q^16+ 75529*q^17+86981*q^18+99139*q^19+111782*q^20+124798*q^21+138104*q^22+151662*q^ 23+165468*q^24+179485*q^25+193666*q^26+207972*q^27+222346*q^28+236654*q^29+ 250730*q^30+264421*q^31+277609*q^32+290270*q^33+302422*q^34+314086*q^35+325288* q^36+336043*q^37+346294*q^38+355976*q^39+365001*q^40+373281*q^41+380785*q^42+ 387563*q^43+393681*q^44+399186*q^45+404095*q^46+408333*q^47+411831*q^48+414517* q^49+416340*q^50+417311*q^51+417434*q^52+416753*q^53+415262*q^54+412928*q^55+ 409646*q^56+405296*q^57+399822*q^58+393211*q^59+385469*q^60+376585*q^61+366492* q^62+355153*q^63+342508*q^64+328564*q^65+313344*q^66+296910*q^67+279317*q^68+ 260649*q^69+240996*q^70+220516*q^71+199412*q^72+177951*q^73+156420*q^74+135131* q^75+114427*q^76+94685*q^77+76285*q^78+59574*q^79+44852*q^80+32326*q^81+22119*q ^82+14216*q^83+8465*q^84+4590*q^85+2211*q^86+915*q^87+310*q^88+79*q^89+13*q^90+ q^91, 1+14*q+92*q^2+390*q^3+1226*q^4+3074*q^5+6433*q^6+11649*q^7+18810*q^8+ 27797*q^9+38457*q^10+50702*q^11+64560*q^12+80168*q^13+97683*q^14+117262*q^15+ 139094*q^16+163384*q^17+190295*q^18+219797*q^19+251628*q^20+285471*q^21+321001* q^22+357933*q^23+396119*q^24+435484*q^25+475962*q^26+517506*q^27+560050*q^28+ 603475*q^29+647639*q^30+692322*q^31+737213*q^32+781887*q^33+825981*q^34+869254* q^35+911573*q^36+952942*q^37+993380*q^38+1032874*q^39+1071335*q^40+1108642*q^41 +1144657*q^42+1179139*q^43+1211893*q^44+1242773*q^45+1271775*q^46+1298939*q^47+ 1324374*q^48+1348108*q^49+1370151*q^50+1390433*q^51+1408841*q^52+1425237*q^53+ 1439509*q^54+1451590*q^55+1461523*q^56+1469373*q^57+1475193*q^58+1478916*q^59+ 1480475*q^60+1479684*q^61+1476438*q^62+1470632*q^63+1462223*q^64+1451160*q^65+ 1437415*q^66+1420891*q^67+1401450*q^68+1378915*q^69+1353083*q^70+1323791*q^71+ 1291032*q^72+1254792*q^73+1215100*q^74+1171925*q^75+1125238*q^76+1075053*q^77+ 1021469*q^78+964684*q^79+904975*q^80+842689*q^81+778223*q^82+712056*q^83+644756 *q^84+577028*q^85+509663*q^86+443544*q^87+379566*q^88+318646*q^89+261693*q^90+ 209577*q^91+163032*q^92+122619*q^93+88656*q^94+61182*q^95+39957*q^96+24426*q^97 +13785*q^98+7055*q^99+3194*q^100+1237*q^101+390*q^102+92*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+ 10*q^7+11*q^6+11*q^5+12*q^4+10*q^3+7*q^2+4*q+1, q^15+5*q^14+11*q^13+18*q^12+23* q^11+26*q^10+30*q^9+32*q^8+32*q^7+31*q^6+29*q^5+25*q^4+18*q^3+11*q^2+5*q+1, q^ 21+6*q^20+16*q^19+30*q^18+44*q^17+54*q^16+65*q^15+76*q^14+84*q^13+89*q^12+93*q^ 11+94*q^10+94*q^9+90*q^8+84*q^7+75*q^6+62*q^5+47*q^4+30*q^3+16*q^2+6*q+1, 1+47* q^25+133*q^22+78*q^24+161*q^21+106*q^23+7*q+22*q^2+22*q^26+47*q^3+82*q^4+121*q^ 5+160*q^6+197*q^7+226*q^8+251*q^9+269*q^10+281*q^11+289*q^12+292*q^13+287*q^14+ 278*q^15+269*q^16+q^28+258*q^17+241*q^18+217*q^19+189*q^20+7*q^27, 1+628*q^25+ 788*q^22+695*q^24+828*q^21+745*q^23+8*q+262*q^30+329*q^29+29*q^34+70*q^33+29*q^ 2+551*q^26+70*q^3+135*q^4+220*q^5+317*q^6+130*q^32+421*q^7+521*q^8+613*q^9+695* q^10+764*q^11+821*q^12+866*q^13+897*q^14+913*q^15+920*q^16+398*q^28+922*q^17+ 914*q^18+893*q^19+865*q^20+473*q^27+196*q^31+8*q^35+q^36, 1+2806*q^25+100*q^42+ 2981*q^22+2885*q^24+3008*q^21+2941*q^23+9*q+9*q^44+2167*q^30+q^45+815*q^37+2320 *q^29+494*q^39+206*q^41+1390*q^34+652*q^38+1606*q^33+37*q^2+2704*q^26+100*q^3+ 212*q^4+378*q^5+591*q^6+1814*q^32+843*q^7+1116*q^8+1395*q^9+1668*q^10+1924*q^11 +2160*q^12+2369*q^13+2549*q^14+2694*q^15+2810*q^16+37*q^43+2456*q^28+2902*q^17+ 2970*q^18+3009*q^19+3021*q^20+2586*q^27+343*q^40+1998*q^31+1183*q^35+991*q^36, 1+10104*q^25+4071*q^42+9944*q^22+10102*q^24+9789*q^21+10048*q^23+10*q+313*q^51+ 2960*q^44+46*q^53+9390*q^30+138*q^52+q^55+10*q^54+2474*q^45+6786*q^37+9605*q^29 +2031*q^46+5780*q^39+4651*q^41+8120*q^34+572*q^50+6298*q^38+8513*q^33+46*q^2+ 10045*q^26+138*q^3+320*q^4+620*q^5+1047*q^6+8855*q^32+1599*q^7+2256*q^8+2987*q^ 9+3765*q^10+4554*q^11+5335*q^12+6086*q^13+6788*q^14+7426*q^15+7993*q^16+3497*q^ 43+891*q^49+9786*q^28+8492*q^17+8925*q^18+9288*q^19+9573*q^20+9937*q^27+5225*q^ 40+9142*q^31+7695*q^35+1247*q^48+7248*q^36+1625*q^47, 1+33383*q^25+25529*q^42+ 30977*q^22+32739*q^24+29851*q^21+31939*q^23+11*q+11830*q^51+22728*q^44+8785*q^ 53+4067*q^57+34330*q^30+10263*q^52+3144*q^58+6186*q^55+1541*q^60+7419*q^54+ 21259*q^45+30837*q^37+34394*q^29+19760*q^46+11*q^65+29031*q^39+915*q^61+26802*q ^41+32994*q^34+13453*q^50+56*q^64+29977*q^38+33517*q^33+5077*q^56+56*q^2+33863* q^26+185*q^3+467*q^4+978*q^5+1776*q^6+33906*q^32+2896*q^7+4341*q^8+6077*q^9+459 *q^62+8058*q^10+10216*q^11+12485*q^12+14807*q^13+17118*q^14+19364*q^15+21500*q^ 16+q^66+24158*q^43+15083*q^49+34347*q^28+23502*q^17+25352*q^18+27039*q^19+28540 *q^20+34180*q^27+27972*q^40+185*q^63+34166*q^31+32354*q^35+16680*q^48+2298*q^59 +31625*q^36+18236*q^47, 1+104480*q^25+109712*q^42+91796*q^22+100710*q^24+86650* q^21+96482*q^23+12*q+80111*q^51+104637*q^44+71182*q^53+52663*q^57+116484*q^30+ 75705*q^52+48035*q^58+61944*q^55+38838*q^60+66585*q^54+101754*q^45+4083*q^71+ 653*q^74+117585*q^37+12714*q^67+114936*q^29+98678*q^46+18610*q^65+7936*q^69+ 115264*q^39+34307*q^61+111847*q^41+119121*q^34+84321*q^50+22052*q^64+116554*q^ 38+118990*q^33+57305*q^56+67*q^2+10206*q^68+107788*q^26+242*q^3+662*q^4+1492*q^ 5+2902*q^6+118503*q^32+5039*q^7+8004*q^8+11824*q^9+29936*q^62+16471*q^10+21858* q^11+27859*q^12+67*q^76+34338*q^13+41138*q^14+48105*q^15+55087*q^16+15501*q^66+ 5892*q^70+12*q^77+q^78+107293*q^43+88275*q^49+112996*q^28+61964*q^17+68635*q^18 +75023*q^19+81048*q^20+110622*q^27+2565*q^72+113702*q^40+1412*q^73+25829*q^63+ 117660*q^31+242*q^75+118906*q^35+91963*q^48+43431*q^59+118376*q^36+95420*q^47, 1+313344*q^25+414517*q^42+260649*q^22+13*q^90+296910*q^24+240996*q^21+279317*q^ 23+13*q+365001*q^51+408333*q^44+346294*q^53+302422*q^57+376585*q^30+355976*q^52 +290270*q^58+325288*q^55+264421*q^60+q^91+336043*q^54+404095*q^45+111782*q^71+ 75529*q^74+415262*q^37+165468*q^67+366492*q^29+399186*q^46+193666*q^65+138104*q ^69+417434*q^39+250730*q^61+416340*q^41+405296*q^34+31902*q^79+373281*q^50+ 207972*q^64+416753*q^38+399822*q^33+314086*q^56+79*q^2+151662*q^68+328564*q^26+ 310*q^3+915*q^4+2211*q^5+4590*q^6+393211*q^32+8465*q^7+14216*q^8+22119*q^9+ 236654*q^62+32326*q^10+44852*q^11+59574*q^12+55368*q^76+76285*q^13+94685*q^14+ 114427*q^15+135131*q^16+179485*q^66+124798*q^70+46700*q^77+38897*q^78+411831*q^ 43+380785*q^49+355153*q^28+156420*q^17+177951*q^18+199412*q^19+220516*q^20+ 342508*q^27+99139*q^72+417311*q^40+86981*q^73+222346*q^63+385469*q^31+64970*q^ 75+409646*q^35+25650*q^80+387563*q^48+10693*q^83+20054*q^81+4125*q^85+2112*q^86 +905*q^87+310*q^88+79*q^89+7008*q^84+15064*q^82+277609*q^59+412928*q^36+393681* q^47, 1+904975*q^25+1470632*q^42+712056*q^22+117262*q^90+842689*q^24+644756*q^ 21+778223*q^23+14*q+18810*q^97+1439509*q^51+1479684*q^44+1408841*q^53+1324374*q ^57+1171925*q^30+50702*q^94+1425237*q^52+1298939*q^58+1370151*q^55+1242773*q^60 +97683*q^91+1390433*q^54+1480475*q^45+825981*q^71+692322*q^74+27797*q^96+11649* q^98+1401450*q^37+993380*q^67+1125238*q^29+1478916*q^46+64560*q^93+6433*q^99+ 1071335*q^65+911573*q^69+1437415*q^39+1211893*q^61+1462223*q^41+1323791*q^34+ 475962*q^79+1451590*q^50+1108642*q^64+3074*q^100+1420891*q^38+1291032*q^33+ 1348108*q^56+92*q^2+952942*q^68+964684*q^26+390*q^3+1237*q^4+3194*q^5+7055*q^6+ 1254792*q^32+13785*q^7+24426*q^8+39957*q^9+38457*q^95+1179139*q^62+1226*q^101+ 61182*q^10+88656*q^11+122619*q^12+603475*q^76+163032*q^13+209577*q^14+261693*q^ 15+318646*q^16+1032874*q^66+869254*q^70+560050*q^77+517506*q^78+1476438*q^43+ 390*q^102+1461523*q^49+1075053*q^28+379566*q^17+443544*q^18+509663*q^19+577028* q^20+1021469*q^27+781887*q^72+1451160*q^40+737213*q^73+92*q^103+1144657*q^63+ 1215100*q^31+647639*q^75+1353083*q^35+14*q^104+435484*q^80+1469373*q^48+q^105+ 321001*q^83+396119*q^81+251628*q^85+219797*q^86+190295*q^87+163384*q^88+139094* q^89+285471*q^84+357933*q^82+1271775*q^59+1378915*q^36+80168*q^92+1475193*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, 73816489] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2]}, is given by [1, 1, 3, 10, 39, 149, 579, 2269, 8969, 35624, 142097, 568534, 2280556, 9166656, 36908275] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2]}, is given by [0, 1, 3, 11, 39, 149, 578, 2270, 8967, 35627, 142091, 568542, 2280537, 9166681, 36908214] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 39, 149, 578, 2269, 8969, 35627, 142091, 568534, 2280556, 9166681, 36908214] ODD:, [0, 1, 3, 11, 39, 149, 579, 2270, 8967, 35624, 142097, 568542, 2280537, 9166656, 36908275] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.012820513, 7.553691275, 10.63785653, 14.27979731, 18.49280776, 23.28886612, 28.67868101, 34.67181086, 41.27680120, 48.50131599, 56.35225439] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.543033500, 2.284246999, 3.180076894, 4.226031798, 5.418113147, 6.752799800, 8.226815270, 9.837031126, 11.58044608, 13.45418978, 15.45552996, 17.58187599] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.38095, 5.21778, 10.1129, 17.8593, 29.3560, 45.6003, 67.6805, 96.7672, 134.107, 181.015, 238.873, 309.122] -5 Skewness: , [Float(undefined), 0., 0., -0.2721915177 10 , -0.01576515099, -0.03646475971, -0.05759078877, -0.07757729740, -0.09593748825, -0.1125548872, -0.1274858059, -0.1408455385, -0.1527723068, -0.1634009790, -0.1728600030] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.234375069, 2.248734141, 2.256138809, 2.268360475, 2.282568655, 2.297046761, 2.311113011, 2.324599892, 2.337455133, 2.349721941, 2.361378683, 2.372472950] end of this data For the equivalence class of patterns, { {[3, 2, 1, 4], [3, 1, 2, 4], [4, 3, 1, 2]}, {[1, 2, 4, 3], [2, 4, 3, 1], [2, 3, 4, 1]}, {[1, 2, 4, 3], [4, 1, 3, 2], [4, 1, 2, 3]}, {[2, 1, 3, 4], [4, 1, 2, 3], [4, 2, 1, 3]}, {[2, 3, 1, 4], [3, 4, 2, 1], [3, 2, 1, 4]}, {[2, 3, 4, 1], [2, 1, 3, 4], [3, 2, 4, 1]}, {[1, 4, 3, 2], [1, 4, 2, 3], [4, 3, 1, 2]}, {[1, 4, 3, 2], [1, 3, 4, 2], [3, 4, 2, 1]}} the member , {[1, 2, 4, 3], [4, 1, 3, 2], [4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 1], {[0, 2, 0]}, {}], [[2, 3, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {3}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [ [2, 4, 3, 1], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 1, 0, 1, 0]}, {4}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+2*q+5*q^2+8*q^3+ 13*q^4+14*q^5+12*q^6+9*q^7+7*q^8+4*q^9+q^10, 1+2*q+5*q^2+8*q^3+15*q^4+23*q^5+29 *q^6+34*q^7+36*q^8+35*q^9+28*q^10+21*q^11+16*q^12+11*q^13+5*q^14+q^15, 1+2*q+5* q^2+8*q^3+15*q^4+25*q^5+35*q^6+49*q^7+67*q^8+80*q^9+87*q^10+91*q^11+92*q^12+91* q^13+81*q^14+65*q^15+49*q^16+37*q^17+27*q^18+16*q^19+6*q^20+q^21, 1+2*q+5*q^2+8 *q^3+15*q^4+25*q^5+37*q^6+55*q^7+79*q^8+104*q^9+132*q^10+164*q^11+194*q^12+219* q^13+237*q^14+244*q^15+242*q^16+237*q^17+231*q^18+214*q^19+184*q^20+148*q^21+ 114*q^22+86*q^23+64*q^24+43*q^25+22*q^26+7*q^27+q^28, 1+2*q+5*q^2+8*q^3+15*q^4+ 25*q^5+37*q^6+57*q^7+85*q^8+116*q^9+154*q^10+202*q^11+260*q^12+321*q^13+386*q^ 14+451*q^15+510*q^16+559*q^17+604*q^18+642*q^19+661*q^20+658*q^21+639*q^22+611* q^23+584*q^24+547*q^25+488*q^26+412*q^27+334*q^28+262*q^29+200*q^30+150*q^31+ 107*q^32+65*q^33+29*q^34+8*q^35+q^36, 1+2*q+5*q^2+8*q^3+15*q^4+25*q^5+37*q^6+57 *q^7+87*q^8+122*q^9+166*q^10+224*q^11+296*q^12+383*q^13+483*q^14+593*q^15+718*q ^16+852*q^17+995*q^18+1143*q^19+1284*q^20+1411*q^21+1516*q^22+1601*q^23+1678*q^ 24+1745*q^25+1781*q^26+1779*q^27+1737*q^28+1662*q^29+1570*q^30+1478*q^31+1379*q ^32+1253*q^33+1092*q^34+916*q^35+748*q^36+596*q^37+462*q^38+350*q^39+257*q^40+ 172*q^41+94*q^42+37*q^43+9*q^44+q^45, 1+2*q+5*q^2+8*q^3+15*q^4+25*q^5+37*q^6+57 *q^7+87*q^8+124*q^9+172*q^10+236*q^11+318*q^12+419*q^13+543*q^14+686*q^15+858*q ^16+1060*q^17+1289*q^18+1542*q^19+1818*q^20+2115*q^21+2424*q^22+2744*q^23+3070* q^24+3384*q^25+3684*q^26+3965*q^27+4212*q^28+4407*q^29+4550*q^30+4663*q^31+4751 *q^32+4801*q^33+4782*q^34+4677*q^35+4499*q^36+4270*q^37+4004*q^38+3731*q^39+ 3460*q^40+3163*q^41+2811*q^42+2418*q^43+2026*q^44+1666*q^45+1344*q^46+1058*q^47 +812*q^48+607*q^49+429*q^50+266*q^51+131*q^52+46*q^53+10*q^54+q^55, 1+2*q+5*q^2 +8*q^3+15*q^4+25*q^5+37*q^6+57*q^7+87*q^8+124*q^9+174*q^10+242*q^11+330*q^12+ 441*q^13+579*q^14+746*q^15+949*q^16+1196*q^17+1495*q^18+1838*q^19+2225*q^20+ 2664*q^21+3161*q^22+3713*q^23+4318*q^24+4967*q^25+5650*q^26+6373*q^27+7124*q^28 +7871*q^29+8599*q^30+9307*q^31+9989*q^32+10644*q^33+11260*q^34+11807*q^35+12250 *q^36+12571*q^37+12773*q^38+12890*q^39+12950*q^40+12935*q^41+12802*q^42+12512*q ^43+12069*q^44+11501*q^45+10839*q^46+10116*q^47+9376*q^48+8646*q^49+7909*q^50+ 7108*q^51+6226*q^52+5320*q^53+4464*q^54+3694*q^55+3010*q^56+2402*q^57+1870*q^58 +1419*q^59+1036*q^60+695*q^61+397*q^62+177*q^63+56*q^64+11*q^65+q^66, 1+2*q+5*q ^2+8*q^3+15*q^4+25*q^5+37*q^6+57*q^7+87*q^8+124*q^9+174*q^10+244*q^11+336*q^12+ 453*q^13+601*q^14+782*q^15+1009*q^16+1287*q^17+1629*q^18+2040*q^19+2519*q^20+ 3073*q^21+3720*q^22+4469*q^23+5325*q^24+6283*q^25+7341*q^26+8516*q^27+9811*q^28 +11202*q^29+12685*q^30+14251*q^31+15894*q^32+17611*q^33+19382*q^34+21171*q^35+ 22949*q^36+24677*q^37+26329*q^38+27907*q^39+29406*q^40+30807*q^41+32094*q^42+ 33246*q^43+34202*q^44+34896*q^45+35300*q^46+35441*q^47+35374*q^48+35143*q^49+ 34762*q^50+34180*q^51+33319*q^52+32143*q^53+30690*q^54+29034*q^55+27237*q^56+ 25337*q^57+23402*q^58+21505*q^59+19651*q^60+17758*q^61+15754*q^62+13677*q^63+ 11657*q^64+9806*q^65+8160*q^66+6704*q^67+5412*q^68+4272*q^69+3289*q^70+2455*q^ 71+1731*q^72+1092*q^73+574*q^74+233*q^75+67*q^76+12*q^77+q^78, 1+2*q+5*q^2+8*q^ 3+15*q^4+25*q^5+37*q^6+57*q^7+87*q^8+124*q^9+174*q^10+244*q^11+338*q^12+459*q^ 13+613*q^14+804*q^15+1045*q^16+1347*q^17+1720*q^18+2174*q^19+2719*q^20+3363*q^ 21+4127*q^22+5030*q^23+6091*q^24+7311*q^25+8699*q^26+10277*q^27+12062*q^28+ 14069*q^29+16300*q^30+18753*q^31+21435*q^32+24356*q^33+27515*q^34+30900*q^35+ 34495*q^36+38259*q^37+42168*q^38+46228*q^39+50416*q^40+54673*q^41+58954*q^42+ 63240*q^43+67483*q^44+71625*q^45+75603*q^46+79343*q^47+82806*q^48+85989*q^49+ 88888*q^50+91502*q^51+93797*q^52+95667*q^53+96976*q^54+97655*q^55+97722*q^56+ 97216*q^57+96205*q^58+94790*q^59+93041*q^60+90922*q^61+88304*q^62+85099*q^63+ 81328*q^64+77101*q^65+72552*q^66+67797*q^67+62921*q^68+58016*q^69+53199*q^70+ 48549*q^71+43988*q^72+39349*q^73+34592*q^74+29885*q^75+25467*q^76+21487*q^77+ 17968*q^78+14864*q^79+12116*q^80+9684*q^81+7561*q^82+5744*q^83+4186*q^84+2823*q ^85+1666*q^86+807*q^87+300*q^88+79*q^89+13*q^90+q^91, 1+2*q+5*q^2+8*q^3+15*q^4+ 25*q^5+37*q^6+57*q^7+87*q^8+124*q^9+174*q^10+244*q^11+338*q^12+461*q^13+619*q^ 14+816*q^15+1067*q^16+1383*q^17+1780*q^18+2265*q^19+2853*q^20+3563*q^21+4415*q^ 22+5433*q^23+6650*q^24+8079*q^25+9737*q^26+11656*q^27+13867*q^28+16394*q^29+ 19277*q^30+22546*q^31+26217*q^32+30311*q^33+34856*q^34+39874*q^35+45394*q^36+ 51406*q^37+57901*q^38+64901*q^39+72428*q^40+80449*q^41+88927*q^42+97846*q^43+ 107178*q^44+116885*q^45+126896*q^46+137122*q^47+147487*q^48+157931*q^49+168401* q^50+178835*q^51+189174*q^52+199336*q^53+209200*q^54+218632*q^55+227485*q^56+ 235604*q^57+242895*q^58+249390*q^59+255142*q^60+260130*q^61+264265*q^62+267402* q^63+269359*q^64+269986*q^65+269227*q^66+267117*q^67+263748*q^68+259251*q^69+ 253808*q^70+247608*q^71+240701*q^72+232938*q^73+224108*q^74+214153*q^75+203236* q^76+191644*q^77+179633*q^78+167388*q^79+155057*q^80+142808*q^81+130839*q^82+ 119317*q^83+108222*q^84+97309*q^85+86357*q^86+75475*q^87+65064*q^88+55509*q^89+ 46980*q^90+39457*q^91+32832*q^92+26980*q^93+21800*q^94+17245*q^95+13305*q^96+ 9930*q^97+7009*q^98+4489*q^99+2473*q^100+1107*q^101+379*q^102+92*q^103+14*q^104 +q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+5*q^4+5*q^3+4*q^2+3*q+1, q^10+2*q^9+5*q^8+8 *q^7+13*q^6+14*q^5+12*q^4+9*q^3+7*q^2+4*q+1, q^15+2*q^14+5*q^13+8*q^12+15*q^11+ 23*q^10+29*q^9+34*q^8+36*q^7+35*q^6+28*q^5+21*q^4+16*q^3+11*q^2+5*q+1, q^21+2*q ^20+5*q^19+8*q^18+15*q^17+25*q^16+35*q^15+49*q^14+67*q^13+80*q^12+87*q^11+91*q^ 10+92*q^9+91*q^8+81*q^7+65*q^6+49*q^5+37*q^4+27*q^3+16*q^2+6*q+1, 1+8*q^25+37*q ^22+15*q^24+55*q^21+25*q^23+7*q+22*q^2+5*q^26+43*q^3+64*q^4+86*q^5+114*q^6+148* q^7+184*q^8+214*q^9+231*q^10+237*q^11+242*q^12+244*q^13+237*q^14+219*q^15+194*q ^16+q^28+164*q^17+132*q^18+104*q^19+79*q^20+2*q^27, 1+202*q^25+386*q^22+260*q^ 24+451*q^21+321*q^23+8*q+37*q^30+57*q^29+5*q^34+8*q^33+29*q^2+154*q^26+65*q^3+ 107*q^4+150*q^5+200*q^6+15*q^32+262*q^7+334*q^8+412*q^9+488*q^10+547*q^11+584*q ^12+611*q^13+639*q^14+658*q^15+661*q^16+85*q^28+642*q^17+604*q^18+559*q^19+510* q^20+116*q^27+25*q^31+2*q^35+q^36, 1+1284*q^25+8*q^42+1601*q^22+1411*q^24+1678* q^21+1516*q^23+9*q+2*q^44+593*q^30+q^45+87*q^37+718*q^29+37*q^39+15*q^41+224*q^ 34+57*q^38+296*q^33+37*q^2+1143*q^26+94*q^3+172*q^4+257*q^5+350*q^6+383*q^32+ 462*q^7+596*q^8+748*q^9+916*q^10+1092*q^11+1253*q^12+1379*q^13+1478*q^14+1570*q ^15+1662*q^16+5*q^43+852*q^28+1737*q^17+1779*q^18+1781*q^19+1745*q^20+995*q^27+ 25*q^40+483*q^31+166*q^35+122*q^36, 1+4550*q^25+419*q^42+4801*q^22+4663*q^24+ 4782*q^21+4751*q^23+10*q+15*q^51+236*q^44+5*q^53+3384*q^30+8*q^52+q^55+2*q^54+ 172*q^45+1289*q^37+3684*q^29+124*q^46+858*q^39+543*q^41+2115*q^34+25*q^50+1060* q^38+2424*q^33+46*q^2+4407*q^26+131*q^3+266*q^4+429*q^5+607*q^6+2744*q^32+812*q ^7+1058*q^8+1344*q^9+1666*q^10+2026*q^11+2418*q^12+2811*q^13+3163*q^14+3460*q^ 15+3731*q^16+318*q^43+37*q^49+3965*q^28+4004*q^17+4270*q^18+4499*q^19+4677*q^20 +4212*q^27+686*q^40+3070*q^31+1818*q^35+57*q^48+1542*q^36+87*q^47, 1+12935*q^25 +4318*q^42+12069*q^22+12802*q^24+11501*q^21+12512*q^23+11*q+746*q^51+3161*q^44+ 441*q^53+124*q^57+12250*q^30+579*q^52+87*q^58+242*q^55+37*q^60+330*q^54+2664*q^ 45+7871*q^37+12571*q^29+2225*q^46+2*q^65+6373*q^39+25*q^61+4967*q^41+9989*q^34+ 949*q^50+5*q^64+7124*q^38+10644*q^33+174*q^56+56*q^2+12950*q^26+177*q^3+397*q^4 +695*q^5+1036*q^6+11260*q^32+1419*q^7+1870*q^8+2402*q^9+15*q^62+3010*q^10+3694* q^11+4464*q^12+5320*q^13+6226*q^14+7108*q^15+7909*q^16+q^66+3713*q^43+1196*q^49 +12773*q^28+8646*q^17+9376*q^18+10116*q^19+10839*q^20+12890*q^27+5650*q^40+8*q^ 63+11807*q^31+9307*q^35+1495*q^48+57*q^59+8599*q^36+1838*q^47, 1+32143*q^25+ 22949*q^42+27237*q^22+30690*q^24+25337*q^21+29034*q^23+12*q+8516*q^51+19382*q^ 44+6283*q^53+3073*q^57+35374*q^30+7341*q^52+2519*q^58+4469*q^55+1629*q^60+5325* q^54+17611*q^45+57*q^71+15*q^74+30807*q^37+244*q^67+35143*q^29+15894*q^46+453*q ^65+124*q^69+27907*q^39+1287*q^61+24677*q^41+34202*q^34+9811*q^50+601*q^64+ 29406*q^38+34896*q^33+3720*q^56+67*q^2+174*q^68+33319*q^26+233*q^3+574*q^4+1092 *q^5+1731*q^6+35300*q^32+2455*q^7+3289*q^8+4272*q^9+1009*q^62+5412*q^10+6704*q^ 11+8160*q^12+5*q^76+9806*q^13+11657*q^14+13677*q^15+15754*q^16+336*q^66+87*q^70 +2*q^77+q^78+21171*q^43+11202*q^49+34762*q^28+17758*q^17+19651*q^18+21505*q^19+ 23402*q^20+34180*q^27+37*q^72+26329*q^40+25*q^73+782*q^63+35441*q^31+8*q^75+ 33246*q^35+12685*q^48+2040*q^59+32094*q^36+14251*q^47, 1+72552*q^25+85989*q^42+ 58016*q^22+2*q^90+67797*q^24+53199*q^21+62921*q^23+13*q+50416*q^51+79343*q^44+ 42168*q^53+27515*q^57+90922*q^30+46228*q^52+24356*q^58+34495*q^55+18753*q^60+q^ 91+38259*q^54+75603*q^45+2719*q^71+1347*q^74+96976*q^37+6091*q^67+88304*q^29+ 71625*q^46+8699*q^65+4127*q^69+93797*q^39+16300*q^61+88888*q^41+97216*q^34+338* q^79+54673*q^50+10277*q^64+95667*q^38+96205*q^33+30900*q^56+79*q^2+5030*q^68+ 77101*q^26+300*q^3+807*q^4+1666*q^5+2823*q^6+94790*q^32+4186*q^7+5744*q^8+7561* q^9+14069*q^62+9684*q^10+12116*q^11+14864*q^12+804*q^76+17968*q^13+21487*q^14+ 25467*q^15+29885*q^16+7311*q^66+3363*q^70+613*q^77+459*q^78+82806*q^43+58954*q^ 49+85099*q^28+34592*q^17+39349*q^18+43988*q^19+48549*q^20+81328*q^27+2174*q^72+ 91502*q^40+1720*q^73+12062*q^63+93041*q^31+1045*q^75+97722*q^35+244*q^80+63240* q^48+87*q^83+174*q^81+37*q^85+25*q^86+15*q^87+8*q^88+5*q^89+57*q^84+124*q^82+ 21435*q^59+97655*q^36+67483*q^47, 1+155057*q^25+267402*q^42+119317*q^22+816*q^ 90+142808*q^24+108222*q^21+130839*q^23+14*q+87*q^97+209200*q^51+260130*q^44+ 189174*q^53+147487*q^57+214153*q^30+244*q^94+199336*q^52+137122*q^58+168401*q^ 55+116885*q^60+619*q^91+178835*q^54+255142*q^45+34856*q^71+22546*q^74+124*q^96+ 57*q^98+263748*q^37+57901*q^67+203236*q^29+249390*q^46+338*q^93+37*q^99+72428*q ^65+45394*q^69+269227*q^39+107178*q^61+269359*q^41+247608*q^34+9737*q^79+218632 *q^50+80449*q^64+25*q^100+267117*q^38+240701*q^33+157931*q^56+92*q^2+51406*q^68 +167388*q^26+379*q^3+1107*q^4+2473*q^5+4489*q^6+232938*q^32+7009*q^7+9930*q^8+ 13305*q^9+174*q^95+97846*q^62+15*q^101+17245*q^10+21800*q^11+26980*q^12+16394*q ^76+32832*q^13+39457*q^14+46980*q^15+55509*q^16+64901*q^66+39874*q^70+13867*q^ 77+11656*q^78+264265*q^43+8*q^102+227485*q^49+191644*q^28+65064*q^17+75475*q^18 +86357*q^19+97309*q^20+179633*q^27+30311*q^72+269986*q^40+26217*q^73+5*q^103+ 88927*q^63+224108*q^31+19277*q^75+253808*q^35+2*q^104+8079*q^80+235604*q^48+q^ 105+4415*q^83+6650*q^81+2853*q^85+2265*q^86+1780*q^87+1383*q^88+1067*q^89+3563* q^84+5433*q^82+126896*q^59+259251*q^36+461*q^92+242895*q^47] The number of permutations avoiding, {[1, 2, 4, 3], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 2, 6, 21, 76, 270, 930, 3114, 10196, 32820, 104283, 328048, 1023854, 3175395, 9797833] The number of EVEN permutations avoiding, {[1, 2, 4, 3], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 1, 3, 11, 39, 135, 465, 1558, 5100, 16410, 52142, 164026, 511931, 1587698, 4898919] The number of ODD permutations avoiding, {[1, 2, 4, 3], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [0, 1, 3, 10, 37, 135, 465, 1556, 5096, 16410, 52141, 164022, 511923, 1587697, 4898914] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 39, 135, 465, 1558, 5100, 16410, 52141, 164026, 511931, 1587697, 4898914] ODD:, [0, 1, 3, 10, 37, 135, 465, 1556, 5096, 16410, 52142, 164022, 511923, 1587698, 4898919] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.197368421, 7.988888889, 11.44193548, 15.56583173, 20.36543743, 25.84351005, 32.00162059, 38.84056601, 46.36059536, 54.56155754, 63.44301071] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.495268426, 2.127636054, 2.877091295, 3.749604210, 4.737184129, 5.827371293, 7.008278966, 8.270095043, 9.605092480, 11.00722426, 12.47170589, 13.99468907] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.23583, 4.52684, 8.27765, 14.0595, 22.4409, 33.9583, 49.1160, 68.3945, 92.2578, 121.159, 155.543, 195.851] Skewness: , [Float(undefined), 0., 0., 0.004136795107, -0.02088984701, -0.06259356142, -0.1013477480, -0.1309118561, -0.1520457935, -0.1669345581, -0.1774095329, -0.1847508666, -0.1897895744, -0.1930888892, -0.1950354045] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.384367494, 2.559729762, 2.626756289, 2.645455640, 2.653864063, 2.664568353, 2.679281912, 2.696733384, 2.715396286, 2.734186521, 2.752415168, 2.769619597] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [2, 4, 3, 1], [2, 3, 1, 4]}, {[2, 3, 1, 4], [4, 3, 2, 1], [4, 2, 1, 3]}, {[1, 2, 3, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [1, 2, 3, 4], [3, 2, 4, 1]}, {[1, 4, 2, 3], [2, 4, 3, 1], [4, 3, 2, 1]}, {[3, 2, 4, 1], [3, 1, 2, 4], [4, 3, 2, 1]}, {[1, 3, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, {[1, 4, 2, 3], [1, 2, 3, 4], [4, 2, 1, 3]}} the member , {[1, 2, 3, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 4, 3, 1], { [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0], [0, 3, 0, 0, 0]}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[3, 2, 1], {[0, 3, 0, 0], [0, 0, 3, 0], [0, 1, 1, 0]}, {2}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[2, 3, 1], {[0, 3, 0, 0], [0, 0, 3, 0], [0, 1, 1, 0]}, {}], [[2, 1, 3], {[0, 3, 0, 0]}, {}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [ [2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 1, 1, 0, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0]}, {}], [[2, 1, 5, 4, 3], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0]}, {4}], [[2, 1, 4, 5, 3], { [0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1], [0, 3, 0, 0, 0]}, {}], [[3, 4, 2, 1], { [0, 0, 1, 1, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0], [0, 0, 0, 3, 0]}, {3}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {2}], [[2, 1], {[0, 3, 0]}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 2, 0, 0]}, {2}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 1, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0], [0, 0, 0, 3, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [ [2, 3, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 2, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [ [3, 2, 5, 4, 1], {[0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 2, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 5, 4], { [0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[2, 3, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 4, 1, 5, 2], %1, {1}], [ [3, 4, 2, 5, 1], {[0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 2, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [2, 4, 1, 5, 3], {[0, 0, 0, 2, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [ [2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [ [2, 1, 3, 5, 4], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[3, 1, 4, 5, 2], %1, {1}], [[3, 2, 4, 5, 1], {[0, 0, 3, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 0, 0, 3, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 2, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 3, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [ [2, 1, 5, 3, 4], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 1, 5, 4, 2], %1, {1}]} %1 := {[0, 0, 2, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 3*q^2+8*q^3+12*q^4+ 15*q^5+13*q^6+11*q^7+7*q^8+4*q^9+q^10, q^3+8*q^4+12*q^5+24*q^6+33*q^7+38*q^8+36 *q^9+33*q^10+26*q^11+19*q^12+11*q^13+5*q^14+q^15, 3*q^5+8*q^6+19*q^7+30*q^8+47* q^9+65*q^10+77*q^11+92*q^12+94*q^13+89*q^14+79*q^15+64*q^16+48*q^17+31*q^18+16* q^19+6*q^20+q^21, 3*q^7+7*q^8+19*q^9+28*q^10+53*q^11+77*q^12+104*q^13+125*q^14+ 151*q^15+181*q^16+207*q^17+223*q^18+228*q^19+215*q^20+188*q^21+157*q^22+120*q^ 23+83*q^24+48*q^25+22*q^26+7*q^27+q^28, q^9+7*q^10+10*q^11+23*q^12+43*q^13+68*q ^14+100*q^15+140*q^16+178*q^17+215*q^18+267*q^19+303*q^20+347*q^21+394*q^22+454 *q^23+509*q^24+540*q^25+541*q^26+507*q^27+452*q^28+379*q^29+300*q^30+215*q^31+ 136*q^32+71*q^33+29*q^34+8*q^35+q^36, 3*q^12+6*q^13+15*q^14+22*q^15+44*q^16+69* q^17+118*q^18+161*q^19+212*q^20+265*q^21+325*q^22+385*q^23+469*q^24+546*q^25+ 618*q^26+688*q^27+767*q^28+862*q^29+983*q^30+1114*q^31+1223*q^32+1283*q^33+1265 *q^34+1195*q^35+1074*q^36+918*q^37+739*q^38+551*q^39+368*q^40+213*q^41+101*q^42 +37*q^43+9*q^44+q^45, 3*q^15+6*q^16+13*q^17+20*q^18+36*q^19+58*q^20+102*q^21+ 152*q^22+215*q^23+288*q^24+354*q^25+446*q^26+530*q^27+636*q^28+720*q^29+839*q^ 30+958*q^31+1113*q^32+1256*q^33+1410*q^34+1532*q^35+1676*q^36+1867*q^37+2118*q^ 38+2405*q^39+2685*q^40+2898*q^41+2988*q^42+2956*q^43+2800*q^44+2546*q^45+2209*q ^46+1813*q^47+1390*q^48+973*q^49+604*q^50+321*q^51+139*q^52+46*q^53+10*q^54+q^ 55, q^18+7*q^19+7*q^20+16*q^21+28*q^22+43*q^23+62*q^24+113*q^25+168*q^26+252*q^ 27+335*q^28+431*q^29+532*q^30+655*q^31+778*q^32+925*q^33+1080*q^34+1246*q^35+ 1383*q^36+1571*q^37+1753*q^38+1989*q^39+2251*q^40+2555*q^41+2846*q^42+3123*q^43 +3384*q^44+3669*q^45+4046*q^46+4534*q^47+5146*q^48+5787*q^49+6366*q^50+6764*q^ 51+6939*q^52+6866*q^53+6551*q^54+6017*q^55+5290*q^56+4422*q^57+3472*q^58+2522*q ^59+1655*q^60+955*q^61+468*q^62+186*q^63+56*q^64+11*q^65+q^66, 3*q^22+6*q^23+12 *q^24+15*q^25+27*q^26+43*q^27+68*q^28+98*q^29+156*q^30+233*q^31+331*q^32+455*q^ 33+578*q^34+717*q^35+875*q^36+1073*q^37+1239*q^38+1456*q^39+1630*q^40+1891*q^41 +2146*q^42+2462*q^43+2732*q^44+3048*q^45+3316*q^46+3660*q^47+4057*q^48+4603*q^ 49+5171*q^50+5797*q^51+6350*q^52+6891*q^53+7415*q^54+8011*q^55+8764*q^56+9733*q ^57+10947*q^58+12333*q^59+13727*q^60+14893*q^61+15696*q^62+16051*q^63+15915*q^ 64+15295*q^65+14179*q^66+12625*q^67+10720*q^68+8604*q^69+6440*q^70+4420*q^71+ 2719*q^72+1461*q^73+663*q^74+243*q^75+67*q^76+12*q^77+q^78, 3*q^26+6*q^27+12*q^ 28+13*q^29+26*q^30+36*q^31+58*q^32+83*q^33+126*q^34+182*q^35+272*q^36+375*q^37+ 519*q^38+688*q^39+877*q^40+1069*q^41+1297*q^42+1536*q^43+1823*q^44+2110*q^45+ 2425*q^46+2711*q^47+3051*q^48+3409*q^49+3857*q^50+4312*q^51+4861*q^52+5371*q^53 +5945*q^54+6456*q^55+7035*q^56+7634*q^57+8411*q^58+9352*q^59+10489*q^60+11741*q ^61+12974*q^62+14096*q^63+15129*q^64+16210*q^65+17437*q^66+18982*q^67+20893*q^ 68+23305*q^69+26154*q^70+29245*q^71+32177*q^72+34600*q^73+36263*q^74+37035*q^75 +36860*q^76+35634*q^77+33340*q^78+30021*q^79+25868*q^80+21160*q^81+16260*q^82+ 11573*q^83+7498*q^84+4328*q^85+2171*q^86+916*q^87+311*q^88+79*q^89+13*q^90+q^91 , q^30+7*q^31+7*q^32+13*q^33+22*q^34+28*q^35+40*q^36+67*q^37+91*q^38+137*q^39+ 193*q^40+268*q^41+366*q^42+513*q^43+700*q^44+909*q^45+1177*q^46+1432*q^47+1755* q^48+2081*q^49+2461*q^50+2800*q^51+3231*q^52+3641*q^53+4174*q^54+4677*q^55+5293 *q^56+5826*q^57+6459*q^58+7030*q^59+7758*q^60+8551*q^61+9606*q^62+10662*q^63+ 11800*q^64+12762*q^65+13758*q^66+14727*q^67+15959*q^68+17373*q^69+19227*q^70+ 21388*q^71+23850*q^72+26360*q^73+28781*q^74+31016*q^75+33147*q^76+35385*q^77+ 37942*q^78+41066*q^79+44900*q^80+49666*q^81+55426*q^82+61953*q^83+68678*q^84+ 74882*q^85+79968*q^86+83528*q^87+85378*q^88+85256*q^89+82925*q^90+78210*q^91+ 71174*q^92+62147*q^93+51716*q^94+40655*q^95+29847*q^96+20174*q^97+12341*q^98+ 6695*q^99+3144*q^100+1238*q^101+391*q^102+92*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+4*q^4+6*q^3+4*q^2+3*q+1, 3*q^8+8*q^7+12*q^6+15* q^5+13*q^4+11*q^3+7*q^2+4*q+1, q^12+8*q^11+12*q^10+24*q^9+33*q^8+38*q^7+36*q^6+ 33*q^5+26*q^4+19*q^3+11*q^2+5*q+1, 3*q^16+8*q^15+19*q^14+30*q^13+47*q^12+65*q^ 11+77*q^10+92*q^9+94*q^8+89*q^7+79*q^6+64*q^5+48*q^4+31*q^3+16*q^2+6*q+1, 3*q^ 21+7*q^20+19*q^19+28*q^18+53*q^17+77*q^16+104*q^15+125*q^14+151*q^13+181*q^12+ 207*q^11+223*q^10+228*q^9+215*q^8+188*q^7+157*q^6+120*q^5+83*q^4+48*q^3+22*q^2+ 7*q+1, 1+10*q^25+68*q^22+23*q^24+100*q^21+43*q^23+8*q+29*q^2+7*q^26+71*q^3+136* q^4+215*q^5+300*q^6+379*q^7+452*q^8+507*q^9+541*q^10+540*q^11+509*q^12+454*q^13 +394*q^14+347*q^15+303*q^16+267*q^17+215*q^18+178*q^19+140*q^20+q^27, 1+212*q^ 25+385*q^22+265*q^24+469*q^21+325*q^23+9*q+22*q^30+44*q^29+3*q^33+37*q^2+161*q^ 26+101*q^3+213*q^4+368*q^5+551*q^6+6*q^32+739*q^7+918*q^8+1074*q^9+1195*q^10+ 1265*q^11+1283*q^12+1223*q^13+1114*q^14+983*q^15+862*q^16+69*q^28+767*q^17+688* q^18+618*q^19+546*q^20+118*q^27+15*q^31, 1+839*q^25+1256*q^22+958*q^24+1410*q^ 21+1113*q^23+10*q+354*q^30+20*q^37+446*q^29+6*q^39+102*q^34+13*q^38+152*q^33+46 *q^2+720*q^26+139*q^3+321*q^4+604*q^5+973*q^6+215*q^32+1390*q^7+1813*q^8+2209*q ^9+2546*q^10+2800*q^11+2956*q^12+2988*q^13+2898*q^14+2685*q^15+2405*q^16+530*q^ 28+2118*q^17+1867*q^18+1676*q^19+1532*q^20+636*q^27+3*q^40+288*q^31+58*q^35+36* q^36, 1+2555*q^25+62*q^42+3384*q^22+2846*q^24+3669*q^21+3123*q^23+11*q+28*q^44+ 1383*q^30+16*q^45+431*q^37+1571*q^29+7*q^46+252*q^39+113*q^41+778*q^34+335*q^38 +925*q^33+56*q^2+2251*q^26+186*q^3+468*q^4+955*q^5+1655*q^6+1080*q^32+2522*q^7+ 3472*q^8+4422*q^9+5290*q^10+6017*q^11+6551*q^12+6866*q^13+6939*q^14+6764*q^15+ 6366*q^16+43*q^43+1753*q^28+5787*q^17+5146*q^18+4534*q^19+4046*q^20+1989*q^27+ 168*q^40+1246*q^31+655*q^35+q^48+532*q^36+7*q^47, 1+6891*q^25+875*q^42+8764*q^ 22+7415*q^24+9733*q^21+8011*q^23+12*q+43*q^51+578*q^44+15*q^53+4057*q^30+27*q^ 52+6*q^55+12*q^54+455*q^45+1891*q^37+4603*q^29+331*q^46+1456*q^39+1073*q^41+ 2732*q^34+68*q^50+1630*q^38+3048*q^33+3*q^56+67*q^2+6350*q^26+243*q^3+663*q^4+ 1461*q^5+2719*q^6+3316*q^32+4420*q^7+6440*q^8+8604*q^9+10720*q^10+12625*q^11+ 14179*q^12+15295*q^13+15915*q^14+16051*q^15+15696*q^16+717*q^43+98*q^49+5171*q^ 28+14893*q^17+13727*q^18+12333*q^19+10947*q^20+5797*q^27+1239*q^40+3660*q^31+ 2462*q^35+156*q^48+2146*q^36+233*q^47, 1+17437*q^25+3409*q^42+23305*q^22+18982* q^24+26154*q^21+20893*q^23+13*q+877*q^51+2711*q^44+519*q^53+126*q^57+11741*q^30 +688*q^52+83*q^58+272*q^55+36*q^60+375*q^54+2425*q^45+5945*q^37+12974*q^29+2110 *q^46+3*q^65+4861*q^39+26*q^61+3857*q^41+7634*q^34+1069*q^50+6*q^64+5371*q^38+ 8411*q^33+182*q^56+79*q^2+16210*q^26+311*q^3+916*q^4+2171*q^5+4328*q^6+9352*q^ 32+7498*q^7+11573*q^8+16260*q^9+13*q^62+21160*q^10+25868*q^11+30021*q^12+33340* q^13+35634*q^14+36860*q^15+37035*q^16+3051*q^43+1297*q^49+14096*q^28+36263*q^17 +34600*q^18+32177*q^19+29245*q^20+15129*q^27+4312*q^40+12*q^63+10489*q^31+7035* q^35+1536*q^48+58*q^59+6456*q^36+1823*q^47, 1+44900*q^25+10662*q^42+61953*q^22+ 49666*q^24+68678*q^21+55426*q^23+14*q+4174*q^51+8551*q^44+3231*q^53+1755*q^57+ 31016*q^30+3641*q^52+1432*q^58+2461*q^55+909*q^60+2800*q^54+7758*q^45+22*q^71+7 *q^74+15959*q^37+91*q^67+33147*q^29+7030*q^46+193*q^65+40*q^69+13758*q^39+700*q ^61+11800*q^41+21388*q^34+4677*q^50+268*q^64+14727*q^38+23850*q^33+2081*q^56+92 *q^2+67*q^68+41066*q^26+391*q^3+1238*q^4+3144*q^5+6695*q^6+26360*q^32+12341*q^7 +20174*q^8+29847*q^9+513*q^62+40655*q^10+51716*q^11+62147*q^12+71174*q^13+78210 *q^14+82925*q^15+85256*q^16+137*q^66+28*q^70+9606*q^43+5293*q^49+35385*q^28+ 85378*q^17+83528*q^18+79968*q^19+74882*q^20+37942*q^27+13*q^72+12762*q^40+7*q^ 73+366*q^63+28781*q^31+q^75+19227*q^35+5826*q^48+1177*q^59+17373*q^36+6459*q^47 ] The number of permutations avoiding, {[1, 2, 3, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 74, 247, 769, 2247, 6238, 16649, 43132, 109257, 272073, 668704, 1626916] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 36, 127, 385, 1119, 3118, 8333, 21587, 54598, 135944, 334435, 813700] The number of ODD permutations avoiding, {[1, 2, 3, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 38, 120, 384, 1128, 3120, 8316, 21545, 54659, 136129, 334269, 813216] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 36, 120, 384, 1119, 3118, 8316, 21545, 54598, 135944, 334269, 813216] ODD:, [0, 1, 3, 12, 38, 127, 385, 1128, 3120, 8333, 21587, 54659, 136129, 334435, 813700] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.142857143, 5.540540541, 8.785425101, 12.94668401, 18.07120605, 24.19397243, 31.34182233, 39.53410461, 48.78328162, 59.09608451, 70.47494706, 82.91938121] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.389954240, 1.868582272, 2.420917224, 3.062167748, 3.794252735, 4.606011334, 5.478769869, 6.391430639, 7.323889043, 8.258924948, 9.182920618, 10.08579810] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.93197, 3.49160, 5.86084, 9.37687, 14.3964, 21.2153, 30.0169, 40.8504, 53.6394, 68.2098, 84.3260, 101.723] Skewness: , [Float(undefined), 0., 0., 0.1693635266, 0.1725783194, 0.08540654430, -0.04662607534, -0.1939287574, -0.3416081148, -0.4840351031, -0.6195849084, -0.7479867699, -0.8694047638, -0.9841329100, -1.092561465] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.227081582, 2.381597542, 2.470395045, 2.522914492, 2.579718517, 2.670116627, 2.807025416, 2.992392209, 3.222663559, 3.492150832, 3.794610754, 4.124321482] end of this data For the equivalence class of patterns, { {[3, 2, 1, 4], [3, 1, 2, 4], [4, 3, 2, 1]}, {[1, 2, 3, 4], [2, 4, 3, 1], [2, 3, 4, 1]}, {[1, 2, 3, 4], [2, 3, 4, 1], [3, 2, 4, 1]}, {[1, 2, 3, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, {[2, 3, 1, 4], [3, 2, 1, 4], [4, 3, 2, 1]}, {[1, 2, 3, 4], [4, 1, 2, 3], [4, 2, 1, 3]}, {[1, 4, 3, 2], [1, 3, 4, 2], [4, 3, 2, 1]}, {[1, 4, 3, 2], [1, 4, 2, 3], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}], [[2, 1], {[0, 2, 0]}, {}], [[2, 3, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {3}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [ [2, 4, 3, 1], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 1, 0, 1, 0]}, {4}], [[2, 3, 4, 1], { [0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 1, 0, 1, 0]}, {1}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 2, 0, 0]}, {2}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, 3*q^2+11*q^3+14*q^4+ 15*q^5+12*q^6+9*q^7+7*q^8+4*q^9+q^10, q^3+10*q^4+21*q^5+32*q^6+38*q^7+41*q^8+37 *q^9+29*q^10+21*q^11+16*q^12+11*q^13+5*q^14+q^15, 3*q^5+13*q^6+32*q^7+55*q^8+66 *q^9+84*q^10+94*q^11+102*q^12+101*q^13+88*q^14+68*q^15+50*q^16+37*q^17+27*q^18+ 16*q^19+6*q^20+q^21, 3*q^7+11*q^8+34*q^9+59*q^10+94*q^11+128*q^12+155*q^13+179* q^14+198*q^15+216*q^16+240*q^17+248*q^18+235*q^19+201*q^20+158*q^21+118*q^22+87 *q^23+64*q^24+43*q^25+22*q^26+7*q^27+q^28, q^9+8*q^10+21*q^11+45*q^12+90*q^13+ 136*q^14+184*q^15+229*q^16+292*q^17+343*q^18+389*q^19+418*q^20+450*q^21+485*q^ 22+530*q^23+568*q^24+572*q^25+527*q^26+450*q^27+361*q^28+276*q^29+205*q^30+151* q^31+107*q^32+65*q^33+29*q^34+8*q^35+q^36, 3*q^12+9*q^13+27*q^14+50*q^15+96*q^ 16+154*q^17+226*q^18+305*q^19+383*q^20+454*q^21+539*q^22+628*q^23+716*q^24+819* q^25+902*q^26+946*q^27+986*q^28+1046*q^29+1145*q^30+1241*q^31+1294*q^32+1268*q^ 33+1157*q^34+993*q^35+813*q^36+637*q^37+481*q^38+356*q^39+258*q^40+172*q^41+94* q^42+37*q^43+9*q^44+q^45, 3*q^15+9*q^16+23*q^17+48*q^18+79*q^19+137*q^20+221*q^ 21+319*q^22+416*q^23+534*q^24+651*q^25+798*q^26+924*q^27+1041*q^28+1156*q^29+ 1304*q^30+1489*q^31+1698*q^32+1863*q^33+1979*q^34+2058*q^35+2128*q^36+2228*q^37 +2404*q^38+2625*q^39+2808*q^40+2859*q^41+2748*q^42+2499*q^43+2168*q^44+1808*q^ 45+1450*q^46+1118*q^47+837*q^48+614*q^49+430*q^50+266*q^51+131*q^52+46*q^53+10* q^54+q^55, q^18+8*q^19+15*q^20+34*q^21+62*q^22+105*q^23+156*q^24+248*q^25+366*q ^26+510*q^27+669*q^28+829*q^29+994*q^30+1179*q^31+1389*q^32+1607*q^33+1816*q^34 +2034*q^35+2246*q^36+2464*q^37+2710*q^38+3014*q^39+3395*q^40+3787*q^41+4099*q^ 42+4280*q^43+4368*q^44+4484*q^45+4684*q^46+5006*q^47+5436*q^48+5885*q^49+6179*q ^50+6177*q^51+5867*q^52+5339*q^53+4687*q^54+3978*q^55+3258*q^56+2568*q^57+1955* q^58+1451*q^59+1044*q^60+696*q^61+397*q^62+177*q^63+56*q^64+11*q^65+q^66, 3*q^ 22+9*q^23+21*q^24+39*q^25+66*q^26+107*q^27+162*q^28+245*q^29+344*q^30+505*q^31+ 699*q^32+908*q^33+1133*q^34+1384*q^35+1644*q^36+1948*q^37+2225*q^38+2508*q^39+ 2796*q^40+3148*q^41+3561*q^42+4024*q^43+4435*q^44+4753*q^45+5057*q^46+5478*q^47 +6066*q^48+6811*q^49+7578*q^50+8248*q^51+8731*q^52+9024*q^53+9200*q^54+9378*q^ 55+9700*q^56+10296*q^57+11142*q^58+12102*q^59+12923*q^60+13320*q^61+13126*q^62+ 12396*q^63+11318*q^64+10048*q^65+8667*q^66+7236*q^67+5826*q^68+4523*q^69+3406*q ^70+2495*q^71+1740*q^72+1093*q^73+574*q^74+233*q^75+67*q^76+12*q^77+q^78, 3*q^ 26+9*q^27+21*q^28+35*q^29+66*q^30+94*q^31+146*q^32+216*q^33+301*q^34+415*q^35+ 586*q^36+807*q^37+1088*q^38+1398*q^39+1723*q^40+2059*q^41+2420*q^42+2824*q^43+ 3293*q^44+3776*q^45+4214*q^46+4635*q^47+5078*q^48+5615*q^49+6279*q^50+7024*q^51 +7825*q^52+8572*q^53+9217*q^54+9807*q^55+10426*q^56+11139*q^57+12075*q^58+13376 *q^59+14944*q^60+16480*q^61+17665*q^62+18372*q^63+18754*q^64+19049*q^65+19441*q ^66+20038*q^67+21032*q^68+22574*q^69+24550*q^70+26506*q^71+27883*q^72+28269*q^ 73+27568*q^74+25986*q^75+23848*q^76+21390*q^77+18717*q^78+15903*q^79+13062*q^80 +10349*q^81+7929*q^82+5901*q^83+4235*q^84+2833*q^85+1667*q^86+807*q^87+300*q^88 +79*q^89+13*q^90+q^91, q^30+8*q^31+15*q^32+28*q^33+53*q^34+81*q^35+118*q^36+167 *q^37+240*q^38+333*q^39+456*q^40+598*q^41+804*q^42+1091*q^43+1468*q^44+1875*q^ 45+2335*q^46+2807*q^47+3314*q^48+3860*q^49+4473*q^50+5081*q^51+5716*q^52+6390*q ^53+7150*q^54+7949*q^55+8787*q^56+9622*q^57+10454*q^58+11287*q^59+12271*q^60+ 13519*q^61+15063*q^62+16689*q^63+18109*q^64+19193*q^65+20062*q^66+20977*q^67+ 22248*q^68+24062*q^69+26478*q^70+29328*q^71+32334*q^72+35052*q^73+37055*q^74+ 38220*q^75+38787*q^76+39199*q^77+39869*q^78+41026*q^79+42869*q^80+45600*q^81+ 49296*q^82+53498*q^83+57176*q^84+59273*q^85+59300*q^86+57423*q^87+54152*q^88+ 49992*q^89+45264*q^90+40109*q^91+34620*q^92+28965*q^93+23411*q^94+18278*q^95+ 13830*q^96+10136*q^97+7068*q^98+4500*q^99+2474*q^100+1107*q^101+379*q^102+92*q^ 103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+5*q^4+5*q^3+4*q^2+3*q+1, 3*q^8+11*q^7+14*q^6+15 *q^5+12*q^4+9*q^3+7*q^2+4*q+1, q^12+10*q^11+21*q^10+32*q^9+38*q^8+41*q^7+37*q^6 +29*q^5+21*q^4+16*q^3+11*q^2+5*q+1, 3*q^16+13*q^15+32*q^14+55*q^13+66*q^12+84*q ^11+94*q^10+102*q^9+101*q^8+88*q^7+68*q^6+50*q^5+37*q^4+27*q^3+16*q^2+6*q+1, 3* q^21+11*q^20+34*q^19+59*q^18+94*q^17+128*q^16+155*q^15+179*q^14+198*q^13+216*q^ 12+240*q^11+248*q^10+235*q^9+201*q^8+158*q^7+118*q^6+87*q^5+64*q^4+43*q^3+22*q^ 2+7*q+1, 1+21*q^25+136*q^22+45*q^24+184*q^21+90*q^23+8*q+29*q^2+8*q^26+65*q^3+ 107*q^4+151*q^5+205*q^6+276*q^7+361*q^8+450*q^9+527*q^10+572*q^11+568*q^12+530* q^13+485*q^14+450*q^15+418*q^16+389*q^17+343*q^18+292*q^19+229*q^20+q^27, 1+383 *q^25+628*q^22+454*q^24+716*q^21+539*q^23+9*q+50*q^30+96*q^29+3*q^33+37*q^2+305 *q^26+94*q^3+172*q^4+258*q^5+356*q^6+9*q^32+481*q^7+637*q^8+813*q^9+993*q^10+ 1157*q^11+1268*q^12+1294*q^13+1241*q^14+1145*q^15+1046*q^16+154*q^28+986*q^17+ 946*q^18+902*q^19+819*q^20+226*q^27+27*q^31, 1+1304*q^25+1863*q^22+1489*q^24+ 1979*q^21+1698*q^23+10*q+651*q^30+48*q^37+798*q^29+9*q^39+221*q^34+23*q^38+319* q^33+46*q^2+1156*q^26+131*q^3+266*q^4+430*q^5+614*q^6+416*q^32+837*q^7+1118*q^8 +1450*q^9+1808*q^10+2168*q^11+2499*q^12+2748*q^13+2859*q^14+2808*q^15+2625*q^16 +924*q^28+2404*q^17+2228*q^18+2128*q^19+2058*q^20+1041*q^27+3*q^40+534*q^31+137 *q^35+79*q^36, 1+3787*q^25+156*q^42+4368*q^22+4099*q^24+4484*q^21+4280*q^23+11* q+62*q^44+2246*q^30+34*q^45+829*q^37+2464*q^29+15*q^46+510*q^39+248*q^41+1389*q ^34+669*q^38+1607*q^33+56*q^2+3395*q^26+177*q^3+397*q^4+696*q^5+1044*q^6+1816*q ^32+1451*q^7+1955*q^8+2568*q^9+3258*q^10+3978*q^11+4687*q^12+5339*q^13+5867*q^ 14+6177*q^15+6179*q^16+105*q^43+2710*q^28+5885*q^17+5436*q^18+5006*q^19+4684*q^ 20+3014*q^27+366*q^40+2034*q^31+1179*q^35+q^48+994*q^36+8*q^47, 1+9024*q^25+ 1644*q^42+9700*q^22+9200*q^24+10296*q^21+9378*q^23+12*q+107*q^51+1133*q^44+39*q ^53+6066*q^30+66*q^52+9*q^55+21*q^54+908*q^45+3148*q^37+6811*q^29+699*q^46+2508 *q^39+1948*q^41+4435*q^34+162*q^50+2796*q^38+4753*q^33+3*q^56+67*q^2+8731*q^26+ 233*q^3+574*q^4+1093*q^5+1740*q^6+5057*q^32+2495*q^7+3406*q^8+4523*q^9+5826*q^ 10+7236*q^11+8667*q^12+10048*q^13+11318*q^14+12396*q^15+13126*q^16+1384*q^43+ 245*q^49+7578*q^28+13320*q^17+12923*q^18+12102*q^19+11142*q^20+8248*q^27+2225*q ^40+5478*q^31+4024*q^35+344*q^48+3561*q^36+505*q^47, 1+19441*q^25+5615*q^42+ 22574*q^22+20038*q^24+24550*q^21+21032*q^23+13*q+1723*q^51+4635*q^44+1088*q^53+ 301*q^57+16480*q^30+1398*q^52+216*q^58+586*q^55+94*q^60+807*q^54+4214*q^45+9217 *q^37+17665*q^29+3776*q^46+3*q^65+7825*q^39+66*q^61+6279*q^41+11139*q^34+2059*q ^50+9*q^64+8572*q^38+12075*q^33+415*q^56+79*q^2+19049*q^26+300*q^3+807*q^4+1667 *q^5+2833*q^6+13376*q^32+4235*q^7+5901*q^8+7929*q^9+35*q^62+10349*q^10+13062*q^ 11+15903*q^12+18717*q^13+21390*q^14+23848*q^15+25986*q^16+5078*q^43+2420*q^49+ 18372*q^28+27568*q^17+28269*q^18+27883*q^19+26506*q^20+18754*q^27+7024*q^40+21* q^63+14944*q^31+10426*q^35+2824*q^48+146*q^59+9807*q^36+3293*q^47, 1+42869*q^25 +16689*q^42+53498*q^22+45600*q^24+57176*q^21+49296*q^23+14*q+7150*q^51+13519*q^ 44+5716*q^53+3314*q^57+38220*q^30+6390*q^52+2807*q^58+4473*q^55+1875*q^60+5081* q^54+12271*q^45+53*q^71+8*q^74+22248*q^37+240*q^67+38787*q^29+11287*q^46+456*q^ 65+118*q^69+20062*q^39+1468*q^61+18109*q^41+29328*q^34+7949*q^50+598*q^64+20977 *q^38+32334*q^33+3860*q^56+92*q^2+167*q^68+41026*q^26+379*q^3+1107*q^4+2474*q^5 +4500*q^6+35052*q^32+7068*q^7+10136*q^8+13830*q^9+1091*q^62+18278*q^10+23411*q^ 11+28965*q^12+34620*q^13+40109*q^14+45264*q^15+49992*q^16+333*q^66+81*q^70+ 15063*q^43+8787*q^49+39199*q^28+54152*q^17+57423*q^18+59300*q^19+59273*q^20+ 39869*q^27+28*q^72+19193*q^40+15*q^73+804*q^63+37055*q^31+q^75+26478*q^35+9622* q^48+2335*q^59+24062*q^36+10454*q^47] The number of permutations avoiding, {[1, 2, 3, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 2, 6, 21, 76, 263, 843, 2501, 6941, 18245, 45928, 111721, 264482, 612707, 1394929] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 1, 3, 10, 37, 133, 425, 1247, 3462, 9129, 22981, 55850, 132211, 306369, 697513] The number of ODD permutations avoiding, {[1, 2, 3, 4], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [0, 1, 3, 11, 39, 130, 418, 1254, 3479, 9116, 22947, 55871, 132271, 306338, 697416] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 37, 130, 418, 1247, 3462, 9116, 22947, 55850, 132211, 306338, 697416] ODD:, [0, 1, 3, 11, 39, 133, 425, 1254, 3479, 9129, 22981, 55871, 132271, 306369, 697513] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.095238095, 5.355263158, 8.395437262, 12.30604982, 17.13954418, 22.92969313, 29.70161688, 37.47463421, 46.26291387, 56.07587284, 66.91879806, 78.79367624] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.411003085, 1.903624987, 2.467179642, 3.128531532, 3.897440650, 4.768365874, 5.727853808, 6.759682859, 7.847710699, 8.977419935, 10.13670052, 11.31611277] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.99093, 3.62379, 6.08698, 9.78771, 15.1900, 22.7373, 32.8083, 45.6933, 61.5866, 80.5941, 102.753, 128.054] Skewness: , [Float(undefined), 0., 0., 0.2379317809, 0.3320176058, 0.2827068050, 0.1581390121, 0.01427653543, -0.1237966512, -0.2493551555, -0.3624185313, -0.4646048256, -0.5575021762, -0.6423398448, -0.7200681405] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.142917187, 2.352523878, 2.485041998, 2.510697706, 2.500341299, 2.506069756, 2.545214907, 2.617840160, 2.718546420, 2.841129808, 2.979803617, 3.129731778] end of this data For the equivalence class of patterns, { {[1, 2, 4, 3], [2, 4, 3, 1], [2, 3, 1, 4]}, {[1, 4, 2, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [3, 4, 2, 1], [4, 1, 3, 2]}, {[1, 2, 4, 3], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 4, 2, 3], [2, 4, 3, 1], [4, 3, 1, 2]}, {[2, 3, 1, 4], [3, 4, 2, 1], [4, 2, 1, 3]}, {[1, 3, 4, 2], [2, 1, 3, 4], [3, 2, 4, 1]}, {[3, 2, 4, 1], [3, 1, 2, 4], [4, 3, 1, 2]}} the member , {[1, 4, 2, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[4, 1, 2, 3], {}, {3}], [[4, 2, 3, 1], %1, {1}], [[4, 5, 2, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[4, 1, 3, 2], %1, {3}], [[1, 2, 3], {}, {2}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[3, 4, 1, 2], {}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[4, 5, 1, 2, 3], {}, {4}], [[3, 5, 1, 2, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 0, 0, 1]}, {}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1]}, {3}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {2}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[4, 5, 1, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2, 5], {[0, 0, 0, 0, 0, 1]}, {1}], [[3, 4, 2, 1], %1, {1}], [[3, 1, 2], {}, {}], [[1, 4, 3, 2], %1, {2}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+2*q+3*q^2+7*q^3+ 12*q^4+14*q^5+13*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+2*q+3*q^2+5*q^3+10*q^4+16*q^5+ 24*q^6+33*q^7+37*q^8+36*q^9+32*q^10+26*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+2*q+ 3*q^2+5*q^3+8*q^4+12*q^5+21*q^6+29*q^7+45*q^8+62*q^9+75*q^10+88*q^11+97*q^12+94 *q^13+90*q^14+79*q^15+63*q^16+48*q^17+31*q^18+16*q^19+6*q^20+q^21, 1+2*q+3*q^2+ 5*q^3+8*q^4+10*q^5+18*q^6+26*q^7+33*q^8+50*q^9+73*q^10+100*q^11+130*q^12+161*q^ 13+188*q^14+214*q^15+234*q^16+245*q^17+248*q^18+241*q^19+220*q^20+192*q^21+157* q^22+119*q^23+83*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+2*q+3*q^2+5*q^3+8*q^4+10*q ^5+16*q^6+23*q^7+33*q^8+43*q^9+57*q^10+78*q^11+108*q^12+149*q^13+194*q^14+243*q ^15+304*q^16+366*q^17+427*q^18+485*q^19+534*q^20+573*q^21+603*q^22+631*q^23+641 *q^24+634*q^25+598*q^26+540*q^27+468*q^28+387*q^29+300*q^30+214*q^31+136*q^32+ 71*q^33+29*q^34+8*q^35+q^36, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+30*q^ 8+43*q^9+54*q^10+72*q^11+92*q^12+120*q^13+158*q^14+210*q^15+266*q^16+338*q^17+ 430*q^18+526*q^19+640*q^20+762*q^21+871*q^22+1003*q^23+1125*q^24+1237*q^25+1348 *q^26+1433*q^27+1495*q^28+1566*q^29+1622*q^30+1652*q^31+1652*q^32+1590*q^33+ 1473*q^34+1324*q^35+1147*q^36+953*q^37+752*q^38+551*q^39+367*q^40+213*q^41+101* q^42+37*q^43+9*q^44+q^45, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+ 40*q^9+54*q^10+69*q^11+90*q^12+115*q^13+140*q^14+184*q^15+236*q^16+292*q^17+366 *q^18+456*q^19+562*q^20+700*q^21+858*q^22+1033*q^23+1239*q^24+1453*q^25+1672*q^ 26+1930*q^27+2169*q^28+2415*q^29+2679*q^30+2935*q^31+3170*q^32+3394*q^33+3585*q ^34+3747*q^35+3911*q^36+4056*q^37+4187*q^38+4273*q^39+4278*q^40+4166*q^41+3948* q^42+3632*q^43+3245*q^44+2817*q^45+2354*q^46+1877*q^47+1409*q^48+973*q^49+603*q ^50+321*q^51+139*q^52+46*q^53+10*q^54+q^55, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q ^6+21*q^7+28*q^8+38*q^9+51*q^10+69*q^11+87*q^12+113*q^13+139*q^14+176*q^15+219* q^16+273*q^17+340*q^18+420*q^19+512*q^20+619*q^21+751*q^22+910*q^23+1103*q^24+ 1328*q^25+1595*q^26+1896*q^27+2232*q^28+2610*q^29+3008*q^30+3452*q^31+3916*q^32 +4404*q^33+4915*q^34+5444*q^35+5974*q^36+6489*q^37+7038*q^38+7560*q^39+8076*q^ 40+8576*q^41+9028*q^42+9438*q^43+9812*q^44+10173*q^45+10537*q^46+10850*q^47+ 11042*q^48+11069*q^49+10873*q^50+10425*q^51+9772*q^52+8943*q^53+7979*q^54+6927* q^55+5820*q^56+4686*q^57+3577*q^58+2548*q^59+1655*q^60+954*q^61+468*q^62+186*q^ 63+56*q^64+11*q^65+q^66, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38 *q^9+49*q^10+66*q^11+87*q^12+110*q^13+137*q^14+175*q^15+215*q^16+266*q^17+327*q ^18+398*q^19+490*q^20+601*q^21+719*q^22+856*q^23+1023*q^24+1213*q^25+1445*q^26+ 1724*q^27+2026*q^28+2394*q^29+2818*q^30+3290*q^31+3840*q^32+4459*q^33+5125*q^34 +5875*q^35+6682*q^36+7516*q^37+8441*q^38+9408*q^39+10390*q^40+11470*q^41+12576* q^42+13673*q^43+14840*q^44+15969*q^45+17082*q^46+18250*q^47+19423*q^48+20570*q^ 49+21733*q^50+22797*q^51+23802*q^52+24792*q^53+25687*q^54+26559*q^55+27400*q^56 +28107*q^57+28553*q^58+28663*q^59+28272*q^60+27369*q^61+25983*q^62+24166*q^63+ 22032*q^64+19648*q^65+17075*q^66+14398*q^67+11693*q^68+9052*q^69+6600*q^70+4454 *q^71+2719*q^72+1460*q^73+663*q^74+243*q^75+67*q^76+12*q^77+q^78, 1+2*q+3*q^2+5 *q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64*q^11+84*q^12+110*q^13+ 134*q^14+173*q^15+214*q^16+262*q^17+324*q^18+395*q^19+474*q^20+579*q^21+703*q^ 22+841*q^23+1009*q^24+1191*q^25+1399*q^26+1650*q^27+1942*q^28+2266*q^29+2654*q^ 30+3095*q^31+3585*q^32+4173*q^33+4817*q^34+5535*q^35+6393*q^36+7339*q^37+8392*q ^38+9601*q^39+10881*q^40+12253*q^41+13792*q^42+15408*q^43+17110*q^44+18978*q^45 +20907*q^46+22913*q^47+25095*q^48+27309*q^49+29619*q^50+32040*q^51+34461*q^52+ 36915*q^53+39451*q^54+41953*q^55+44461*q^56+47103*q^57+49761*q^58+52494*q^59+ 55264*q^60+57882*q^61+60356*q^62+62746*q^63+65034*q^64+67248*q^65+69388*q^66+ 71308*q^67+72876*q^68+73922*q^69+74184*q^70+73446*q^71+71580*q^72+68579*q^73+ 64577*q^74+59784*q^75+54355*q^76+48440*q^77+42160*q^78+35666*q^79+29161*q^80+ 22851*q^81+16978*q^82+11804*q^83+7541*q^84+4328*q^85+2170*q^86+916*q^87+311*q^ 88+79*q^89+13*q^90+q^91, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38 *q^9+49*q^10+64*q^11+82*q^12+107*q^13+134*q^14+170*q^15+212*q^16+261*q^17+320*q ^18+392*q^19+475*q^20+573*q^21+687*q^22+825*q^23+992*q^24+1181*q^25+1396*q^26+ 1641*q^27+1925*q^28+2248*q^29+2617*q^30+3042*q^31+3519*q^32+4072*q^33+4693*q^34 +5397*q^35+6179*q^36+7059*q^37+8064*q^38+9182*q^39+10440*q^40+11856*q^41+13438* q^42+15214*q^43+17193*q^44+19343*q^45+21705*q^46+24257*q^47+27023*q^48+29953*q^ 49+33123*q^50+36520*q^51+40071*q^52+43913*q^53+47937*q^54+52120*q^55+56562*q^56 +61125*q^57+65851*q^58+70822*q^59+75956*q^60+81291*q^61+86788*q^62+92276*q^63+ 97817*q^64+103477*q^65+109226*q^66+115204*q^67+121397*q^68+127779*q^69+134327*q ^70+140992*q^71+147470*q^72+153700*q^73+159607*q^74+165205*q^75+170698*q^76+ 176038*q^77+181075*q^78+185594*q^79+189238*q^80+191543*q^81+192198*q^82+190705* q^83+186712*q^84+180186*q^85+171297*q^86+160410*q^87+147969*q^88+134284*q^89+ 119607*q^90+104217*q^91+88448*q^92+72731*q^93+57564*q^94+43458*q^95+30945*q^96+ 20494*q^97+12394*q^98+6695*q^99+3143*q^100+1238*q^101+391*q^102+92*q^103+14*q^ 104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+4*q^4+6*q^3+4*q^2+3*q+1, q^10+2*q^9+3*q^8+7 *q^7+12*q^6+14*q^5+13*q^4+11*q^3+7*q^2+4*q+1, q^15+2*q^14+3*q^13+5*q^12+10*q^11 +16*q^10+24*q^9+33*q^8+37*q^7+36*q^6+32*q^5+26*q^4+19*q^3+11*q^2+5*q+1, q^21+2* q^20+3*q^19+5*q^18+8*q^17+12*q^16+21*q^15+29*q^14+45*q^13+62*q^12+75*q^11+88*q^ 10+97*q^9+94*q^8+90*q^7+79*q^6+63*q^5+48*q^4+31*q^3+16*q^2+6*q+1, 1+5*q^25+18*q ^22+8*q^24+26*q^21+10*q^23+7*q+22*q^2+3*q^26+48*q^3+83*q^4+119*q^5+157*q^6+192* q^7+220*q^8+241*q^9+248*q^10+245*q^11+234*q^12+214*q^13+188*q^14+161*q^15+130*q ^16+q^28+100*q^17+73*q^18+50*q^19+33*q^20+2*q^27, 1+78*q^25+194*q^22+108*q^24+ 243*q^21+149*q^23+8*q+16*q^30+23*q^29+3*q^34+5*q^33+29*q^2+57*q^26+71*q^3+136*q ^4+214*q^5+300*q^6+8*q^32+387*q^7+468*q^8+540*q^9+598*q^10+634*q^11+641*q^12+ 631*q^13+603*q^14+573*q^15+534*q^16+33*q^28+485*q^17+427*q^18+366*q^19+304*q^20 +43*q^27+10*q^31+2*q^35+q^36, 1+640*q^25+5*q^42+1003*q^22+762*q^24+1125*q^21+ 871*q^23+9*q+2*q^44+210*q^30+q^45+30*q^37+266*q^29+16*q^39+8*q^41+72*q^34+21*q^ 38+92*q^33+37*q^2+526*q^26+101*q^3+213*q^4+367*q^5+551*q^6+120*q^32+752*q^7+953 *q^8+1147*q^9+1324*q^10+1473*q^11+1590*q^12+1652*q^13+1652*q^14+1622*q^15+1566* q^16+3*q^43+338*q^28+1495*q^17+1433*q^18+1348*q^19+1237*q^20+430*q^27+10*q^40+ 158*q^31+54*q^35+43*q^36, 1+2679*q^25+115*q^42+3394*q^22+2935*q^24+3585*q^21+ 3170*q^23+10*q+8*q^51+69*q^44+3*q^53+1453*q^30+5*q^52+q^55+2*q^54+54*q^45+366*q ^37+1672*q^29+40*q^46+236*q^39+140*q^41+700*q^34+10*q^50+292*q^38+858*q^33+46*q ^2+2415*q^26+139*q^3+321*q^4+603*q^5+973*q^6+1033*q^32+1409*q^7+1877*q^8+2354*q ^9+2817*q^10+3245*q^11+3632*q^12+3948*q^13+4166*q^14+4278*q^15+4273*q^16+90*q^ 43+16*q^49+1930*q^28+4187*q^17+4056*q^18+3911*q^19+3747*q^20+2169*q^27+184*q^40 +1239*q^31+562*q^35+21*q^48+456*q^36+28*q^47, 1+8576*q^25+1103*q^42+9812*q^22+ 9028*q^24+10173*q^21+9438*q^23+11*q+176*q^51+751*q^44+113*q^53+38*q^57+5974*q^ 30+139*q^52+28*q^58+69*q^55+16*q^60+87*q^54+619*q^45+2610*q^37+6489*q^29+512*q^ 46+2*q^65+1896*q^39+10*q^61+1328*q^41+3916*q^34+219*q^50+3*q^64+2232*q^38+4404* q^33+51*q^56+56*q^2+8076*q^26+186*q^3+468*q^4+954*q^5+1655*q^6+4915*q^32+2548*q ^7+3577*q^8+4686*q^9+8*q^62+5820*q^10+6927*q^11+7979*q^12+8943*q^13+9772*q^14+ 10425*q^15+10873*q^16+q^66+910*q^43+273*q^49+7038*q^28+11069*q^17+11042*q^18+ 10850*q^19+10537*q^20+7560*q^27+1595*q^40+5*q^63+5444*q^31+3452*q^35+340*q^48+ 21*q^59+3008*q^36+420*q^47, 1+24792*q^25+6682*q^42+27400*q^22+25687*q^24+28107* q^21+26559*q^23+12*q+1724*q^51+5125*q^44+1213*q^53+601*q^57+19423*q^30+1445*q^ 52+490*q^58+856*q^55+327*q^60+1023*q^54+4459*q^45+21*q^71+8*q^74+11470*q^37+66* q^67+20570*q^29+3840*q^46+110*q^65+38*q^69+9408*q^39+266*q^61+7516*q^41+14840*q ^34+2026*q^50+137*q^64+10390*q^38+15969*q^33+719*q^56+67*q^2+49*q^68+23802*q^26 +243*q^3+663*q^4+1460*q^5+2719*q^6+17082*q^32+4454*q^7+6600*q^8+9052*q^9+215*q^ 62+11693*q^10+14398*q^11+17075*q^12+3*q^76+19648*q^13+22032*q^14+24166*q^15+ 25983*q^16+87*q^66+28*q^70+2*q^77+q^78+5875*q^43+2394*q^49+21733*q^28+27369*q^ 17+28272*q^18+28663*q^19+28553*q^20+22797*q^27+16*q^72+8441*q^40+10*q^73+175*q^ 63+18250*q^31+5*q^75+13673*q^35+2818*q^48+398*q^59+12576*q^36+3290*q^47, 1+ 69388*q^25+27309*q^42+73922*q^22+2*q^90+71308*q^24+74184*q^21+72876*q^23+13*q+ 10881*q^51+22913*q^44+8392*q^53+4817*q^57+57882*q^30+9601*q^52+4173*q^58+6393*q ^55+3095*q^60+q^91+7339*q^54+20907*q^45+474*q^71+262*q^74+39451*q^37+1009*q^67+ 60356*q^29+18978*q^46+1399*q^65+703*q^69+34461*q^39+2654*q^61+29619*q^41+47103* q^34+84*q^79+12253*q^50+1650*q^64+36915*q^38+49761*q^33+5535*q^56+79*q^2+841*q^ 68+67248*q^26+311*q^3+916*q^4+2170*q^5+4328*q^6+52494*q^32+7541*q^7+11804*q^8+ 16978*q^9+2266*q^62+22851*q^10+29161*q^11+35666*q^12+173*q^76+42160*q^13+48440* q^14+54355*q^15+59784*q^16+1191*q^66+579*q^70+134*q^77+110*q^78+25095*q^43+ 13792*q^49+62746*q^28+64577*q^17+68579*q^18+71580*q^19+73446*q^20+65034*q^27+ 395*q^72+32040*q^40+324*q^73+1942*q^63+55264*q^31+214*q^75+44461*q^35+64*q^80+ 15408*q^48+28*q^83+49*q^81+16*q^85+10*q^86+8*q^87+5*q^88+3*q^89+21*q^84+38*q^82 +3585*q^59+41953*q^36+17110*q^47, 1+189238*q^25+92276*q^42+190705*q^22+170*q^90 +191543*q^24+186712*q^21+192198*q^23+14*q+28*q^97+47937*q^51+81291*q^44+40071*q ^53+27023*q^57+165205*q^30+64*q^94+43913*q^52+24257*q^58+33123*q^55+19343*q^60+ 134*q^91+36520*q^54+75956*q^45+4693*q^71+3042*q^74+38*q^96+21*q^98+121397*q^37+ 8064*q^67+170698*q^29+70822*q^46+82*q^93+16*q^99+10440*q^65+6179*q^69+109226*q^ 39+17193*q^61+97817*q^41+140992*q^34+1396*q^79+52120*q^50+11856*q^64+10*q^100+ 115204*q^38+147470*q^33+29953*q^56+92*q^2+7059*q^68+185594*q^26+391*q^3+1238*q^ 4+3143*q^5+6695*q^6+153700*q^32+12394*q^7+20494*q^8+30945*q^9+49*q^95+15214*q^ 62+8*q^101+43458*q^10+57564*q^11+72731*q^12+2248*q^76+88448*q^13+104217*q^14+ 119607*q^15+134284*q^16+9182*q^66+5397*q^70+1925*q^77+1641*q^78+86788*q^43+5*q^ 102+56562*q^49+176038*q^28+147969*q^17+160410*q^18+171297*q^19+180186*q^20+ 181075*q^27+4072*q^72+103477*q^40+3519*q^73+3*q^103+13438*q^63+159607*q^31+2617 *q^75+134327*q^35+2*q^104+1181*q^80+61125*q^48+q^105+687*q^83+992*q^81+475*q^85 +392*q^86+320*q^87+261*q^88+212*q^89+573*q^84+825*q^82+21705*q^59+127779*q^36+ 107*q^92+65851*q^47] The number of permutations avoiding, {[1, 4, 2, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 75, 261, 876, 2839, 8923, 27329, 81923, 241257, 700150, 2007431, 5698047] The number of EVEN permutations avoiding, {[1, 4, 2, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, is given by [1, 1, 3, 10, 37, 131, 440, 1419, 4461, 13660, 40960, 120632, 350071, 1003715, 2849024] The number of ODD permutations avoiding, {[1, 4, 2, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, is given by [0, 1, 3, 11, 38, 130, 436, 1420, 4462, 13669, 40963, 120625, 350079, 1003716, 2849023] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 37, 130, 436, 1419, 4461, 13669, 40963, 120632, 350071, 1003716, 2849023] ODD:, [0, 1, 3, 11, 38, 131, 440, 1420, 4462, 13660, 40960, 120625, 350079, 1003715, 2849024] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.095238095, 5.386666667, 8.429118774, 12.25799087, 16.90383938, 22.39538272, 28.75966922, 36.02178875, 44.20458266, 53.32843391, 63.41114987, 74.46793419] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.476958326, 2.071349533, 2.758434434, 3.546445590, 4.436263360, 5.424190381, 6.503509714, 7.665459885, 8.899919064, 10.19592616, 11.54210758, 12.92704547] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.18141, 4.29049, 7.60896, 12.5773, 19.6804, 29.4218, 42.2956, 58.7593, 79.2086, 103.957, 133.220, 167.109] Skewness: , [Float(undefined), 0., 0., -0.07547205670, -0.1629439397, -0.2441061880, -0.3129554440, -0.3726558460, -0.4274258116, -0.4801982568, -0.5325574282, -0.5852284822, -0.6384229782, -0.6920809066, -0.7459811518] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.496285849, 2.742718655, 2.882050837, 2.955695724, 2.996683779, 3.027393594, 3.061185966, 3.105204243, 3.163036768, 3.236116664, 3.324911823, 3.429094612] end of this data For the equivalence class of patterns, { {[1, 3, 4, 2], [2, 4, 3, 1], [2, 3, 1, 4]}, {[2, 3, 1, 4], [3, 2, 4, 1], [4, 2, 1, 3]}, {[3, 2, 4, 1], [3, 1, 2, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [2, 3, 1, 4], [3, 2, 4, 1]}, {[1, 3, 4, 2], [2, 4, 3, 1], [4, 1, 3, 2]}, {[1, 4, 2, 3], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 4, 2, 3], [3, 1, 2, 4], [4, 2, 1, 3]}, {[1, 4, 2, 3], [2, 4, 3, 1], [4, 1, 3, 2]}} the member , {[1, 3, 4, 2], [2, 4, 3, 1], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3], {}, {}], [[3, 2, 4, 1], {}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[3, 1, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[1, 0, 0, 0, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0]}, {}], [[1, 4, 2, 3], %1, {3}], [[1, 3, 2, 4], %1, {1}], [[2, 4, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], %1, {3}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[3, 2, 1], {}, {2}]} %1 := {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+3*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3 +12*q^4+11*q^5+11*q^6+7*q^7+6*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+25*q^4+28*q^5 +28*q^6+28*q^7+27*q^8+27*q^9+21*q^10+17*q^11+14*q^12+10*q^13+5*q^14+q^15, 1+6*q +16*q^2+31*q^3+47*q^4+58*q^5+67*q^6+67*q^7+69*q^8+71*q^9+72*q^10+70*q^11+73*q^ 12+67*q^13+62*q^14+49*q^15+40*q^16+35*q^17+25*q^18+15*q^19+6*q^20+q^21, 1+7*q+ 22*q^2+48*q^3+82*q^4+113*q^5+140*q^6+156*q^7+163*q^8+172*q^9+174*q^10+179*q^11+ 184*q^12+189*q^13+194*q^14+200*q^15+197*q^16+198*q^17+189*q^18+169*q^19+145*q^ 20+122*q^21+102*q^22+86*q^23+65*q^24+41*q^25+21*q^26+7*q^27+q^28, 1+8*q+29*q^2+ 71*q^3+135*q^4+207*q^5+277*q^6+333*q^7+371*q^8+398*q^9+420*q^10+434*q^11+450*q^ 12+459*q^13+473*q^14+488*q^15+504*q^16+525*q^17+554*q^18+565*q^19+578*q^20+577* q^21+578*q^22+561*q^23+538*q^24+487*q^25+426*q^26+364*q^27+311*q^28+268*q^29+ 219*q^30+167*q^31+112*q^32+63*q^33+28*q^34+8*q^35+q^36, 1+9*q+37*q^2+101*q^3+ 212*q^4+359*q^5+521*q^6+674*q^7+797*q^8+895*q^9+968*q^10+1028*q^11+1080*q^12+ 1124*q^13+1156*q^14+1199*q^15+1226*q^16+1264*q^17+1314*q^18+1366*q^19+1421*q^20 +1481*q^21+1541*q^22+1612*q^23+1679*q^24+1725*q^25+1742*q^26+1747*q^27+1735*q^ 28+1716*q^29+1653*q^30+1556*q^31+1429*q^32+1267*q^33+1100*q^34+952*q^35+822*q^ 36+704*q^37+574*q^38+434*q^39+301*q^40+182*q^41+92*q^42+36*q^43+9*q^44+q^45, 1+ 10*q+46*q^2+139*q^3+320*q^4+594*q^5+935*q^6+1300*q^7+1637*q^8+1928*q^9+2167*q^ 10+2360*q^11+2531*q^12+2672*q^13+2796*q^14+2914*q^15+3019*q^16+3115*q^17+3229*q ^18+3328*q^19+3436*q^20+3545*q^21+3672*q^22+3825*q^23+4011*q^24+4191*q^25+4377* q^26+4547*q^27+4737*q^28+4931*q^29+5128*q^30+5266*q^31+5388*q^32+5429*q^33+5430 *q^34+5374*q^35+5305*q^36+5161*q^37+4949*q^38+4627*q^39+4239*q^40+3806*q^41+ 3353*q^42+2930*q^43+2552*q^44+2213*q^45+1871*q^46+1515*q^47+1149*q^48+806*q^49+ 512*q^50+282*q^51+129*q^52+45*q^53+10*q^54+q^55, 1+11*q+56*q^2+186*q^3+467*q^4+ 944*q^5+1608*q^6+2400*q^7+3221*q^8+3996*q^9+4683*q^10+5271*q^11+5782*q^12+6230* q^13+6614*q^14+6969*q^15+7290*q^16+7591*q^17+7894*q^18+8199*q^19+8486*q^20+8773 *q^21+9038*q^22+9340*q^23+9674*q^24+10033*q^25+10409*q^26+10808*q^27+11224*q^28 +11716*q^29+12268*q^30+12850*q^31+13441*q^32+14006*q^33+14530*q^34+15051*q^35+ 15584*q^36+16126*q^37+16598*q^38+16964*q^39+17173*q^40+17258*q^41+17202*q^42+ 17038*q^43+16750*q^44+16340*q^45+15755*q^46+14963*q^47+13966*q^48+12798*q^49+ 11547*q^50+10280*q^51+9071*q^52+7975*q^53+6964*q^54+6002*q^55+5027*q^56+4032*q^ 57+3065*q^58+2174*q^59+1419*q^60+831*q^61+420*q^62+175*q^63+55*q^64+11*q^65+q^ 66, 1+12*q+67*q^2+243*q^3+662*q^4+1449*q^5+2662*q^6+4258*q^7+6088*q^8+7975*q^9+ 9780*q^10+11418*q^11+12886*q^12+14195*q^13+15355*q^14+16400*q^15+17358*q^16+ 18227*q^17+19082*q^18+19910*q^19+20727*q^20+21539*q^21+22322*q^22+23110*q^23+ 23952*q^24+24791*q^25+25651*q^26+26507*q^27+27364*q^28+28284*q^29+29354*q^30+ 30510*q^31+31786*q^32+33094*q^33+34443*q^34+35834*q^35+37367*q^36+39027*q^37+ 40823*q^38+42602*q^39+44352*q^40+45987*q^41+47583*q^42+49082*q^43+50611*q^44+ 52064*q^45+53449*q^46+54548*q^47+55364*q^48+55725*q^49+55768*q^50+55397*q^51+ 54759*q^52+53793*q^53+52488*q^54+50731*q^55+48522*q^56+45780*q^57+42608*q^58+ 39105*q^59+35417*q^60+31764*q^61+28253*q^62+25008*q^63+22017*q^64+19180*q^65+ 16383*q^66+13588*q^67+10829*q^68+8223*q^69+5889*q^70+3921*q^71+2389*q^72+1297*q ^73+605*q^74+231*q^75+66*q^76+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+915*q^4+2158* q^5+4260*q^6+7287*q^7+11087*q^8+15350*q^9+19743*q^10+23989*q^11+27955*q^12+ 31602*q^13+34913*q^14+37930*q^15+40689*q^16+43225*q^17+45619*q^18+47918*q^19+ 50142*q^20+52350*q^21+54480*q^22+56617*q^23+58827*q^24+61096*q^25+63394*q^26+ 65716*q^27+67965*q^28+70232*q^29+72611*q^30+75138*q^31+77811*q^32+80587*q^33+ 83373*q^34+86272*q^35+89342*q^36+92770*q^37+96560*q^38+100652*q^39+104878*q^40+ 109209*q^41+113604*q^42+118141*q^43+122904*q^44+127967*q^45+133282*q^46+138739* q^47+144113*q^48+149244*q^49+154087*q^50+158709*q^51+163160*q^52+167530*q^53+ 171701*q^54+175539*q^55+178793*q^56+181240*q^57+182660*q^58+183088*q^59+182443* q^60+180839*q^61+178297*q^62+174885*q^63+170557*q^64+165142*q^65+158480*q^66+ 150529*q^67+141422*q^68+131343*q^69+120628*q^70+109674*q^71+98857*q^72+88531*q^ 73+78841*q^74+69830*q^75+61320*q^76+53072*q^77+44930*q^78+36924*q^79+29267*q^80 +22205*q^81+15998*q^82+10826*q^83+6786*q^84+3872*q^85+1958*q^86+847*q^87+298*q^ 88+78*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1237*q^4+3130*q^5+6615*q^6+12071 *q^7+19513*q^8+28552*q^9+38564*q^10+48880*q^11+59002*q^12+68653*q^13+77685*q^14 +86065*q^15+93828*q^16+101010*q^17+107744*q^18+114137*q^19+120274*q^20+126249*q ^21+132072*q^22+137759*q^23+143508*q^24+149354*q^25+155334*q^26+161430*q^27+ 167542*q^28+173659*q^29+179936*q^30+186407*q^31+193161*q^32+200073*q^33+206994* q^34+213881*q^35+220866*q^36+228162*q^37+236079*q^38+244564*q^39+253500*q^40+ 262714*q^41+272245*q^42+282065*q^43+292519*q^44+303686*q^45+315813*q^46+328716* q^47+342267*q^48+356059*q^49+370042*q^50+384115*q^51+398566*q^52+413489*q^53+ 429141*q^54+445276*q^55+461850*q^56+478239*q^57+494219*q^58+509362*q^59+523827* q^60+537443*q^61+550594*q^62+563068*q^63+575076*q^64+585921*q^65+595289*q^66+ 602322*q^67+606877*q^68+608448*q^69+607224*q^70+603035*q^71+596250*q^72+586747* q^73+574714*q^74+559952*q^75+542369*q^76+521600*q^77+497526*q^78+470248*q^79+ 440321*q^80+408449*q^81+375454*q^82+342328*q^83+309897*q^84+278938*q^85+249760* q^86+222372*q^87+196475*q^88+171582*q^89+147306*q^90+123612*q^91+100812*q^92+ 79499*q^93+60273*q^94+43617*q^95+29875*q^96+19151*q^97+11330*q^98+6073*q^99+ 2872*q^100+1157*q^101+377*q^102+91*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+6*q^3+3*q^2+3*q+1, q^10+4*q^9+7*q^8+ 11*q^7+12*q^6+11*q^5+11*q^4+7*q^3+6*q^2+4*q+1, q^15+5*q^14+11*q^13+19*q^12+25*q ^11+28*q^10+28*q^9+28*q^8+27*q^7+27*q^6+21*q^5+17*q^4+14*q^3+10*q^2+5*q+1, q^21 +6*q^20+16*q^19+31*q^18+47*q^17+58*q^16+67*q^15+67*q^14+69*q^13+71*q^12+72*q^11 +70*q^10+73*q^9+67*q^8+62*q^7+49*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+48*q^ 25+140*q^22+82*q^24+156*q^21+113*q^23+7*q+21*q^2+22*q^26+41*q^3+65*q^4+86*q^5+ 102*q^6+122*q^7+145*q^8+169*q^9+189*q^10+198*q^11+197*q^12+200*q^13+194*q^14+ 189*q^15+184*q^16+q^28+179*q^17+174*q^18+172*q^19+163*q^20+7*q^27, 1+434*q^25+ 473*q^22+450*q^24+488*q^21+459*q^23+8*q+277*q^30+333*q^29+29*q^34+71*q^33+28*q^ 2+420*q^26+63*q^3+112*q^4+167*q^5+219*q^6+135*q^32+268*q^7+311*q^8+364*q^9+426* q^10+487*q^11+538*q^12+561*q^13+578*q^14+577*q^15+578*q^16+371*q^28+565*q^17+ 554*q^18+525*q^19+504*q^20+398*q^27+207*q^31+8*q^35+q^36, 1+1421*q^25+101*q^42+ 1612*q^22+1481*q^24+1679*q^21+1541*q^23+9*q+9*q^44+1199*q^30+q^45+797*q^37+1226 *q^29+521*q^39+212*q^41+1028*q^34+674*q^38+1080*q^33+36*q^2+1366*q^26+92*q^3+ 182*q^4+301*q^5+434*q^6+1124*q^32+574*q^7+704*q^8+822*q^9+952*q^10+1100*q^11+ 1267*q^12+1429*q^13+1556*q^14+1653*q^15+1716*q^16+37*q^43+1264*q^28+1735*q^17+ 1747*q^18+1742*q^19+1725*q^20+1314*q^27+359*q^40+1156*q^31+968*q^35+895*q^36, 1 +5128*q^25+2672*q^42+5429*q^22+5266*q^24+5430*q^21+5388*q^23+10*q+320*q^51+2360 *q^44+46*q^53+4191*q^30+139*q^52+q^55+10*q^54+2167*q^45+3229*q^37+4377*q^29+ 1928*q^46+3019*q^39+2796*q^41+3545*q^34+594*q^50+3115*q^38+3672*q^33+45*q^2+ 4931*q^26+129*q^3+282*q^4+512*q^5+806*q^6+3825*q^32+1149*q^7+1515*q^8+1871*q^9+ 2213*q^10+2552*q^11+2930*q^12+3353*q^13+3806*q^14+4239*q^15+4627*q^16+2531*q^43 +935*q^49+4547*q^28+4949*q^17+5161*q^18+5305*q^19+5374*q^20+4737*q^27+2914*q^40 +4011*q^31+3436*q^35+1300*q^48+3328*q^36+1637*q^47, 1+17258*q^25+9674*q^42+ 16750*q^22+17202*q^24+16340*q^21+17038*q^23+11*q+6969*q^51+9038*q^44+6230*q^53+ 3996*q^57+15584*q^30+6614*q^52+3221*q^58+5271*q^55+1608*q^60+5782*q^54+8773*q^ 45+11716*q^37+16126*q^29+8486*q^46+11*q^65+10808*q^39+944*q^61+10033*q^41+13441 *q^34+7290*q^50+56*q^64+11224*q^38+14006*q^33+4683*q^56+55*q^2+17173*q^26+175*q ^3+420*q^4+831*q^5+1419*q^6+14530*q^32+2174*q^7+3065*q^8+4032*q^9+467*q^62+5027 *q^10+6002*q^11+6964*q^12+7975*q^13+9071*q^14+10280*q^15+11547*q^16+q^66+9340*q ^43+7591*q^49+16598*q^28+12798*q^17+13966*q^18+14963*q^19+15755*q^20+16964*q^27 +10409*q^40+186*q^63+15051*q^31+12850*q^35+7894*q^48+2400*q^59+12268*q^36+8199* q^47, 1+53793*q^25+37367*q^42+48522*q^22+52488*q^24+45780*q^21+50731*q^23+12*q+ 26507*q^51+34443*q^44+24791*q^53+21539*q^57+55364*q^30+25651*q^52+20727*q^58+ 23110*q^55+19082*q^60+23952*q^54+33094*q^45+4258*q^71+662*q^74+45987*q^37+11418 *q^67+55725*q^29+31786*q^46+14195*q^65+7975*q^69+42602*q^39+18227*q^61+39027*q^ 41+50611*q^34+27364*q^50+15355*q^64+44352*q^38+52064*q^33+22322*q^56+66*q^2+ 9780*q^68+54759*q^26+231*q^3+605*q^4+1297*q^5+2389*q^6+53449*q^32+3921*q^7+5889 *q^8+8223*q^9+17358*q^62+10829*q^10+13588*q^11+16383*q^12+67*q^76+19180*q^13+ 22017*q^14+25008*q^15+28253*q^16+12886*q^66+6088*q^70+12*q^77+q^78+35834*q^43+ 28284*q^49+55768*q^28+31764*q^17+35417*q^18+39105*q^19+42608*q^20+55397*q^27+ 2662*q^72+40823*q^40+1449*q^73+16400*q^63+54548*q^31+243*q^75+49082*q^35+29354* q^48+19910*q^59+47583*q^36+30510*q^47, 1+158480*q^25+149244*q^42+131343*q^22+13 *q^90+150529*q^24+120628*q^21+141422*q^23+13*q+104878*q^51+138739*q^44+96560*q^ 53+83373*q^57+180839*q^30+100652*q^52+80587*q^58+89342*q^55+75138*q^60+q^91+ 92770*q^54+133282*q^45+50142*q^71+43225*q^74+171701*q^37+58827*q^67+178297*q^29 +127967*q^46+63394*q^65+54480*q^69+163160*q^39+72611*q^61+154087*q^41+181240*q^ 34+27955*q^79+109209*q^50+65716*q^64+167530*q^38+182660*q^33+86272*q^56+78*q^2+ 56617*q^68+165142*q^26+298*q^3+847*q^4+1958*q^5+3872*q^6+183088*q^32+6786*q^7+ 10826*q^8+15998*q^9+70232*q^62+22205*q^10+29267*q^11+36924*q^12+37930*q^76+ 44930*q^13+53072*q^14+61320*q^15+69830*q^16+61096*q^66+52350*q^70+34913*q^77+ 31602*q^78+144113*q^43+113604*q^49+174885*q^28+78841*q^17+88531*q^18+98857*q^19 +109674*q^20+170557*q^27+47918*q^72+158709*q^40+45619*q^73+67965*q^63+182443*q^ 31+40689*q^75+178793*q^35+23989*q^80+118141*q^48+11087*q^83+19743*q^81+4260*q^ 85+2158*q^86+915*q^87+311*q^88+79*q^89+7287*q^84+15350*q^82+77811*q^59+175539*q ^36+122904*q^47, 1+440321*q^25+563068*q^42+342328*q^22+86065*q^90+408449*q^24+ 309897*q^21+375454*q^23+14*q+19513*q^97+429141*q^51+537443*q^44+398566*q^53+ 342267*q^57+559952*q^30+48880*q^94+413489*q^52+328716*q^58+370042*q^55+303686*q ^60+77685*q^91+384115*q^54+523827*q^45+206994*q^71+186407*q^74+28552*q^96+12071 *q^98+606877*q^37+236079*q^67+542369*q^29+509362*q^46+59002*q^93+6615*q^99+ 253500*q^65+220866*q^69+595289*q^39+292519*q^61+575076*q^41+603035*q^34+155334* q^79+445276*q^50+262714*q^64+3130*q^100+602322*q^38+596250*q^33+356059*q^56+91* q^2+228162*q^68+470248*q^26+377*q^3+1157*q^4+2872*q^5+6073*q^6+586747*q^32+ 11330*q^7+19151*q^8+29875*q^9+38564*q^95+282065*q^62+1237*q^101+43617*q^10+ 60273*q^11+79499*q^12+173659*q^76+100812*q^13+123612*q^14+147306*q^15+171582*q^ 16+244564*q^66+213881*q^70+167542*q^77+161430*q^78+550594*q^43+391*q^102+461850 *q^49+521600*q^28+196475*q^17+222372*q^18+249760*q^19+278938*q^20+497526*q^27+ 200073*q^72+585921*q^40+193161*q^73+92*q^103+272245*q^63+574714*q^31+179936*q^ 75+607224*q^35+14*q^104+149354*q^80+478239*q^48+q^105+132072*q^83+143508*q^81+ 120274*q^85+114137*q^86+107744*q^87+101010*q^88+93828*q^89+126249*q^84+137759*q ^82+315813*q^59+608448*q^36+68653*q^92+494219*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [2, 4, 3, 1], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 75, 267, 948, 3367, 11988, 42842, 153783, 554624, 2009904, 7318260, 26768537] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [2, 4, 3, 1], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 38, 132, 478, 1680, 6005, 21410, 76929, 277283, 1005083, 3659043, 13384721] The number of ODD permutations avoiding, {[1, 3, 4, 2], [2, 4, 3, 1], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 37, 135, 470, 1687, 5983, 21432, 76854, 277341, 1004821, 3659217, 13383816] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 38, 135, 470, 1680, 6005, 21432, 76854, 277283, 1005083, 3659217, 13383816] ODD:, [0, 1, 3, 12, 37, 132, 478, 1687, 5983, 21410, 76929, 277341, 1004821, 3659043, 13384721] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.840000000, 7.217228464, 10.14978903, 13.69913870, 17.91966967, 22.85808319, 28.55324711, 35.03642287, 42.33186809, 50.45773490, 59.42713037] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.526782813, 2.274730753, 3.224090470, 4.374929771, 5.717594797, 7.236539528, 8.912187406, 10.72268184, 12.64563609, 14.65966348, 16.74551692, 18.88677953] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.33107, 5.17440, 10.3948, 19.1400, 32.6909, 52.3675, 79.4271, 114.976, 159.912, 214.906, 280.412, 356.710] Skewness: , [Float(undefined), 0., 0., 0.08022105171, 0.1433940019, 0.1433377923, 0.09812251554, 0.02847201694, -0.05273123298, -0.1382472784, -0.2239758673, -0.3074906600, -0.3872841605, -0.4624288714, -0.5323389225] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.326456034, 2.332241401, 2.271871896, 2.210347994, 2.170394850, 2.159880788, 2.180176268, 2.229633746, 2.305038752, 2.402424765, 2.517547501, 2.646012608] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1]}, {[1, 4, 3, 2], [1, 2, 3, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [2, 3, 4, 1], [4, 3, 2, 1]}, {[1, 3, 2, 4], [4, 1, 2, 3], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1, 4, 3, 5], { [0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {5}], [[2, 1], {}, {}], [[2, 1, 3], {}, {}], [[3, 2, 1], {[0, 2, 1, 0], [1, 0, 1, 0], [0, 0, 0, 1]}, {}], [[2, 3, 4, 1], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {}], [ [2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[3, 2, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[3, 5, 2, 4, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 2, 1, 0, 0]}, {4}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 2, 1, 0, 0]}, {4}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0]}, {}], [[4, 3, 1, 2], { [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {1}], [[4, 2, 1, 3], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 2, 1, 0]}, {2}], [[4, 3, 2, 1], %4, {1}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {2}], [[3, 1, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {1}], [ [2, 3, 5, 1, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {3}], [[3, 4, 5, 1, 2], %5, {1}], [[2, 3, 4, 1, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 4, 5, 2, 1], %1, {1}], [[2, 4, 5, 1, 3], {[0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[3, 4, 1, 5, 2], %5, {1}], [[3, 4, 2, 5, 1], %1, {1}], [[2, 4, 1, 5, 3], { [0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[2, 3, 1, 5, 4], {[1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[2, 1, 5, 3, 4], %2, {1}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1]}, {}], [[3, 1, 5, 4, 2], %3, {1}], [[2, 1, 5, 4, 3], {[0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [1, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 2, 5, 4, 1], %1, {3}], [[2, 1, 4, 5, 3], {[0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [1, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 1, 4, 5, 2], %5, {1}], [[3, 2, 4, 5, 1], %1, {3}], [[2, 4, 1, 3, 5], { [0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {5}], [[3, 2, 4, 1], %4, {3}], [[1, 3, 2], {[0, 2, 1, 0]}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {}], [[2, 1, 4, 3], {[1, 0, 0, 1, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[3, 1, 2], {[0, 2, 1, 0], [1, 0, 0, 0]}, {}], [[2, 3, 1], {[1, 0, 1, 0]}, {}], [[2, 5, 4, 1, 3], {[0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[3, 5, 4, 1, 2], %3, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], %4, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 1, 0], [0, 2, 1, 0, 0]}, {1}], [[3, 5, 1, 4, 2], %3, {1}], [[2, 5, 1, 3, 4], %2, {1}], [[2, 5, 1, 4, 3], { [0, 1, 1, 1, 0, 0], [0, 0, 2, 1, 0, 0], [0, 2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[2, 4, 3, 1, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 5, 3, 1, 4], %2, {2}], [[3, 5, 4, 2, 1], %1, {1}], [[2, 1, 3, 5, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[2, 3, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1, 3, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}]} %1 := {[0, 2, 1, 0, 0, 0], [1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} %2 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]} %3 := {[0, 2, 1, 0, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]} %4 := {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [0, 2, 1, 0, 0]} %5 := {[0, 2, 1, 0, 0, 0], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+5*q^2+5*q^3+5*q^4+2*q^5+q^6, 3*q^2+9*q^3+13*q^4+ 16*q^5+13*q^6+8*q^7+5*q^8+2*q^9+q^10, q^3+6*q^4+14*q^5+25*q^6+36*q^7+33*q^8+30* q^9+23*q^10+16*q^11+8*q^12+5*q^13+2*q^14+q^15, q^5+4*q^6+13*q^7+26*q^8+42*q^9+ 55*q^10+66*q^11+68*q^12+61*q^13+50*q^14+37*q^15+26*q^16+16*q^17+8*q^18+5*q^19+2 *q^20+q^21, q^8+4*q^9+12*q^10+26*q^11+43*q^12+60*q^13+84*q^14+98*q^15+121*q^16+ 114*q^17+113*q^18+95*q^19+78*q^20+57*q^21+40*q^22+26*q^23+16*q^24+8*q^25+5*q^26 +2*q^27+q^28, q^12+4*q^13+13*q^14+26*q^15+43*q^16+60*q^17+89*q^18+114*q^19+144* q^20+169*q^21+187*q^22+190*q^23+187*q^24+164*q^25+140*q^26+112*q^27+85*q^28+60* q^29+40*q^30+26*q^31+16*q^32+8*q^33+5*q^34+2*q^35+q^36, q^17+4*q^18+14*q^19+28* q^20+43*q^21+60*q^22+88*q^23+118*q^24+160*q^25+185*q^26+237*q^27+255*q^28+291*q ^29+289*q^30+287*q^31+260*q^32+237*q^33+191*q^34+157*q^35+119*q^36+88*q^37+60*q ^38+40*q^39+26*q^40+16*q^41+8*q^42+5*q^43+2*q^44+q^45, q^23+4*q^24+15*q^25+30*q ^26+45*q^27+60*q^28+88*q^29+116*q^30+164*q^31+198*q^32+250*q^33+298*q^34+349*q^ 35+381*q^36+421*q^37+420*q^38+420*q^39+394*q^40+357*q^41+310*q^42+264*q^43+208* q^44+164*q^45+122*q^46+88*q^47+60*q^48+40*q^49+26*q^50+16*q^51+8*q^52+5*q^53+2* q^54+q^55, q^30+4*q^31+16*q^32+32*q^33+47*q^34+62*q^35+88*q^36+116*q^37+162*q^ 38+200*q^39+263*q^40+303*q^41+392*q^42+426*q^43+506*q^44+540*q^45+589*q^46+584* q^47+596*q^48+555*q^49+525*q^50+464*q^51+407*q^52+337*q^53+281*q^54+215*q^55+ 167*q^56+122*q^57+88*q^58+60*q^59+40*q^60+26*q^61+16*q^62+8*q^63+5*q^64+2*q^65+ q^66, q^38+4*q^39+17*q^40+34*q^41+49*q^42+64*q^43+90*q^44+116*q^45+163*q^46+196 *q^47+266*q^48+312*q^49+395*q^50+460*q^51+549*q^52+608*q^53+704*q^54+735*q^55+ 793*q^56+800*q^57+803*q^58+766*q^59+731*q^60+660*q^61+595*q^62+514*q^63+434*q^ 64+354*q^65+288*q^66+218*q^67+167*q^68+122*q^69+88*q^70+60*q^71+40*q^72+26*q^73 +16*q^74+8*q^75+5*q^76+2*q^77+q^78, q^47+4*q^48+18*q^49+36*q^50+51*q^51+66*q^52 +92*q^53+118*q^54+164*q^55+196*q^56+263*q^57+312*q^58+406*q^59+453*q^60+584*q^ 61+636*q^62+771*q^63+829*q^64+940*q^65+973*q^66+1059*q^67+1043*q^68+1065*q^69+ 1022*q^70+985*q^71+901*q^72+836*q^73+730*q^74+645*q^75+541*q^76+451*q^77+361*q^ 78+291*q^79+218*q^80+167*q^81+122*q^82+88*q^83+60*q^84+40*q^85+26*q^86+16*q^87+ 8*q^88+5*q^89+2*q^90+q^91, q^57+4*q^58+19*q^59+38*q^60+53*q^61+68*q^62+94*q^63+ 120*q^64+167*q^65+196*q^66+265*q^67+306*q^68+409*q^69+458*q^70+577*q^71+660*q^ 72+799*q^73+876*q^74+1035*q^75+1094*q^76+1228*q^77+1273*q^78+1352*q^79+1350*q^ 80+1380*q^81+1324*q^82+1287*q^83+1204*q^84+1120*q^85+1006*q^86+906*q^87+780*q^ 88+672*q^89+558*q^90+458*q^91+364*q^92+291*q^93+218*q^94+167*q^95+122*q^96+88*q ^97+60*q^98+40*q^99+26*q^100+16*q^101+8*q^102+5*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+5*q^4+5*q^3+5*q^2+2*q+1, 3*q^8+9*q^7+13*q^6+16* q^5+13*q^4+8*q^3+5*q^2+2*q+1, q^12+6*q^11+14*q^10+25*q^9+36*q^8+33*q^7+30*q^6+ 23*q^5+16*q^4+8*q^3+5*q^2+2*q+1, q^16+4*q^15+13*q^14+26*q^13+42*q^12+55*q^11+66 *q^10+68*q^9+61*q^8+50*q^7+37*q^6+26*q^5+16*q^4+8*q^3+5*q^2+2*q+1, q^20+4*q^19+ 12*q^18+26*q^17+43*q^16+60*q^15+84*q^14+98*q^13+121*q^12+114*q^11+113*q^10+95*q ^9+78*q^8+57*q^7+40*q^6+26*q^5+16*q^4+8*q^3+5*q^2+2*q+1, 1+13*q^22+q^24+26*q^21 +4*q^23+2*q+5*q^2+8*q^3+16*q^4+26*q^5+40*q^6+60*q^7+85*q^8+112*q^9+140*q^10+164 *q^11+187*q^12+190*q^13+187*q^14+169*q^15+144*q^16+114*q^17+89*q^18+60*q^19+43* q^20, 1+28*q^25+88*q^22+43*q^24+118*q^21+60*q^23+2*q+5*q^2+14*q^26+8*q^3+16*q^4 +26*q^5+40*q^6+60*q^7+88*q^8+119*q^9+157*q^10+191*q^11+237*q^12+260*q^13+287*q^ 14+289*q^15+291*q^16+q^28+255*q^17+237*q^18+185*q^19+160*q^20+4*q^27, 1+116*q^ 25+250*q^22+164*q^24+298*q^21+198*q^23+2*q+15*q^30+30*q^29+5*q^2+88*q^26+8*q^3+ 16*q^4+26*q^5+40*q^6+q^32+60*q^7+88*q^8+122*q^9+164*q^10+208*q^11+264*q^12+310* q^13+357*q^14+394*q^15+420*q^16+45*q^28+420*q^17+421*q^18+381*q^19+349*q^20+60* q^27+4*q^31, 1+303*q^25+506*q^22+392*q^24+540*q^21+426*q^23+2*q+88*q^30+116*q^ 29+16*q^34+32*q^33+5*q^2+263*q^26+8*q^3+16*q^4+26*q^5+40*q^6+47*q^32+60*q^7+88* q^8+122*q^9+167*q^10+215*q^11+281*q^12+337*q^13+407*q^14+464*q^15+525*q^16+162* q^28+555*q^17+596*q^18+584*q^19+589*q^20+200*q^27+62*q^31+4*q^35+q^36, 1+608*q^ 25+793*q^22+704*q^24+800*q^21+735*q^23+2*q+266*q^30+34*q^37+312*q^29+4*q^39+90* q^34+17*q^38+116*q^33+5*q^2+549*q^26+8*q^3+16*q^4+26*q^5+40*q^6+163*q^32+60*q^7 +88*q^8+122*q^9+167*q^10+218*q^11+288*q^12+354*q^13+434*q^14+514*q^15+595*q^16+ 395*q^28+660*q^17+731*q^18+766*q^19+803*q^20+460*q^27+q^40+196*q^31+64*q^35+49* q^36, 1+973*q^25+18*q^42+1065*q^22+1059*q^24+1022*q^21+1043*q^23+2*q+q^44+584*q ^30+118*q^37+636*q^29+66*q^39+36*q^41+263*q^34+92*q^38+312*q^33+5*q^2+940*q^26+ 8*q^3+16*q^4+26*q^5+40*q^6+406*q^32+60*q^7+88*q^8+122*q^9+167*q^10+218*q^11+291 *q^12+361*q^13+451*q^14+541*q^15+645*q^16+4*q^43+771*q^28+730*q^17+836*q^18+901 *q^19+985*q^20+829*q^27+51*q^40+453*q^31+196*q^35+164*q^36, 1+1350*q^25+94*q^42 +1287*q^22+1380*q^24+1204*q^21+1324*q^23+2*q+53*q^44+1035*q^30+38*q^45+306*q^37 +1094*q^29+19*q^46+196*q^39+120*q^41+577*q^34+265*q^38+660*q^33+5*q^2+1352*q^26 +8*q^3+16*q^4+26*q^5+40*q^6+799*q^32+60*q^7+88*q^8+122*q^9+167*q^10+218*q^11+ 291*q^12+364*q^13+458*q^14+558*q^15+672*q^16+68*q^43+1228*q^28+780*q^17+906*q^ 18+1006*q^19+1120*q^20+1273*q^27+167*q^40+876*q^31+458*q^35+q^48+409*q^36+4*q^ 47] The number of permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [1, 2, 6, 21, 70, 200, 481, 1004, 1886, 3270, 5325, 8246, 12254, 17596, 24545] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [1, 1, 3, 11, 35, 97, 239, 514, 951, 1605, 2637, 4190, 6195, 8657, 12115] The number of ODD permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [0, 1, 3, 10, 35, 103, 242, 490, 935, 1665, 2688, 4056, 6059, 8939, 12430] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 35, 103, 242, 514, 951, 1665, 2688, 4190, 6195, 8939, 12430] ODD:, [0, 1, 3, 10, 35, 97, 239, 490, 935, 1605, 2637, 4056, 6059, 8657, 12115] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.242857143, 8.215000000, 12.12889813, 17.06772908, 23.04082715, 30.04250765, 38.06572770, 47.10477807, 57.15545944, 68.21476472, 80.28054594] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.361938061, 1.768257531, 2.251394012, 2.789201152, 3.343751148, 3.899034352, 4.451130891, 4.999849675, 5.545913050, 6.090115725, 6.633109946, 7.175381058] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.85488, 3.12673, 5.06878, 7.77964, 11.1807, 15.2025, 19.8126, 24.9985, 30.7572, 37.0895, 43.9981, 51.4861] Skewness: , [Float(undefined), 0., 0., 0.2532187345, 0.3579472262, 0.3413739225, 0.2621321785, 0.1764987886, 0.1059753046, 0.05214395683, 0.01206508584, -0.01754692549, -0.03941042534, -0.05556734325, -0.06753603442] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.323997824, 2.733571592, 2.848956410, 2.836352650, 2.820106329, 2.819631624, 2.828762655, 2.841428961, 2.854138187, 2.865463174, 2.874912538, 2.882490831] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 1, 4], [3, 4, 1, 2]}, {[3, 4, 1, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 1, 3, 2]}, {[1, 4, 2, 3], [3, 4, 1, 2], [4, 2, 1, 3]}, {[1, 3, 4, 2], [3, 4, 1, 2], [3, 2, 4, 1]}, {[1, 4, 2, 3], [2, 4, 3, 1], [2, 1, 4, 3]}, {[2, 3, 1, 4], [2, 1, 4, 3], [4, 2, 1, 3]}, {[2, 1, 4, 3], [3, 2, 4, 1], [3, 1, 2, 4]}} the member , {[1, 3, 4, 2], [2, 1, 4, 3], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 2, 1], {}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[4, 2, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {4}], [[4, 3, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[3, 4, 2, 1], {}, {1}], [[4, 3, 2, 1], {}, {2}], [[1, 3, 2], {}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 4, 3, 1], {}, {3}], [[1, 4, 3, 2], {}, {3}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 2, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+3*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+4*q^2+8*q^3+ 10*q^4+12*q^5+13*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+5*q^2+10*q^3+12*q^4+20*q^5+ 24*q^6+28*q^7+33*q^8+35*q^9+32*q^10+27*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*q+ 6*q^2+12*q^3+14*q^4+23*q^5+34*q^6+42*q^7+52*q^8+65*q^9+77*q^10+87*q^11+96*q^12+ 97*q^13+94*q^14+84*q^15+67*q^16+50*q^17+31*q^18+16*q^19+6*q^20+q^21, 1+7*q+7*q^ 2+14*q^3+16*q^4+26*q^5+38*q^6+54*q^7+69*q^8+89*q^9+112*q^10+134*q^11+170*q^12+ 197*q^13+220*q^14+248*q^15+271*q^16+287*q^17+290*q^18+278*q^19+252*q^20+219*q^ 21+175*q^22+129*q^23+86*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+8*q^2+16*q^3+18 *q^4+29*q^5+42*q^6+59*q^7+83*q^8+110*q^9+141*q^10+176*q^11+224*q^12+276*q^13+ 339*q^14+402*q^15+469*q^16+542*q^17+620*q^18+686*q^19+755*q^20+811*q^21+853*q^ 22+882*q^23+879*q^24+843*q^25+776*q^26+684*q^27+576*q^28+461*q^29+341*q^30+232* q^31+140*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+9*q+9*q^2+18*q^3+20*q^4+32*q^5+46* q^6+64*q^7+89*q^8+127*q^9+166*q^10+210*q^11+274*q^12+339*q^13+429*q^14+529*q^15 +649*q^16+780*q^17+926*q^18+1075*q^19+1243*q^20+1443*q^21+1637*q^22+1832*q^23+ 2024*q^24+2207*q^25+2377*q^26+2530*q^27+2650*q^28+2736*q^29+2758*q^30+2712*q^31 +2596*q^32+2405*q^33+2151*q^34+1858*q^35+1541*q^36+1220*q^37+911*q^38+628*q^39+ 395*q^40+218*q^41+101*q^42+37*q^43+9*q^44+q^45, 1+10*q+10*q^2+20*q^3+22*q^4+35* q^5+50*q^6+69*q^7+95*q^8+135*q^9+186*q^10+239*q^11+314*q^12+397*q^13+502*q^14+ 633*q^15+791*q^16+971*q^17+1197*q^18+1434*q^19+1703*q^20+2020*q^21+2376*q^22+ 2753*q^23+3190*q^24+3645*q^25+4127*q^26+4634*q^27+5165*q^28+5708*q^29+6263*q^30 +6793*q^31+7276*q^32+7736*q^33+8134*q^34+8454*q^35+8689*q^36+8792*q^37+8740*q^ 38+8515*q^39+8097*q^40+7516*q^41+6792*q^42+5958*q^43+5062*q^44+4155*q^45+3266*q ^46+2441*q^47+1709*q^48+1102*q^49+643*q^50+327*q^51+139*q^52+46*q^53+10*q^54+q^ 55, 1+11*q+11*q^2+22*q^3+24*q^4+38*q^5+54*q^6+74*q^7+101*q^8+143*q^9+196*q^10+ 262*q^11+348*q^12+443*q^13+569*q^14+717*q^15+909*q^16+1131*q^17+1410*q^18+1720* q^19+2104*q^20+2546*q^21+3042*q^22+3609*q^23+4256*q^24+4993*q^25+5810*q^26+6731 *q^27+7741*q^28+8844*q^29+10014*q^30+11260*q^31+12600*q^32+14030*q^33+15498*q^ 34+17028*q^35+18569*q^36+20119*q^37+21632*q^38+23087*q^39+24434*q^40+25673*q^41 +26744*q^42+27601*q^43+28219*q^44+28514*q^45+28431*q^46+27939*q^47+26978*q^48+ 25564*q^49+23728*q^50+21536*q^51+19095*q^52+16501*q^53+13853*q^54+11261*q^55+ 8813*q^56+6584*q^57+4654*q^58+3067*q^59+1855*q^60+1008*q^61+475*q^62+186*q^63+ 56*q^64+11*q^65+q^66, 1+12*q+12*q^2+24*q^3+26*q^4+41*q^5+58*q^6+79*q^7+107*q^8+ 151*q^9+206*q^10+274*q^11+375*q^12+482*q^13+622*q^14+794*q^15+1004*q^16+1264*q^ 17+1590*q^18+1957*q^19+2418*q^20+2971*q^21+3620*q^22+4355*q^23+5221*q^24+6203*q ^25+7342*q^26+8648*q^27+10138*q^28+11806*q^29+13695*q^30+15749*q^31+18038*q^32+ 20562*q^33+23294*q^34+26266*q^35+29503*q^36+32978*q^37+36667*q^38+40538*q^39+ 44568*q^40+48758*q^41+53127*q^42+57601*q^43+62124*q^44+66652*q^45+71084*q^46+ 75364*q^47+79417*q^48+83149*q^49+86484*q^50+89350*q^51+91626*q^52+93223*q^53+ 93960*q^54+93717*q^55+92396*q^56+89898*q^57+86206*q^58+81374*q^59+75476*q^60+ 68726*q^61+61341*q^62+53592*q^63+45743*q^64+38033*q^65+30681*q^66+23904*q^67+ 17867*q^68+12706*q^69+8512*q^70+5296*q^71+3012*q^72+1530*q^73+671*q^74+243*q^75 +67*q^76+12*q^77+q^78, 1+13*q+13*q^2+26*q^3+28*q^4+44*q^5+62*q^6+84*q^7+113*q^8 +159*q^9+216*q^10+286*q^11+390*q^12+513*q^13+667*q^14+855*q^15+1091*q^16+1371*q ^17+1740*q^18+2157*q^19+2681*q^20+3317*q^21+4083*q^22+4965*q^23+6033*q^24+7266* q^25+8683*q^26+10354*q^27+12278*q^28+14490*q^29+17028*q^30+19893*q^31+23121*q^ 32+26826*q^33+30932*q^34+35491*q^35+40562*q^36+46173*q^37+52325*q^38+59053*q^39 +66369*q^40+74326*q^41+82936*q^42+92176*q^43+101951*q^44+112406*q^45+123392*q^ 46+134920*q^47+146985*q^48+159514*q^49+172355*q^50+185531*q^51+198889*q^52+ 212380*q^53+225845*q^54+239060*q^55+251807*q^56+264001*q^57+275354*q^58+285673* q^59+294766*q^60+302378*q^61+308233*q^62+312097*q^63+313622*q^64+312504*q^65+ 308467*q^66+301267*q^67+290863*q^68+277319*q^69+260791*q^70+241673*q^71+220442* q^72+197685*q^73+174091*q^74+150314*q^75+126993*q^76+104701*q^77+83939*q^78+ 65145*q^79+48693*q^80+34806*q^81+23596*q^82+15012*q^83+8840*q^84+4739*q^85+2258 *q^86+925*q^87+311*q^88+79*q^89+13*q^90+q^91, 1+14*q+14*q^2+28*q^3+30*q^4+47*q^ 5+66*q^6+89*q^7+119*q^8+167*q^9+226*q^10+298*q^11+405*q^12+531*q^13+703*q^14+ 907*q^15+1160*q^16+1469*q^17+1861*q^18+2324*q^19+2903*q^20+3608*q^21+4462*q^22+ 5470*q^23+6693*q^24+8122*q^25+9828*q^26+11815*q^27+14142*q^28+16838*q^29+19977* q^30+23567*q^31+27711*q^32+32467*q^33+37895*q^34+44033*q^35+51006*q^36+58827*q^ 37+67595*q^38+77329*q^39+88148*q^40+100192*q^41+113535*q^42+128200*q^43+144250* q^44+161787*q^45+180857*q^46+201497*q^47+223746*q^48+247620*q^49+273211*q^50+ 300527*q^51+329596*q^52+360398*q^53+392871*q^54+426818*q^55+462188*q^56+498909* q^57+536808*q^58+575803*q^59+615660*q^60+656173*q^61+697114*q^62+738084*q^63+ 778691*q^64+818582*q^65+857234*q^66+894112*q^67+928714*q^68+960489*q^69+988875* q^70+1013350*q^71+1033197*q^72+1047784*q^73+1056376*q^74+1058218*q^75+1052551*q ^76+1038813*q^77+1016432*q^78+985200*q^79+945165*q^80+896726*q^81+840680*q^82+ 778129*q^83+710414*q^84+639209*q^85+566182*q^86+493071*q^87+421535*q^88+353061* q^89+288972*q^90+230403*q^91+178255*q^92+133202*q^93+95603*q^94+65435*q^95+ 42349*q^96+25636*q^97+14318*q^98+7252*q^99+3251*q^100+1248*q^101+391*q^102+92*q ^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+4*q^8+8 *q^7+10*q^6+12*q^5+13*q^4+11*q^3+7*q^2+4*q+1, q^15+5*q^14+5*q^13+10*q^12+12*q^ 11+20*q^10+24*q^9+28*q^8+33*q^7+35*q^6+32*q^5+27*q^4+19*q^3+11*q^2+5*q+1, q^21+ 6*q^20+6*q^19+12*q^18+14*q^17+23*q^16+34*q^15+42*q^14+52*q^13+65*q^12+77*q^11+ 87*q^10+96*q^9+97*q^8+94*q^7+84*q^6+67*q^5+50*q^4+31*q^3+16*q^2+6*q+1, 1+14*q^ 25+38*q^22+16*q^24+54*q^21+26*q^23+7*q+22*q^2+7*q^26+48*q^3+86*q^4+129*q^5+175* q^6+219*q^7+252*q^8+278*q^9+290*q^10+287*q^11+271*q^12+248*q^13+220*q^14+197*q^ 15+170*q^16+q^28+134*q^17+112*q^18+89*q^19+69*q^20+7*q^27, 1+176*q^25+339*q^22+ 224*q^24+402*q^21+276*q^23+8*q+42*q^30+59*q^29+8*q^34+16*q^33+29*q^2+141*q^26+ 71*q^3+140*q^4+232*q^5+341*q^6+18*q^32+461*q^7+576*q^8+684*q^9+776*q^10+843*q^ 11+879*q^12+882*q^13+853*q^14+811*q^15+755*q^16+83*q^28+686*q^17+620*q^18+542*q ^19+469*q^20+110*q^27+29*q^31+8*q^35+q^36, 1+1243*q^25+18*q^42+1832*q^22+1443*q ^24+2024*q^21+1637*q^23+9*q+9*q^44+529*q^30+q^45+89*q^37+649*q^29+46*q^39+20*q^ 41+210*q^34+64*q^38+274*q^33+37*q^2+1075*q^26+101*q^3+218*q^4+395*q^5+628*q^6+ 339*q^32+911*q^7+1220*q^8+1541*q^9+1858*q^10+2151*q^11+2405*q^12+2596*q^13+2712 *q^14+2758*q^15+2736*q^16+9*q^43+780*q^28+2650*q^17+2530*q^18+2377*q^19+2207*q^ 20+926*q^27+32*q^40+429*q^31+166*q^35+127*q^36, 1+6263*q^25+397*q^42+7736*q^22+ 6793*q^24+8134*q^21+7276*q^23+10*q+22*q^51+239*q^44+10*q^53+3645*q^30+20*q^52+q ^55+10*q^54+186*q^45+1197*q^37+4127*q^29+135*q^46+791*q^39+502*q^41+2020*q^34+ 35*q^50+971*q^38+2376*q^33+46*q^2+5708*q^26+139*q^3+327*q^4+643*q^5+1102*q^6+ 2753*q^32+1709*q^7+2441*q^8+3266*q^9+4155*q^10+5062*q^11+5958*q^12+6792*q^13+ 7516*q^14+8097*q^15+8515*q^16+314*q^43+50*q^49+4634*q^28+8740*q^17+8792*q^18+ 8689*q^19+8454*q^20+5165*q^27+633*q^40+3190*q^31+1703*q^35+69*q^48+1434*q^36+95 *q^47, 1+25673*q^25+4256*q^42+28219*q^22+26744*q^24+28514*q^21+27601*q^23+11*q+ 717*q^51+3042*q^44+443*q^53+143*q^57+18569*q^30+569*q^52+101*q^58+262*q^55+54*q ^60+348*q^54+2546*q^45+8844*q^37+20119*q^29+2104*q^46+11*q^65+6731*q^39+38*q^61 +4993*q^41+12600*q^34+909*q^50+11*q^64+7741*q^38+14030*q^33+196*q^56+56*q^2+ 24434*q^26+186*q^3+475*q^4+1008*q^5+1855*q^6+15498*q^32+3067*q^7+4654*q^8+6584* q^9+24*q^62+8813*q^10+11261*q^11+13853*q^12+16501*q^13+19095*q^14+21536*q^15+ 23728*q^16+q^66+3609*q^43+1131*q^49+21632*q^28+25564*q^17+26978*q^18+27939*q^19 +28431*q^20+23087*q^27+5810*q^40+22*q^63+17028*q^31+11260*q^35+1410*q^48+74*q^ 59+10014*q^36+1720*q^47, 1+93223*q^25+29503*q^42+92396*q^22+93960*q^24+89898*q^ 21+93717*q^23+12*q+8648*q^51+23294*q^44+6203*q^53+2971*q^57+79417*q^30+7342*q^ 52+2418*q^58+4355*q^55+1590*q^60+5221*q^54+20562*q^45+79*q^71+26*q^74+48758*q^ 37+274*q^67+83149*q^29+18038*q^46+482*q^65+151*q^69+40538*q^39+1264*q^61+32978* q^41+62124*q^34+10138*q^50+622*q^64+44568*q^38+66652*q^33+3620*q^56+67*q^2+206* q^68+91626*q^26+243*q^3+671*q^4+1530*q^5+3012*q^6+71084*q^32+5296*q^7+8512*q^8+ 12706*q^9+1004*q^62+17867*q^10+23904*q^11+30681*q^12+12*q^76+38033*q^13+45743*q ^14+53592*q^15+61341*q^16+375*q^66+107*q^70+12*q^77+q^78+26266*q^43+11806*q^49+ 86484*q^28+68726*q^17+75476*q^18+81374*q^19+86206*q^20+89350*q^27+58*q^72+36667 *q^40+41*q^73+794*q^63+75364*q^31+24*q^75+57601*q^35+13695*q^48+1957*q^59+53127 *q^36+15749*q^47, 1+308467*q^25+159514*q^42+277319*q^22+13*q^90+301267*q^24+ 260791*q^21+290863*q^23+13*q+66369*q^51+134920*q^44+52325*q^53+30932*q^57+ 302378*q^30+59053*q^52+26826*q^58+40562*q^55+19893*q^60+q^91+46173*q^54+123392* q^45+2681*q^71+1371*q^74+225845*q^37+6033*q^67+308233*q^29+112406*q^46+8683*q^ 65+4083*q^69+198889*q^39+17028*q^61+172355*q^41+264001*q^34+390*q^79+74326*q^50 +10354*q^64+212380*q^38+275354*q^33+35491*q^56+79*q^2+4965*q^68+312504*q^26+311 *q^3+925*q^4+2258*q^5+4739*q^6+285673*q^32+8840*q^7+15012*q^8+23596*q^9+14490*q ^62+34806*q^10+48693*q^11+65145*q^12+855*q^76+83939*q^13+104701*q^14+126993*q^ 15+150314*q^16+7266*q^66+3317*q^70+667*q^77+513*q^78+146985*q^43+82936*q^49+ 312097*q^28+174091*q^17+197685*q^18+220442*q^19+241673*q^20+313622*q^27+2157*q^ 72+185531*q^40+1740*q^73+12278*q^63+294766*q^31+1091*q^75+251807*q^35+286*q^80+ 92176*q^48+113*q^83+216*q^81+62*q^85+44*q^86+28*q^87+26*q^88+13*q^89+84*q^84+ 159*q^82+23121*q^59+239060*q^36+101951*q^47, 1+945165*q^25+738084*q^42+778129*q ^22+907*q^90+896726*q^24+710414*q^21+840680*q^23+14*q+119*q^97+392871*q^51+ 656173*q^44+329596*q^53+223746*q^57+1058218*q^30+298*q^94+360398*q^52+201497*q^ 58+273211*q^55+161787*q^60+703*q^91+300527*q^54+615660*q^45+37895*q^71+23567*q^ 74+167*q^96+89*q^98+928714*q^37+67595*q^67+1052551*q^29+575803*q^46+405*q^93+66 *q^99+88148*q^65+51006*q^69+857234*q^39+144250*q^61+778691*q^41+1013350*q^34+ 9828*q^79+426818*q^50+100192*q^64+47*q^100+894112*q^38+1033197*q^33+247620*q^56 +92*q^2+58827*q^68+985200*q^26+391*q^3+1248*q^4+3251*q^5+7252*q^6+1047784*q^32+ 14318*q^7+25636*q^8+42349*q^9+226*q^95+128200*q^62+30*q^101+65435*q^10+95603*q^ 11+133202*q^12+16838*q^76+178255*q^13+230403*q^14+288972*q^15+353061*q^16+77329 *q^66+44033*q^70+14142*q^77+11815*q^78+697114*q^43+28*q^102+462188*q^49+1038813 *q^28+421535*q^17+493071*q^18+566182*q^19+639209*q^20+1016432*q^27+32467*q^72+ 818582*q^40+27711*q^73+14*q^103+113535*q^63+1056376*q^31+19977*q^75+988875*q^35 +14*q^104+8122*q^80+498909*q^48+q^105+4462*q^83+6693*q^81+2903*q^85+2324*q^86+ 1861*q^87+1469*q^88+1160*q^89+3608*q^84+5470*q^82+180857*q^59+960489*q^36+531*q ^92+536808*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 75, 268, 961, 3467, 12591, 46012, 169088, 624478, 2316582, 8627816, 32247951] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 36, 131, 478, 1730, 6295, 23002, 84549, 312225, 1158300, 4313848, 16123980] The number of ODD permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 39, 137, 483, 1737, 6296, 23010, 84539, 312253, 1158282, 4313968, 16123971] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 36, 137, 483, 1730, 6295, 23010, 84539, 312225, 1158300, 4313968, 16123971] ODD:, [0, 1, 3, 12, 39, 131, 478, 1737, 6296, 23002, 84549, 312253, 1158282, 4313848, 16123980] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.240000000, 8.152985075, 11.82622268, 16.27833862, 21.52013343, 27.56050595, 34.40791186, 42.07035796, 50.55523439, 59.86926216, 70.01852753] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.526782813, 2.226147644, 3.019048736, 3.876069109, 4.784493365, 5.741448608, 6.747102874, 7.801518622, 8.903977564, 10.05318777, 11.24759341, 12.48558071] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.33107, 4.95573, 9.11466, 15.0239, 22.8914, 32.9642, 45.5234, 60.8637, 79.2808, 101.067, 126.508, 155.890] Skewness: , [Float(undefined), 0., 0., -0.08021543222, -0.1941592167, -0.2946622194, -0.3703734745, -0.4237964444, -0.4612881746, -0.4887073934, -0.5100780244, -0.5277424635, -0.5429548683, -0.5563997819, -0.5684299213] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.326456034, 2.465591407, 2.617414321, 2.764233182, 2.885292511, 2.976261244, 3.042815583, 3.092870909, 3.132634081, 3.165958015, 3.195140236, 3.221240714] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 1, 4], [3, 2, 1, 4]}, {[1, 3, 4, 2], [4, 1, 3, 2], [4, 1, 2, 3]}, {[1, 4, 3, 2], [1, 3, 4, 2], [3, 2, 4, 1]}, {[1, 4, 2, 3], [2, 4, 3, 1], [2, 3, 4, 1]}, {[3, 2, 1, 4], [3, 1, 2, 4], [4, 1, 3, 2]}, {[2, 3, 1, 4], [4, 1, 2, 3], [4, 2, 1, 3]}, {[2, 3, 4, 1], [3, 2, 4, 1], [3, 1, 2, 4]}, {[1, 4, 3, 2], [1, 4, 2, 3], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [4, 1, 3, 2], [4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}], [[2, 1], {[0, 2, 0]}, {}], [[1, 4, 2, 3], %1, {1}], [[2, 4, 1, 3], %1, {1}], [[3, 4, 1, 2], %1, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [ [3, 4, 2, 1], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 1, 0, 1, 0]}, {3}], [[2, 4, 3, 1], { [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 1, 0, 1, 0]}, {3}], [[2, 3, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 2, 0, 0, 0]}, {1}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+9*q^3+ 10*q^4+12*q^5+11*q^6+9*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+16*q^3+20*q^4+23*q^5+ 27*q^6+28*q^7+29*q^8+27*q^9+25*q^10+20*q^11+16*q^12+11*q^13+5*q^14+q^15, 1+6*q+ 16*q^2+27*q^3+37*q^4+45*q^5+55*q^6+61*q^7+68*q^8+74*q^9+76*q^10+73*q^11+74*q^12 +69*q^13+62*q^14+54*q^15+45*q^16+36*q^17+27*q^18+16*q^19+6*q^20+q^21, 1+7*q+22* q^2+43*q^3+65*q^4+85*q^5+105*q^6+125*q^7+146*q^8+167*q^9+179*q^10+190*q^11+201* q^12+210*q^13+209*q^14+203*q^15+196*q^16+188*q^17+175*q^18+160*q^19+138*q^20+ 118*q^21+99*q^22+81*q^23+63*q^24+43*q^25+22*q^26+7*q^27+q^28, 1+8*q+29*q^2+65*q ^3+109*q^4+154*q^5+198*q^6+244*q^7+289*q^8+342*q^9+394*q^10+442*q^11+479*q^12+ 513*q^13+539*q^14+565*q^15+585*q^16+596*q^17+595*q^18+586*q^19+566*q^20+542*q^ 21+511*q^22+483*q^23+448*q^24+407*q^25+358*q^26+306*q^27+258*q^28+217*q^29+180* q^30+144*q^31+106*q^32+65*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^2+94*q^3+175*q^4 +268*q^5+364*q^6+464*q^7+566*q^8+677*q^9+796*q^10+923*q^11+1054*q^12+1175*q^13+ 1281*q^14+1375*q^15+1456*q^16+1534*q^17+1618*q^18+1681*q^19+1719*q^20+1731*q^21 +1722*q^22+1709*q^23+1694*q^24+1659*q^25+1592*q^26+1507*q^27+1415*q^28+1324*q^ 29+1236*q^30+1139*q^31+1038*q^32+921*q^33+795*q^34+673*q^35+566*q^36+475*q^37+ 397*q^38+324*q^39+250*q^40+171*q^41+94*q^42+37*q^43+9*q^44+q^45, 1+10*q+46*q^2+ 131*q^3+270*q^4+449*q^5+649*q^6+862*q^7+1086*q^8+1322*q^9+1581*q^10+1861*q^11+ 2167*q^12+2484*q^13+2811*q^14+3120*q^15+3403*q^16+3665*q^17+3930*q^18+4184*q^19 +4419*q^20+4624*q^21+4800*q^22+4948*q^23+5066*q^24+5136*q^25+5161*q^26+5144*q^ 27+5101*q^28+5028*q^29+4927*q^30+4790*q^31+4621*q^32+4430*q^33+4210*q^34+3950*q ^35+3670*q^36+3395*q^37+3139*q^38+2885*q^39+2626*q^40+2349*q^41+2054*q^42+1754* q^43+1478*q^44+1241*q^45+1041*q^46+872*q^47+721*q^48+574*q^49+421*q^50+265*q^51 +131*q^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+177*q^3+402*q^4+726*q^5+1121*q^6+ 1562*q^7+2039*q^8+2546*q^9+3096*q^10+3692*q^11+4352*q^12+5076*q^13+5856*q^14+ 6662*q^15+7468*q^16+8254*q^17+9035*q^18+9798*q^19+10529*q^20+11248*q^21+11949*q ^22+12620*q^23+13263*q^24+13851*q^25+14345*q^26+14722*q^27+15018*q^28+15263*q^ 29+15459*q^30+15582*q^31+15613*q^32+15551*q^33+15392*q^34+15117*q^35+14743*q^36 +14324*q^37+13895*q^38+13434*q^39+12915*q^40+12315*q^41+11633*q^42+10895*q^43+ 10151*q^44+9406*q^45+8658*q^46+7940*q^47+7256*q^48+6599*q^49+5933*q^50+5241*q^ 51+4535*q^52+3855*q^53+3243*q^54+2721*q^55+2282*q^56+1913*q^57+1593*q^58+1295*q ^59+995*q^60+686*q^61+396*q^62+177*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+233 *q^3+580*q^4+1136*q^5+1877*q^6+2757*q^7+3744*q^8+4818*q^9+5982*q^10+7250*q^11+ 8641*q^12+10179*q^13+11874*q^14+13737*q^15+15713*q^16+17748*q^17+19807*q^18+ 21886*q^19+23955*q^20+26012*q^21+28037*q^22+30064*q^23+32126*q^24+34188*q^25+ 36178*q^26+38021*q^27+39679*q^28+41197*q^29+42599*q^30+43853*q^31+44956*q^32+ 45924*q^33+46706*q^34+47263*q^35+47599*q^36+47698*q^37+47599*q^38+47342*q^39+ 46921*q^40+46333*q^41+45537*q^42+44515*q^43+43316*q^44+41984*q^45+40507*q^46+ 38893*q^47+37225*q^48+35550*q^49+33836*q^50+32007*q^51+30026*q^52+27929*q^53+ 25812*q^54+23772*q^55+21835*q^56+19980*q^57+18192*q^58+16492*q^59+14851*q^60+ 13219*q^61+11571*q^62+9954*q^63+8447*q^64+7110*q^65+5966*q^66+5003*q^67+4195*q^ 68+3506*q^69+2888*q^70+2290*q^71+1681*q^72+1082*q^73+573*q^74+233*q^75+67*q^76+ 12*q^77+q^78, 1+13*q+79*q^2+300*q^3+814*q^4+1725*q^5+3051*q^6+4738*q^7+6719*q^8 +8943*q^9+11386*q^10+14062*q^11+16995*q^12+20238*q^13+23818*q^14+27786*q^15+ 32150*q^16+36885*q^17+41899*q^18+47125*q^19+52465*q^20+57863*q^21+63292*q^22+ 68782*q^23+74371*q^24+80087*q^25+85902*q^26+91725*q^27+97437*q^28+102996*q^29+ 108351*q^30+113471*q^31+118344*q^32+122954*q^33+127214*q^34+131097*q^35+134608* q^36+137763*q^37+140570*q^38+143006*q^39+144967*q^40+146421*q^41+147317*q^42+ 147627*q^43+147374*q^44+146683*q^45+145587*q^46+144056*q^47+142120*q^48+139828* q^49+137141*q^50+133998*q^51+130376*q^52+126325*q^53+121940*q^54+117326*q^55+ 112545*q^56+107643*q^57+102630*q^58+97498*q^59+92264*q^60+86986*q^61+81638*q^62 +76188*q^63+70651*q^64+65115*q^65+59712*q^66+54582*q^67+49805*q^68+45322*q^69+ 41061*q^70+36975*q^71+33025*q^72+29153*q^73+25364*q^74+21759*q^75+18469*q^76+ 15570*q^77+13078*q^78+10969*q^79+9198*q^80+7701*q^81+6394*q^82+5178*q^83+3971*q ^84+2763*q^85+1655*q^86+806*q^87+300*q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+ 379*q^3+1115*q^4+2549*q^5+4823*q^6+7931*q^7+11780*q^8+16267*q^9+21314*q^10+ 26905*q^11+33067*q^12+39876*q^13+47406*q^14+55751*q^15+65011*q^16+75228*q^17+ 86404*q^18+98485*q^19+111319*q^20+124675*q^21+138397*q^22+152407*q^23+166773*q^ 24+181541*q^25+196714*q^26+212260*q^27+228072*q^28+244088*q^29+260279*q^30+ 276528*q^31+292646*q^32+308453*q^33+323740*q^34+338395*q^35+352406*q^36+365815* q^37+378654*q^38+391000*q^39+402752*q^40+413712*q^41+423679*q^42+432473*q^43+ 439958*q^44+446199*q^45+451281*q^46+455180*q^47+457892*q^48+459564*q^49+460239* q^50+459802*q^51+458046*q^52+454869*q^53+450273*q^54+444467*q^55+437618*q^56+ 429809*q^57+421135*q^58+411676*q^59+401445*q^60+390457*q^61+378722*q^62+366175* q^63+352768*q^64+338621*q^65+323949*q^66+309056*q^67+294186*q^68+279433*q^69+ 264766*q^70+250137*q^71+235440*q^72+220592*q^73+205690*q^74+190950*q^75+176521* q^76+162469*q^77+148849*q^78+135790*q^79+123495*q^80+112089*q^81+101522*q^82+ 91562*q^83+82008*q^84+72764*q^85+63834*q^86+55324*q^87+47424*q^88+40308*q^89+ 34053*q^90+28650*q^91+24047*q^92+20167*q^93+16899*q^94+14095*q^95+11572*q^96+ 9149*q^97+6734*q^98+4418*q^99+2461*q^100+1106*q^101+379*q^102+92*q^103+14*q^104 +q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+9 *q^7+10*q^6+12*q^5+11*q^4+9*q^3+7*q^2+4*q+1, q^15+5*q^14+11*q^13+16*q^12+20*q^ 11+23*q^10+27*q^9+28*q^8+29*q^7+27*q^6+25*q^5+20*q^4+16*q^3+11*q^2+5*q+1, q^21+ 6*q^20+16*q^19+27*q^18+37*q^17+45*q^16+55*q^15+61*q^14+68*q^13+74*q^12+76*q^11+ 73*q^10+74*q^9+69*q^8+62*q^7+54*q^6+45*q^5+36*q^4+27*q^3+16*q^2+6*q+1, 1+43*q^ 25+105*q^22+65*q^24+125*q^21+85*q^23+7*q+22*q^2+22*q^26+43*q^3+63*q^4+81*q^5+99 *q^6+118*q^7+138*q^8+160*q^9+175*q^10+188*q^11+196*q^12+203*q^13+209*q^14+210*q ^15+201*q^16+q^28+190*q^17+179*q^18+167*q^19+146*q^20+7*q^27, 1+442*q^25+539*q^ 22+479*q^24+565*q^21+513*q^23+8*q+198*q^30+244*q^29+29*q^34+65*q^33+29*q^2+394* q^26+65*q^3+106*q^4+144*q^5+180*q^6+109*q^32+217*q^7+258*q^8+306*q^9+358*q^10+ 407*q^11+448*q^12+483*q^13+511*q^14+542*q^15+566*q^16+289*q^28+586*q^17+595*q^ 18+596*q^19+585*q^20+342*q^27+154*q^31+8*q^35+q^36, 1+1719*q^25+94*q^42+1709*q^ 22+1731*q^24+1694*q^21+1722*q^23+9*q+9*q^44+1375*q^30+q^45+566*q^37+1456*q^29+ 364*q^39+175*q^41+923*q^34+464*q^38+1054*q^33+37*q^2+1681*q^26+94*q^3+171*q^4+ 250*q^5+324*q^6+1175*q^32+397*q^7+475*q^8+566*q^9+673*q^10+795*q^11+921*q^12+ 1038*q^13+1139*q^14+1236*q^15+1324*q^16+37*q^43+1534*q^28+1415*q^17+1507*q^18+ 1592*q^19+1659*q^20+1618*q^27+268*q^40+1281*q^31+796*q^35+677*q^36, 1+4927*q^25 +2484*q^42+4430*q^22+4790*q^24+4210*q^21+4621*q^23+10*q+270*q^51+1861*q^44+46*q ^53+5136*q^30+131*q^52+q^55+10*q^54+1581*q^45+3930*q^37+5161*q^29+1322*q^46+ 3403*q^39+2811*q^41+4624*q^34+449*q^50+3665*q^38+4800*q^33+46*q^2+5028*q^26+131 *q^3+265*q^4+421*q^5+574*q^6+4948*q^32+721*q^7+872*q^8+1041*q^9+1241*q^10+1478* q^11+1754*q^12+2054*q^13+2349*q^14+2626*q^15+2885*q^16+2167*q^43+649*q^49+5144* q^28+3139*q^17+3395*q^18+3670*q^19+3950*q^20+5101*q^27+3120*q^40+5066*q^31+4419 *q^35+862*q^48+4184*q^36+1086*q^47, 1+12315*q^25+13263*q^42+10151*q^22+11633*q^ 24+9406*q^21+10895*q^23+11*q+6662*q^51+11949*q^44+5076*q^53+2546*q^57+14743*q^ 30+5856*q^52+2039*q^58+3692*q^55+1121*q^60+4352*q^54+11248*q^45+15263*q^37+ 14324*q^29+10529*q^46+11*q^65+14722*q^39+726*q^61+13851*q^41+15613*q^34+7468*q^ 50+56*q^64+15018*q^38+15551*q^33+3096*q^56+56*q^2+12915*q^26+177*q^3+396*q^4+ 686*q^5+995*q^6+15392*q^32+1295*q^7+1593*q^8+1913*q^9+402*q^62+2282*q^10+2721*q ^11+3243*q^12+3855*q^13+4535*q^14+5241*q^15+5933*q^16+q^66+12620*q^43+8254*q^49 +13895*q^28+6599*q^17+7256*q^18+7940*q^19+8658*q^20+13434*q^27+14345*q^40+177*q ^63+15117*q^31+15582*q^35+9035*q^48+1562*q^59+15459*q^36+9798*q^47, 1+27929*q^ 25+47599*q^42+21835*q^22+25812*q^24+19980*q^21+23772*q^23+12*q+38021*q^51+46706 *q^44+34188*q^53+26012*q^57+37225*q^30+36178*q^52+23955*q^58+30064*q^55+19807*q ^60+32126*q^54+45924*q^45+2757*q^71+580*q^74+46333*q^37+7250*q^67+35550*q^29+ 44956*q^46+10179*q^65+4818*q^69+47342*q^39+17748*q^61+47698*q^41+43316*q^34+ 39679*q^50+11874*q^64+46921*q^38+41984*q^33+28037*q^56+67*q^2+5982*q^68+30026*q ^26+233*q^3+573*q^4+1082*q^5+1681*q^6+40507*q^32+2290*q^7+2888*q^8+3506*q^9+ 15713*q^62+4195*q^10+5003*q^11+5966*q^12+67*q^76+7110*q^13+8447*q^14+9954*q^15+ 11571*q^16+8641*q^66+3744*q^70+12*q^77+q^78+47263*q^43+41197*q^49+33836*q^28+ 13219*q^17+14851*q^18+16492*q^19+18192*q^20+32007*q^27+1877*q^72+47599*q^40+ 1136*q^73+13737*q^63+38893*q^31+233*q^75+44515*q^35+42599*q^48+21886*q^59+45537 *q^36+43853*q^47, 1+59712*q^25+139828*q^42+45322*q^22+13*q^90+54582*q^24+41061* q^21+49805*q^23+13*q+144967*q^51+144056*q^44+140570*q^53+127214*q^57+86986*q^30 +143006*q^52+122954*q^58+134608*q^55+113471*q^60+q^91+137763*q^54+145587*q^45+ 52465*q^71+36885*q^74+121940*q^37+74371*q^67+81638*q^29+146683*q^46+85902*q^65+ 63292*q^69+130376*q^39+108351*q^61+137141*q^41+107643*q^34+16995*q^79+146421*q^ 50+91725*q^64+126325*q^38+102630*q^33+131097*q^56+79*q^2+68782*q^68+65115*q^26+ 300*q^3+806*q^4+1655*q^5+2763*q^6+97498*q^32+3971*q^7+5178*q^8+6394*q^9+102996* q^62+7701*q^10+9198*q^11+10969*q^12+27786*q^76+13078*q^13+15570*q^14+18469*q^15 +21759*q^16+80087*q^66+57863*q^70+23818*q^77+20238*q^78+142120*q^43+147317*q^49 +76188*q^28+25364*q^17+29153*q^18+33025*q^19+36975*q^20+70651*q^27+47125*q^72+ 133998*q^40+41899*q^73+97437*q^63+92264*q^31+32150*q^75+112545*q^35+14062*q^80+ 147627*q^48+6719*q^83+11386*q^81+3051*q^85+1725*q^86+814*q^87+300*q^88+79*q^89+ 4738*q^84+8943*q^82+118344*q^59+117326*q^36+147374*q^47, 1+123495*q^25+366175*q ^42+91562*q^22+55751*q^90+112089*q^24+82008*q^21+101522*q^23+14*q+11780*q^97+ 450273*q^51+390457*q^44+458046*q^53+457892*q^57+190950*q^30+26905*q^94+454869*q ^52+455180*q^58+460239*q^55+446199*q^60+47406*q^91+459802*q^54+401445*q^45+ 323740*q^71+276528*q^74+16267*q^96+7931*q^98+294186*q^37+378654*q^67+176521*q^ 29+411676*q^46+33067*q^93+4823*q^99+402752*q^65+352406*q^69+323949*q^39+439958* q^61+352768*q^41+250137*q^34+196714*q^79+444467*q^50+413712*q^64+2549*q^100+ 309056*q^38+235440*q^33+459564*q^56+92*q^2+365815*q^68+135790*q^26+379*q^3+1106 *q^4+2461*q^5+4418*q^6+220592*q^32+6734*q^7+9149*q^8+11572*q^9+21314*q^95+ 432473*q^62+1115*q^101+14095*q^10+16899*q^11+20167*q^12+244088*q^76+24047*q^13+ 28650*q^14+34053*q^15+40308*q^16+391000*q^66+338395*q^70+228072*q^77+212260*q^ 78+378722*q^43+379*q^102+437618*q^49+162469*q^28+47424*q^17+55324*q^18+63834*q^ 19+72764*q^20+148849*q^27+308453*q^72+338621*q^40+292646*q^73+92*q^103+423679*q ^63+205690*q^31+260279*q^75+264766*q^35+14*q^104+181541*q^80+429809*q^48+q^105+ 138397*q^83+166773*q^81+111319*q^85+98485*q^86+86404*q^87+75228*q^88+65011*q^89 +124675*q^84+152407*q^82+451281*q^59+279433*q^36+39876*q^92+421135*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 2, 6, 21, 75, 265, 929, 3249, 11362, 39746, 139060, 486549, 1702349, 5956172, 20839367] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [1, 1, 3, 10, 37, 134, 467, 1622, 5675, 19875, 69540, 243279, 851167, 2978061, 10419664] The number of ODD permutations avoiding, {[1, 3, 4, 2], [4, 1, 3, 2], [4, 1, 2, 3]}, is given by [0, 1, 3, 11, 38, 131, 462, 1627, 5687, 19871, 69520, 243270, 851182, 2978111, 10419703] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 37, 131, 462, 1622, 5675, 19871, 69520, 243279, 851167, 2978111, 10419703] ODD:, [0, 1, 3, 11, 38, 134, 467, 1627, 5687, 19875, 69540, 243270, 851182, 2978061, 10419664] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.013333333, 7.520754717, 10.49300323, 13.91535857, 17.78727337, 22.11258995, 26.89381562, 32.13155098, 37.82559863, 43.97572082, 50.58182319] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.543033500, 2.300685887, 3.235496433, 4.327821298, 5.552062652, 6.889558261, 8.330422754, 9.869448436, 11.50257372, 13.22567805, 15.03465592, 16.92569413] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.38095, 5.29316, 10.4684, 18.7300, 30.8254, 47.4660, 69.3959, 97.4060, 132.309, 174.919, 226.041, 286.479] -5 Skewness: , [Float(undefined), 0., 0., -0.2721915177 10 , -0.01629180244, -0.01575121141, 0.0007636317604, 0.02173192156, 0.03938600310, 0.05149827762, 0.05887906244, 0.06305371576, 0.06523098050, 0.06616370058, 0.06629111550] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.234375069, 2.256361326, 2.258599101, 2.276372961, 2.311658369, 2.355854487, 2.400561019, 2.441344196, 2.477140054, 2.508401526, 2.536024711, 2.560673123] end of this data For the equivalence class of patterns, { {[1, 3, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, {[1, 4, 3, 2], [3, 1, 2, 4], [4, 2, 3, 1]}, {[1, 4, 3, 2], [2, 3, 1, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [2, 3, 4, 1], [4, 2, 1, 3]}, {[1, 3, 2, 4], [2, 3, 4, 1], [4, 1, 3, 2]}, {[1, 4, 2, 3], [3, 2, 1, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [2, 4, 3, 1], [4, 1, 2, 3]}, {[1, 3, 2, 4], [3, 2, 4, 1], [4, 1, 2, 3]}} the member , {[1, 3, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {3}], [ [3, 1, 4, 2, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 2, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[3, 5, 4, 1, 2], { [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {4}], [[4, 3, 1, 2], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 1, 0]}, {3}], [[2, 1, 3], {[1, 1, 0, 1]}, {}], [[4, 3, 2, 1], %1, {1}], [[4, 2, 1, 3], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [1, 1, 0, 0, 1]}, {1}], [[2, 5, 3, 1, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], %1, {1}], [[2, 5, 4, 1, 3], { [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[3, 5, 4, 2, 1], {[1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 2, 5, 4, 1], { [1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {2}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 1, 5, 2, 4], { [0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {2}], [[4, 1, 5, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {3}], [ [2, 1, 5, 4, 3], {[1, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {2}], [[1, 3, 2], {[1, 0, 1, 1], [0, 1, 0, 1]}, {}], [[3, 2, 1], {[1, 0, 1, 0], [0, 0, 0, 1]}, {}], [[4, 1, 5, 2, 3], { [0, 0, 1, 0, 1, 0], [0, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [[1, 4, 2, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 1]}, {3}], [[3, 2, 4, 1], %1, {3}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1], [1, 0, 0, 1, 0]}, {}], [[3, 1, 2], {[0, 1, 0, 1], [1, 0, 0, 0]}, {2}], [[3, 1, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 3, 1], {[1, 0, 1, 0]}, {}], [[2, 4, 3, 1, 5], {[0, 0, 0, 0, 0, 0]}, {1}]} %1 := {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [1, 0, 1, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+7*q^2+9*q^3+ 10*q^4+13*q^5+12*q^6+8*q^7+5*q^8+2*q^9+q^10, 1+5*q+11*q^2+16*q^3+19*q^4+20*q^5+ 27*q^6+28*q^7+24*q^8+26*q^9+21*q^10+15*q^11+8*q^12+5*q^13+2*q^14+q^15, 1+6*q+16 *q^2+27*q^3+35*q^4+39*q^5+41*q^6+53*q^7+60*q^8+57*q^9+53*q^10+53*q^11+58*q^12+ 55*q^13+44*q^14+33*q^15+24*q^16+15*q^17+8*q^18+5*q^19+2*q^20+q^21, 1+7*q+22*q^2 +43*q^3+62*q^4+74*q^5+80*q^6+85*q^7+104*q^8+123*q^9+126*q^10+121*q^11+114*q^12+ 117*q^13+124*q^14+122*q^15+119*q^16+117*q^17+109*q^18+90*q^19+70*q^20+51*q^21+ 36*q^22+24*q^23+15*q^24+8*q^25+5*q^26+2*q^27+q^28, 1+8*q+29*q^2+65*q^3+105*q^4+ 136*q^5+154*q^6+165*q^7+177*q^8+208*q^9+248*q^10+269*q^11+268*q^12+257*q^13+249 *q^14+261*q^15+277*q^16+273*q^17+261*q^18+250*q^19+256*q^20+266*q^21+261*q^22+ 238*q^23+205*q^24+167*q^25+133*q^26+105*q^27+77*q^28+54*q^29+36*q^30+24*q^31+15 *q^32+8*q^33+5*q^34+2*q^35+q^36, 1+9*q+37*q^2+94*q^3+170*q^4+241*q^5+290*q^6+ 319*q^7+342*q^8+370*q^9+424*q^10+500*q^11+560*q^12+584*q^13+575*q^14+554*q^15+ 558*q^16+577*q^17+612*q^18+625*q^19+608*q^20+571*q^21+553*q^22+565*q^23+579*q^ 24+582*q^25+578*q^26+573*q^27+545*q^28+497*q^29+430*q^30+373*q^31+311*q^32+248* q^33+191*q^34+148*q^35+112*q^36+80*q^37+54*q^38+36*q^39+24*q^40+15*q^41+8*q^42+ 5*q^43+2*q^44+q^45, 1+10*q+46*q^2+131*q^3+264*q^4+411*q^5+531*q^6+609*q^7+661*q ^8+712*q^9+776*q^10+876*q^11+1015*q^12+1155*q^13+1244*q^14+1267*q^15+1246*q^16+ 1234*q^17+1245*q^18+1285*q^19+1356*q^20+1417*q^21+1411*q^22+1346*q^23+1285*q^24 +1260*q^25+1289*q^26+1317*q^27+1306*q^28+1280*q^29+1261*q^30+1265*q^31+1269*q^ 32+1261*q^33+1217*q^34+1122*q^35+999*q^36+883*q^37+768*q^38+653*q^39+537*q^40+ 438*q^41+354*q^42+272*q^43+206*q^44+155*q^45+115*q^46+80*q^47+54*q^48+36*q^49+ 24*q^50+15*q^51+8*q^52+5*q^53+2*q^54+q^55, 1+11*q+56*q^2+177*q^3+395*q^4+675*q^ 5+942*q^6+1140*q^7+1270*q^8+1373*q^9+1488*q^10+1631*q^11+1824*q^12+2082*q^13+ 2374*q^14+2613*q^15+2742*q^16+2769*q^17+2754*q^18+2756*q^19+2798*q^20+2890*q^21 +3042*q^22+3178*q^23+3232*q^24+3164*q^25+3046*q^26+2954*q^27+2925*q^28+2969*q^ 29+3031*q^30+3051*q^31+2985*q^32+2892*q^33+2821*q^34+2822*q^35+2835*q^36+2851*q ^37+2861*q^38+2846*q^39+2762*q^40+2590*q^41+2392*q^42+2188*q^43+1987*q^44+1760* q^45+1525*q^46+1304*q^47+1110*q^48+919*q^49+751*q^50+602*q^51+481*q^52+378*q^53 +287*q^54+213*q^55+158*q^56+115*q^57+80*q^58+54*q^59+36*q^60+24*q^61+15*q^62+8* q^63+5*q^64+2*q^65+q^66, 1+12*q+67*q^2+233*q^3+572*q^4+1070*q^5+1617*q^6+2082*q ^7+2410*q^8+2643*q^9+2861*q^10+3119*q^11+3431*q^12+3817*q^13+4307*q^14+4888*q^ 15+5444*q^16+5849*q^17+6052*q^18+6106*q^19+6129*q^20+6196*q^21+6332*q^22+6551*q ^23+6843*q^24+7156*q^25+7353*q^26+7346*q^27+7201*q^28+7007*q^29+6874*q^30+6836* q^31+6906*q^32+7059*q^33+7154*q^34+7082*q^35+6873*q^36+6646*q^37+6521*q^38+6516 *q^39+6550*q^40+6537*q^41+6470*q^42+6423*q^43+6408*q^44+6377*q^45+6288*q^46+ 6107*q^47+5820*q^48+5458*q^49+5078*q^50+4681*q^51+4258*q^52+3798*q^53+3347*q^54 +2918*q^55+2530*q^56+2171*q^57+1834*q^58+1526*q^59+1251*q^60+1017*q^61+816*q^62 +645*q^63+505*q^64+393*q^65+294*q^66+216*q^67+158*q^68+115*q^69+80*q^70+54*q^71 +36*q^72+24*q^73+15*q^74+8*q^75+5*q^76+2*q^77+q^78, 1+13*q+79*q^2+300*q^3+805*q ^4+1642*q^5+2687*q^6+3699*q^7+4492*q^8+5053*q^9+5504*q^10+5980*q^11+6550*q^12+ 7221*q^13+8010*q^14+8964*q^15+10104*q^16+11305*q^17+12339*q^18+13036*q^19+13383 *q^20+13553*q^21+13732*q^22+13999*q^23+14383*q^24+14878*q^25+15484*q^26+16143*q ^27+16699*q^28+16950*q^29+16868*q^30+16566*q^31+16265*q^32+16099*q^33+16090*q^ 34+16260*q^35+16547*q^36+16789*q^37+16783*q^38+16498*q^39+16035*q^40+15607*q^41 +15356*q^42+15331*q^43+15413*q^44+15427*q^45+15272*q^46+14992*q^47+14755*q^48+ 14591*q^49+14485*q^50+14398*q^51+14359*q^52+14272*q^53+13989*q^54+13446*q^55+ 12776*q^56+12066*q^57+11321*q^58+10508*q^59+9654*q^60+8796*q^61+7912*q^62+7059* q^63+6252*q^64+5501*q^65+4769*q^66+4091*q^67+3470*q^68+2945*q^69+2475*q^70+2045 *q^71+1667*q^72+1349*q^73+1082*q^74+859*q^75+669*q^76+520*q^77+400*q^78+297*q^ 79+216*q^80+158*q^81+115*q^82+80*q^83+54*q^84+36*q^85+24*q^86+15*q^87+8*q^88+5* q^89+2*q^90+q^91, 1+14*q+92*q^2+379*q^3+1105*q^4+2447*q^5+4329*q^6+6386*q^7+ 8191*q^8+9545*q^9+10557*q^10+11484*q^11+12530*q^12+13771*q^13+15201*q^14+16832* q^15+18731*q^16+20977*q^17+23465*q^18+25846*q^19+27747*q^20+28984*q^21+29706*q^ 22+30260*q^23+30890*q^24+31710*q^25+32707*q^26+33833*q^27+35113*q^28+36552*q^29 +37938*q^30+38890*q^31+39174*q^32+38902*q^33+38427*q^34+38077*q^35+37979*q^36+ 38095*q^37+38441*q^38+39003*q^39+39557*q^40+39788*q^41+39515*q^42+38804*q^43+ 37896*q^44+37089*q^45+36594*q^46+36529*q^47+36744*q^48+36852*q^49+36615*q^50+ 36025*q^51+35265*q^52+34625*q^53+34201*q^54+33895*q^55+33573*q^56+33246*q^57+ 33026*q^58+32890*q^59+32686*q^60+32205*q^61+31363*q^62+30240*q^63+28942*q^64+ 27551*q^65+26145*q^66+24686*q^67+23057*q^68+21268*q^69+19423*q^70+17685*q^71+ 16045*q^72+14451*q^73+12878*q^74+11363*q^75+9953*q^76+8680*q^77+7528*q^78+6486* q^79+5523*q^80+4647*q^81+3880*q^82+3237*q^83+2686*q^84+2186*q^85+1765*q^86+1414 *q^87+1125*q^88+883*q^89+684*q^90+527*q^91+403*q^92+297*q^93+216*q^94+158*q^95+ 115*q^96+80*q^97+54*q^98+36*q^99+24*q^100+15*q^101+8*q^102+5*q^103+2*q^104+q^ 105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+9 *q^7+10*q^6+13*q^5+12*q^4+8*q^3+5*q^2+2*q+1, q^15+5*q^14+11*q^13+16*q^12+19*q^ 11+20*q^10+27*q^9+28*q^8+24*q^7+26*q^6+21*q^5+15*q^4+8*q^3+5*q^2+2*q+1, q^21+6* q^20+16*q^19+27*q^18+35*q^17+39*q^16+41*q^15+53*q^14+60*q^13+57*q^12+53*q^11+53 *q^10+58*q^9+55*q^8+44*q^7+33*q^6+24*q^5+15*q^4+8*q^3+5*q^2+2*q+1, 1+43*q^25+80 *q^22+62*q^24+85*q^21+74*q^23+2*q+5*q^2+22*q^26+8*q^3+15*q^4+24*q^5+36*q^6+51*q ^7+70*q^8+90*q^9+109*q^10+117*q^11+119*q^12+122*q^13+124*q^14+117*q^15+114*q^16 +q^28+121*q^17+126*q^18+123*q^19+104*q^20+7*q^27, 1+269*q^25+249*q^22+268*q^24+ 261*q^21+257*q^23+2*q+154*q^30+165*q^29+29*q^34+65*q^33+5*q^2+248*q^26+8*q^3+15 *q^4+24*q^5+36*q^6+105*q^32+54*q^7+77*q^8+105*q^9+133*q^10+167*q^11+205*q^12+ 238*q^13+261*q^14+266*q^15+256*q^16+177*q^28+250*q^17+261*q^18+273*q^19+277*q^ 20+208*q^27+136*q^31+8*q^35+q^36, 1+608*q^25+94*q^42+565*q^22+571*q^24+579*q^21 +553*q^23+2*q+9*q^44+554*q^30+q^45+342*q^37+558*q^29+290*q^39+170*q^41+500*q^34 +319*q^38+560*q^33+5*q^2+625*q^26+8*q^3+15*q^4+24*q^5+36*q^6+584*q^32+54*q^7+80 *q^8+112*q^9+148*q^10+191*q^11+248*q^12+311*q^13+373*q^14+430*q^15+497*q^16+37* q^43+577*q^28+545*q^17+573*q^18+578*q^19+582*q^20+612*q^27+241*q^40+575*q^31+ 424*q^35+370*q^36, 1+1261*q^25+1155*q^42+1261*q^22+1265*q^24+1217*q^21+1269*q^ 23+2*q+264*q^51+876*q^44+46*q^53+1260*q^30+131*q^52+q^55+10*q^54+776*q^45+1245* q^37+1289*q^29+712*q^46+1246*q^39+1244*q^41+1417*q^34+411*q^50+1234*q^38+1411*q ^33+5*q^2+1280*q^26+8*q^3+15*q^4+24*q^5+36*q^6+1346*q^32+54*q^7+80*q^8+115*q^9+ 155*q^10+206*q^11+272*q^12+354*q^13+438*q^14+537*q^15+653*q^16+1015*q^43+531*q^ 49+1317*q^28+768*q^17+883*q^18+999*q^19+1122*q^20+1306*q^27+1267*q^40+1285*q^31 +1356*q^35+609*q^48+1285*q^36+661*q^47, 1+2590*q^25+3232*q^42+1987*q^22+2392*q^ 24+1760*q^21+2188*q^23+2*q+2613*q^51+3042*q^44+2082*q^53+1373*q^57+2835*q^30+ 2374*q^52+1270*q^58+1631*q^55+942*q^60+1824*q^54+2890*q^45+2969*q^37+2851*q^29+ 2798*q^46+11*q^65+2954*q^39+675*q^61+3164*q^41+2985*q^34+2742*q^50+56*q^64+2925 *q^38+2892*q^33+1488*q^56+5*q^2+2762*q^26+8*q^3+15*q^4+24*q^5+36*q^6+2821*q^32+ 54*q^7+80*q^8+115*q^9+395*q^62+158*q^10+213*q^11+287*q^12+378*q^13+481*q^14+602 *q^15+751*q^16+q^66+3178*q^43+2769*q^49+2861*q^28+919*q^17+1110*q^18+1304*q^19+ 1525*q^20+2846*q^27+3046*q^40+177*q^63+2822*q^31+3051*q^35+2754*q^48+1140*q^59+ 3031*q^36+2756*q^47, 1+3798*q^25+6873*q^42+2530*q^22+3347*q^24+2171*q^21+2918*q ^23+2*q+7346*q^51+7154*q^44+7156*q^53+6196*q^57+5820*q^30+7353*q^52+6129*q^58+ 6551*q^55+6052*q^60+6843*q^54+7059*q^45+2082*q^71+572*q^74+6537*q^37+3119*q^67+ 5458*q^29+6906*q^46+3817*q^65+2643*q^69+6516*q^39+5849*q^61+6646*q^41+6408*q^34 +7201*q^50+4307*q^64+6550*q^38+6377*q^33+6332*q^56+5*q^2+2861*q^68+4258*q^26+8* q^3+15*q^4+24*q^5+36*q^6+6288*q^32+54*q^7+80*q^8+115*q^9+5444*q^62+158*q^10+216 *q^11+294*q^12+67*q^76+393*q^13+505*q^14+645*q^15+816*q^16+3431*q^66+2410*q^70+ 12*q^77+q^78+7082*q^43+7007*q^49+5078*q^28+1017*q^17+1251*q^18+1526*q^19+1834*q ^20+4681*q^27+1617*q^72+6521*q^40+1070*q^73+4888*q^63+6107*q^31+233*q^75+6423*q ^35+6874*q^48+6106*q^59+6470*q^36+6836*q^47, 1+4769*q^25+14591*q^42+2945*q^22+ 13*q^90+4091*q^24+2475*q^21+3470*q^23+2*q+16035*q^51+14992*q^44+16783*q^53+ 16090*q^57+8796*q^30+16498*q^52+16099*q^58+16547*q^55+16566*q^60+q^91+16789*q^ 54+15272*q^45+13383*q^71+11305*q^74+13989*q^37+14383*q^67+7912*q^29+15427*q^46+ 15484*q^65+13732*q^69+14359*q^39+16868*q^61+14485*q^41+12066*q^34+6550*q^79+ 15607*q^50+16143*q^64+14272*q^38+11321*q^33+16260*q^56+5*q^2+13999*q^68+5501*q^ 26+8*q^3+15*q^4+24*q^5+36*q^6+10508*q^32+54*q^7+80*q^8+115*q^9+16950*q^62+158*q ^10+216*q^11+297*q^12+8964*q^76+400*q^13+520*q^14+669*q^15+859*q^16+14878*q^66+ 13553*q^70+8010*q^77+7221*q^78+14755*q^43+15356*q^49+7059*q^28+1082*q^17+1349*q ^18+1667*q^19+2045*q^20+6252*q^27+13036*q^72+14398*q^40+12339*q^73+16699*q^63+ 9654*q^31+10104*q^75+12776*q^35+5980*q^80+15331*q^48+4492*q^83+5504*q^81+2687*q ^85+1642*q^86+805*q^87+300*q^88+79*q^89+3699*q^84+5053*q^82+16265*q^59+13446*q^ 36+15413*q^47, 1+5523*q^25+30240*q^42+3237*q^22+16832*q^90+4647*q^24+2686*q^21+ 3880*q^23+2*q+8191*q^97+34201*q^51+32205*q^44+35265*q^53+36744*q^57+11363*q^30+ 11484*q^94+34625*q^52+36529*q^58+36615*q^55+37089*q^60+15201*q^91+36025*q^54+ 32686*q^45+38427*q^71+38890*q^74+9545*q^96+6386*q^98+23057*q^37+38441*q^67+9953 *q^29+32890*q^46+12530*q^93+4329*q^99+39557*q^65+37979*q^69+26145*q^39+37896*q^ 61+28942*q^41+17685*q^34+32707*q^79+33895*q^50+39788*q^64+2447*q^100+24686*q^38 +16045*q^33+36852*q^56+5*q^2+38095*q^68+6486*q^26+8*q^3+15*q^4+24*q^5+36*q^6+ 14451*q^32+54*q^7+80*q^8+115*q^9+10557*q^95+38804*q^62+1105*q^101+158*q^10+216* q^11+297*q^12+36552*q^76+403*q^13+527*q^14+684*q^15+883*q^16+39003*q^66+38077*q ^70+35113*q^77+33833*q^78+31363*q^43+379*q^102+33573*q^49+8680*q^28+1125*q^17+ 1414*q^18+1765*q^19+2186*q^20+7528*q^27+38902*q^72+27551*q^40+39174*q^73+92*q^ 103+39515*q^63+12878*q^31+37938*q^75+19423*q^35+14*q^104+31710*q^80+33246*q^48+ q^105+29706*q^83+30890*q^81+27747*q^85+25846*q^86+23465*q^87+20977*q^88+18731*q ^89+28984*q^84+30260*q^82+36594*q^59+21268*q^36+13771*q^92+33026*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [1, 2, 6, 21, 72, 229, 686, 1972, 5514, 15131, 40986, 110013, 293376, 778678, 2059646] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [1, 1, 3, 11, 36, 113, 342, 988, 2758, 7564, 20490, 55012, 146692, 389335, 1029815] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, is given by [0, 1, 3, 10, 36, 116, 344, 984, 2756, 7567, 20496, 55001, 146684, 389343, 1029831] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 36, 116, 344, 988, 2758, 7567, 20496, 55012, 146692, 389343, 1029831] ODD:, [0, 1, 3, 10, 36, 113, 342, 984, 2756, 7564, 20490, 55001, 146684, 389335, 1029815] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.805555556, 7.048034934, 9.676384840, 12.68255578, 16.05404425, 19.77655145, 23.83584639, 28.21898321, 32.91482603, 37.91416863, 43.20961903] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.495268426, 2.177147018, 3.054434542, 4.124900819, 5.373347188, 6.782649444, 8.337137758, 10.02301015, 11.82811855, 13.74167204, 15.75401700, 17.85650049] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.23583, 4.73997, 9.32957, 17.0148, 28.8729, 46.0043, 69.5079, 100.461, 139.904, 188.834, 248.189, 318.855] Skewness: , [Float(undefined), 0., 0., -0.004130812756, 0.01562074159, 0.03173716373, 0.03832193441, 0.04032957115, 0.04211441525, 0.04576554560, 0.05190565896, 0.06039767491, 0.07079661737, 0.08256552507, 0.09520064406] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.384387498, 2.462537180, 2.409534563, 2.341267686, 2.289165626, 2.255575451, 2.236518594, 2.228069845, 2.227385285, 2.232463287, 2.242045097, 2.255152532] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 1, 4], [4, 1, 2, 3]}, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 1, 3, 2]}, {[2, 3, 4, 1], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 4, 3, 2], [3, 2, 4, 1], [3, 1, 2, 4]}, {[1, 4, 3, 2], [2, 3, 1, 4], [4, 2, 1, 3]}, {[1, 4, 2, 3], [2, 4, 3, 1], [3, 2, 1, 4]}, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 1, 2, 3]}, {[1, 4, 2, 3], [2, 3, 4, 1], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [3, 2, 1, 4], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], %2, {1}], [[2, 1, 4, 3, 5], %1, {1}], [[3, 1, 4, 2, 5], %1, {1}], [[2, 1, 3], {}, {}], [[1, 3, 2], {[0, 1, 0, 1]}, {}], [[1, 4, 2, 3], %2, {3}], [[3, 2, 4, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 1, 5, 4, 3], {[0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[4, 3, 1, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {3}], [[2, 5, 4, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[1, 4, 3, 2, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 5, 3, 2, 4], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[1, 5, 4, 2, 3], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [ [1, 5, 4, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}], [[2, 1, 5, 3, 4], %1, {4}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[4, 3, 2, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 0, 0, 1]}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {3}], [[2, 4, 1, 3], %2, {4}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[3, 2, 5, 4, 1], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {}], [[2, 1, 3, 4], %2, {3}], [[4, 2, 5, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 5, 2, 4], %1, {2}], [[4, 2, 1, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {4}]} %1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]} %2 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+9*q^3+ 9*q^4+10*q^5+11*q^6+9*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+16*q^3+18*q^4+17*q^5+ 18*q^6+21*q^7+21*q^8+23*q^9+22*q^10+22*q^11+16*q^12+11*q^13+5*q^14+q^15, 1+6*q+ 16*q^2+27*q^3+34*q^4+35*q^5+32*q^6+32*q^7+35*q^8+39*q^9+39*q^10+40*q^11+46*q^12 +50*q^13+49*q^14+52*q^15+47*q^16+40*q^17+27*q^18+16*q^19+6*q^20+q^21, 1+7*q+22* q^2+43*q^3+61*q^4+69*q^5+67*q^6+60*q^7+57*q^8+61*q^9+66*q^10+69*q^11+71*q^12+69 *q^13+69*q^14+88*q^15+92*q^16+99*q^17+103*q^18+115*q^19+118*q^20+116*q^21+107*q ^22+92*q^23+69*q^24+43*q^25+22*q^26+7*q^27+q^28, 1+8*q+29*q^2+65*q^3+104*q^4+ 130*q^5+136*q^6+127*q^7+112*q^8+103*q^9+107*q^10+116*q^11+121*q^12+126*q^13+121 *q^14+118*q^15+118*q^16+135*q^17+146*q^18+156*q^19+165*q^20+187*q^21+197*q^22+ 232*q^23+242*q^24+254*q^25+263*q^26+274*q^27+264*q^28+246*q^29+212*q^30+168*q^ 31+114*q^32+65*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^2+94*q^3+169*q^4+234*q^5+ 266*q^6+263*q^7+239*q^8+209*q^9+189*q^10+188*q^11+202*q^12+219*q^13+230*q^14+ 224*q^15+213*q^16+199*q^17+196*q^18+220*q^19+238*q^20+264*q^21+260*q^22+267*q^ 23+287*q^24+321*q^25+364*q^26+408*q^27+440*q^28+473*q^29+512*q^30+549*q^31+585* q^32+606*q^33+615*q^34+624*q^35+611*q^36+567*q^37+494*q^38+400*q^39+291*q^40+ 181*q^41+94*q^42+37*q^43+9*q^44+q^45, 1+10*q+46*q^2+131*q^3+263*q^4+403*q^5+500 *q^6+529*q^7+502*q^8+448*q^9+391*q^10+349*q^11+334*q^12+351*q^13+387*q^14+423*q ^15+429*q^16+411*q^17+375*q^18+339*q^19+321*q^20+365*q^21+404*q^22+445*q^23+462 *q^24+449*q^25+429*q^26+465*q^27+497*q^28+571*q^29+625*q^30+692*q^31+718*q^32+ 797*q^33+870*q^34+966*q^35+1047*q^36+1114*q^37+1165*q^38+1246*q^39+1306*q^40+ 1386*q^41+1431*q^42+1453*q^43+1442*q^44+1415*q^45+1310*q^46+1158*q^47+950*q^48+ 720*q^49+483*q^50+277*q^51+131*q^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+177*q^3 +394*q^4+666*q^5+903*q^6+1029*q^7+1031*q^8+950*q^9+839*q^10+732*q^11+647*q^12+ 601*q^13+612*q^14+674*q^15+762*q^16+812*q^17+803*q^18+743*q^19+667*q^20+594*q^ 21+558*q^22+595*q^23+670*q^24+768*q^25+818*q^26+827*q^27+776*q^28+750*q^29+745* q^30+790*q^31+922*q^32+1028*q^33+1090*q^34+1143*q^35+1183*q^36+1245*q^37+1385*q ^38+1551*q^39+1720*q^40+1879*q^41+1994*q^42+2070*q^43+2215*q^44+2373*q^45+2544* q^46+2703*q^47+2839*q^48+2994*q^49+3150*q^50+3290*q^51+3389*q^52+3430*q^53+3386 *q^54+3265*q^55+3054*q^56+2705*q^57+2265*q^58+1755*q^59+1243*q^60+773*q^61+410* q^62+177*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+233*q^3+571*q^4+1060*q^5+1569 *q^6+1932*q^7+2060*q^8+1981*q^9+1789*q^10+1571*q^11+1370*q^12+1203*q^13+1093*q^ 14+1076*q^15+1169*q^16+1346*q^17+1496*q^18+1548*q^19+1480*q^20+1349*q^21+1203*q ^22+1072*q^23+975*q^24+998*q^25+1108*q^26+1299*q^27+1458*q^28+1538*q^29+1495*q^ 30+1385*q^31+1268*q^32+1244*q^33+1299*q^34+1507*q^35+1721*q^36+1907*q^37+1937*q ^38+1965*q^39+1944*q^40+2026*q^41+2169*q^42+2417*q^43+2708*q^44+3020*q^45+3213* q^46+3398*q^47+3546*q^48+3768*q^49+4049*q^50+4378*q^51+4646*q^52+4928*q^53+5138 *q^54+5426*q^55+5757*q^56+6188*q^57+6580*q^58+6961*q^59+7250*q^60+7615*q^61+ 7851*q^62+8055*q^63+8074*q^64+7958*q^65+7620*q^66+7110*q^67+6350*q^68+5376*q^69 +4266*q^70+3123*q^71+2069*q^72+1198*q^73+589*q^74+233*q^75+67*q^76+12*q^77+q^78 , 1+13*q+79*q^2+300*q^3+804*q^4+1631*q^5+2629*q^6+3501*q^7+3992*q^8+4041*q^9+ 3770*q^10+3360*q^11+2941*q^12+2563*q^13+2241*q^14+2004*q^15+1915*q^16+2032*q^17 +2344*q^18+2690*q^19+2911*q^20+2912*q^21+2738*q^22+2466*q^23+2186*q^24+1927*q^ 25+1732*q^26+1704*q^27+1853*q^28+2194*q^29+2581*q^30+2833*q^31+2846*q^32+2710*q ^33+2439*q^34+2226*q^35+2138*q^36+2192*q^37+2483*q^38+2907*q^39+3304*q^40+3510* q^41+3548*q^42+3431*q^43+3321*q^44+3374*q^45+3570*q^46+3910*q^47+4407*q^48+4859 *q^49+5214*q^50+5450*q^51+5625*q^52+5836*q^53+6254*q^54+6744*q^55+7343*q^56+ 7901*q^57+8322*q^58+8619*q^59+8972*q^60+9390*q^61+9959*q^62+10645*q^63+11328*q^ 64+12015*q^65+12740*q^66+13443*q^67+14280*q^68+15211*q^69+16145*q^70+16969*q^71 +17695*q^72+18266*q^73+18819*q^74+19081*q^75+19090*q^76+18690*q^77+17899*q^78+ 16618*q^79+14921*q^80+12760*q^81+10310*q^82+7764*q^83+5370*q^84+3335*q^85+1804* q^86+824*q^87+300*q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+379*q^3+1104*q^4+ 2435*q^5+4260*q^6+6130*q^7+7493*q^8+8033*q^9+7811*q^10+7130*q^11+6301*q^12+5504 *q^13+4793*q^14+4179*q^15+3699*q^16+3452*q^17+3556*q^18+4049*q^19+4740*q^20+ 5346*q^21+5612*q^22+5486*q^23+5054*q^24+4507*q^25+3966*q^26+3484*q^27+3103*q^28 +2973*q^29+3147*q^30+3710*q^31+4458*q^32+5120*q^33+5404*q^34+5304*q^35+4914*q^ 36+4421*q^37+3979*q^38+3715*q^39+3693*q^40+4149*q^41+4873*q^42+5745*q^43+6408*q ^44+6665*q^45+6432*q^46+6109*q^47+5762*q^48+5735*q^49+6027*q^50+6628*q^51+7377* q^52+8254*q^53+8910*q^54+9274*q^55+9410*q^56+9465*q^57+9568*q^58+10133*q^59+ 10949*q^60+12076*q^61+13236*q^62+14167*q^63+14681*q^64+15063*q^65+15380*q^66+ 15999*q^67+16933*q^68+18074*q^69+19198*q^70+20336*q^71+21189*q^72+22165*q^73+ 23210*q^74+24664*q^75+26267*q^76+28223*q^77+29962*q^78+31848*q^79+33614*q^80+ 35535*q^81+37401*q^82+39461*q^83+41240*q^84+42834*q^85+44002*q^86+44947*q^87+ 45381*q^88+45159*q^89+44068*q^90+42068*q^91+39059*q^92+35105*q^93+30292*q^94+ 24825*q^95+19137*q^96+13691*q^97+8951*q^98+5224*q^99+2647*q^100+1126*q^101+379* q^102+92*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+9 *q^7+9*q^6+10*q^5+11*q^4+9*q^3+7*q^2+4*q+1, q^15+5*q^14+11*q^13+16*q^12+18*q^11 +17*q^10+18*q^9+21*q^8+21*q^7+23*q^6+22*q^5+22*q^4+16*q^3+11*q^2+5*q+1, q^21+6* q^20+16*q^19+27*q^18+34*q^17+35*q^16+32*q^15+32*q^14+35*q^13+39*q^12+39*q^11+40 *q^10+46*q^9+50*q^8+49*q^7+52*q^6+47*q^5+40*q^4+27*q^3+16*q^2+6*q+1, 1+43*q^25+ 67*q^22+61*q^24+60*q^21+69*q^23+7*q+22*q^2+22*q^26+43*q^3+69*q^4+92*q^5+107*q^6 +116*q^7+118*q^8+115*q^9+103*q^10+99*q^11+92*q^12+88*q^13+69*q^14+69*q^15+71*q^ 16+q^28+69*q^17+66*q^18+61*q^19+57*q^20+7*q^27, 1+116*q^25+121*q^22+121*q^24+ 118*q^21+126*q^23+8*q+136*q^30+127*q^29+29*q^34+65*q^33+29*q^2+107*q^26+65*q^3+ 114*q^4+168*q^5+212*q^6+104*q^32+246*q^7+264*q^8+274*q^9+263*q^10+254*q^11+242* q^12+232*q^13+197*q^14+187*q^15+165*q^16+112*q^28+156*q^17+146*q^18+135*q^19+ 118*q^20+103*q^27+130*q^31+8*q^35+q^36, 1+238*q^25+94*q^42+267*q^22+264*q^24+ 287*q^21+260*q^23+9*q+9*q^44+224*q^30+q^45+239*q^37+213*q^29+266*q^39+169*q^41+ 188*q^34+263*q^38+202*q^33+37*q^2+220*q^26+94*q^3+181*q^4+291*q^5+400*q^6+219*q ^32+494*q^7+567*q^8+611*q^9+624*q^10+615*q^11+606*q^12+585*q^13+549*q^14+512*q^ 15+473*q^16+37*q^43+199*q^28+440*q^17+408*q^18+364*q^19+321*q^20+196*q^27+234*q ^40+230*q^31+189*q^35+209*q^36, 1+625*q^25+351*q^42+797*q^22+692*q^24+870*q^21+ 718*q^23+10*q+263*q^51+349*q^44+46*q^53+449*q^30+131*q^52+q^55+10*q^54+391*q^45 +375*q^37+429*q^29+448*q^46+429*q^39+387*q^41+365*q^34+403*q^50+411*q^38+404*q^ 33+46*q^2+571*q^26+131*q^3+277*q^4+483*q^5+720*q^6+445*q^32+950*q^7+1158*q^8+ 1310*q^9+1415*q^10+1442*q^11+1453*q^12+1431*q^13+1386*q^14+1306*q^15+1246*q^16+ 334*q^43+500*q^49+465*q^28+1165*q^17+1114*q^18+1047*q^19+966*q^20+497*q^27+423* q^40+462*q^31+321*q^35+529*q^48+339*q^36+502*q^47, 1+1879*q^25+670*q^42+2215*q^ 22+1994*q^24+2373*q^21+2070*q^23+11*q+674*q^51+558*q^44+601*q^53+950*q^57+1183* q^30+612*q^52+1031*q^58+732*q^55+903*q^60+647*q^54+594*q^45+750*q^37+1245*q^29+ 667*q^46+11*q^65+827*q^39+666*q^61+768*q^41+922*q^34+762*q^50+56*q^64+776*q^38+ 1028*q^33+839*q^56+56*q^2+1720*q^26+177*q^3+410*q^4+773*q^5+1243*q^6+1090*q^32+ 1755*q^7+2265*q^8+2705*q^9+394*q^62+3054*q^10+3265*q^11+3386*q^12+3430*q^13+ 3389*q^14+3290*q^15+3150*q^16+q^66+595*q^43+812*q^49+1385*q^28+2994*q^17+2839*q ^18+2703*q^19+2544*q^20+1551*q^27+818*q^40+177*q^63+1143*q^31+790*q^35+803*q^48 +1029*q^59+745*q^36+743*q^47, 1+4928*q^25+1721*q^42+5757*q^22+5138*q^24+6188*q^ 21+5426*q^23+12*q+1299*q^51+1299*q^44+998*q^53+1349*q^57+3546*q^30+1108*q^52+ 1480*q^58+1072*q^55+1496*q^60+975*q^54+1244*q^45+1932*q^71+571*q^74+2026*q^37+ 1571*q^67+3768*q^29+1268*q^46+1203*q^65+1981*q^69+1965*q^39+1346*q^61+1907*q^41 +2708*q^34+1458*q^50+1093*q^64+1944*q^38+3020*q^33+1203*q^56+67*q^2+1789*q^68+ 4646*q^26+233*q^3+589*q^4+1198*q^5+2069*q^6+3213*q^32+3123*q^7+4266*q^8+5376*q^ 9+1169*q^62+6350*q^10+7110*q^11+7620*q^12+67*q^76+7958*q^13+8074*q^14+8055*q^15 +7851*q^16+1370*q^66+2060*q^70+12*q^77+q^78+1507*q^43+1538*q^49+4049*q^28+7615* q^17+7250*q^18+6961*q^19+6580*q^20+4378*q^27+1569*q^72+1937*q^40+1060*q^73+1076 *q^63+3398*q^31+233*q^75+2417*q^35+1495*q^48+1548*q^59+2169*q^36+1385*q^47, 1+ 12740*q^25+4859*q^42+15211*q^22+13*q^90+13443*q^24+16145*q^21+14280*q^23+13*q+ 3304*q^51+3910*q^44+2483*q^53+2439*q^57+9390*q^30+2907*q^52+2710*q^58+2138*q^55 +2833*q^60+q^91+2192*q^54+3570*q^45+2911*q^71+2032*q^74+6254*q^37+2186*q^67+ 9959*q^29+3374*q^46+1732*q^65+2738*q^69+5625*q^39+2581*q^61+5214*q^41+7901*q^34 +2941*q^79+3510*q^50+1704*q^64+5836*q^38+8322*q^33+2226*q^56+79*q^2+2466*q^68+ 12015*q^26+300*q^3+824*q^4+1804*q^5+3335*q^6+8619*q^32+5370*q^7+7764*q^8+10310* q^9+2194*q^62+12760*q^10+14921*q^11+16618*q^12+2004*q^76+17899*q^13+18690*q^14+ 19090*q^15+19081*q^16+1927*q^66+2912*q^70+2241*q^77+2563*q^78+4407*q^43+3548*q^ 49+10645*q^28+18819*q^17+18266*q^18+17695*q^19+16969*q^20+11328*q^27+2690*q^72+ 5450*q^40+2344*q^73+1853*q^63+8972*q^31+1915*q^75+7343*q^35+3360*q^80+3431*q^48 +3992*q^83+3770*q^81+2629*q^85+1631*q^86+804*q^87+300*q^88+79*q^89+3501*q^84+ 4041*q^82+2846*q^59+6744*q^36+3321*q^47, 1+33614*q^25+14167*q^42+39461*q^22+ 4179*q^90+35535*q^24+41240*q^21+37401*q^23+14*q+7493*q^97+8910*q^51+12076*q^44+ 7377*q^53+5762*q^57+24664*q^30+7130*q^94+8254*q^52+6109*q^58+6027*q^55+6665*q^ 60+4793*q^91+6628*q^54+10949*q^45+5404*q^71+3710*q^74+8033*q^96+6130*q^98+16933 *q^37+3979*q^67+26267*q^29+10133*q^46+6301*q^93+4260*q^99+3693*q^65+4914*q^69+ 15380*q^39+6408*q^61+14681*q^41+20336*q^34+3966*q^79+9274*q^50+4149*q^64+2435*q ^100+15999*q^38+21189*q^33+5735*q^56+92*q^2+4421*q^68+31848*q^26+379*q^3+1126*q ^4+2647*q^5+5224*q^6+22165*q^32+8951*q^7+13691*q^8+19137*q^9+7811*q^95+5745*q^ 62+1104*q^101+24825*q^10+30292*q^11+35105*q^12+2973*q^76+39059*q^13+42068*q^14+ 44068*q^15+45159*q^16+3715*q^66+5304*q^70+3103*q^77+3484*q^78+13236*q^43+379*q^ 102+9410*q^49+28223*q^28+45381*q^17+44947*q^18+44002*q^19+42834*q^20+29962*q^27 +5120*q^72+15063*q^40+4458*q^73+92*q^103+4873*q^63+23210*q^31+3147*q^75+19198*q ^35+14*q^104+4507*q^80+9465*q^48+q^105+5612*q^83+5054*q^81+4740*q^85+4049*q^86+ 3556*q^87+3452*q^88+3699*q^89+5346*q^84+5486*q^82+6432*q^59+18074*q^36+5504*q^ 92+9568*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 72, 228, 670, 1864, 5000, 13099, 33789, 86239, 218432, 550107, 1379348] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [1, 1, 3, 10, 36, 112, 332, 926, 2482, 6542, 16829, 43128, 109016, 275176, 689137] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [0, 1, 3, 11, 36, 116, 338, 938, 2518, 6557, 16960, 43111, 109416, 274931, 690211] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 36, 116, 338, 926, 2482, 6557, 16960, 43128, 109016, 274931, 690211] ODD:, [0, 1, 3, 11, 36, 112, 332, 938, 2518, 6542, 16829, 43111, 109416, 275176, 689137] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.027777778, 7.649122807, 10.95223881, 15.03755365, 20.00200000, 25.92404000, 32.85814910, 40.83863449, 49.88682061, 60.01671129, 71.23788993] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.543033500, 2.345043368, 3.428422112, 4.818879483, 6.510942064, 8.472142350, 10.65144821, 12.99193856, 15.44104116, 17.95476995, 20.49738548, 23.03946580] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.38095, 5.49923, 11.7541, 23.2216, 42.3924, 71.7772, 113.453, 168.790, 238.426, 322.374, 420.143, 530.817] -5 Skewness: , [Float(undefined), 0., 0., -0.2721915177 10 , -0.03338261314, -0.1015632175, -0.1962894473, -0.3115632636, -0.4422151771, -0.5826295130, -0.7275459142, -0.8730377421, -1.016757829, -1.157659204, -1.295546726] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.234375069, 2.177654329, 2.075132828, 2.006517634, 1.996805350, 2.057117437, 2.191937040, 2.399671681, 2.674619834, 3.009872623, 3.399255171, 3.838459435] end of this data For the equivalence class of patterns, { {[2, 4, 1, 3], [2, 3, 4, 1], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 4, 1, 3], [3, 2, 1, 4]}, {[1, 4, 3, 2], [3, 2, 1, 4], [3, 1, 4, 2]}, {[2, 3, 4, 1], [3, 1, 4, 2], [4, 1, 2, 3]}} the member , {[1, 4, 3, 2], [2, 4, 1, 3], [3, 2, 1, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 3, 2], {[0, 1, 0, 0]}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[1, 4, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 1, 1, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {1}], [[4, 1, 2, 5, 3], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {2}], [ [4, 1, 3, 5, 2], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}], [[4, 2, 3, 5, 1], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[2, 1, 3], {[0, 1, 1, 0]}, {}], [ [3, 1, 2, 4, 5], {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 0]}, {4}], [[2, 1, 3, 4], {[0, 0, 1, 1, 0], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {3}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 5, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {2}], [[3, 1, 2], {}, {}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[4, 1, 2, 3], {[0, 1, 1, 0, 0]}, {3}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+3*q^3+5*q^4+3*q^5+q^6, 1+4*q+9*q^2+10*q^3 +6*q^4+9*q^5+12*q^6+9*q^7+9*q^8+4*q^9+q^10, 1+5*q+14*q^2+22*q^3+21*q^4+11*q^5+ 17*q^6+23*q^7+25*q^8+18*q^9+29*q^10+27*q^11+20*q^12+14*q^13+5*q^14+q^15, 1+6*q+ 20*q^2+40*q^3+52*q^4+42*q^5+26*q^6+29*q^7+42*q^8+55*q^9+50*q^10+35*q^11+51*q^12 +67*q^13+72*q^14+65*q^15+70*q^16+56*q^17+37*q^18+20*q^19+6*q^20+q^21, 1+7*q+27* q^2+65*q^3+106*q^4+116*q^5+91*q^6+59*q^7+59*q^8+72*q^9+107*q^10+122*q^11+103*q^ 12+70*q^13+93*q^14+119*q^15+146*q^16+139*q^17+160*q^18+195*q^19+203*q^20+185*q^ 21+175*q^22+150*q^23+106*q^24+61*q^25+27*q^26+7*q^27+q^28, 1+8*q+35*q^2+98*q^3+ 191*q^4+262*q^5+261*q^6+200*q^7+145*q^8+125*q^9+138*q^10+191*q^11+260*q^12+273* q^13+212*q^14+155*q^15+173*q^16+218*q^17+292*q^18+298*q^19+267*q^20+279*q^21+ 395*q^22+482*q^23+527*q^24+506*q^25+525*q^26+560*q^27+524*q^28+463*q^29+399*q^ 30+295*q^31+185*q^32+93*q^33+35*q^34+8*q^35+q^36, 1+9*q+44*q^2+140*q^3+316*q^4+ 518*q^5+631*q^6+587*q^7+463*q^8+351*q^9+295*q^10+279*q^11+358*q^12+501*q^13+625 *q^14+597*q^15+470*q^16+353*q^17+344*q^18+410*q^19+563*q^20+663*q^21+647*q^22+ 525*q^23+531*q^24+672*q^25+872*q^26+1050*q^27+1159*q^28+1229*q^29+1338*q^30+ 1461*q^31+1553*q^32+1539*q^33+1504*q^34+1496*q^35+1478*q^36+1273*q^37+1064*q^38 +820*q^39+540*q^40+302*q^41+134*q^42+44*q^43+9*q^44+q^45, 1+10*q+54*q^2+192*q^3 +491*q^4+932*q^5+1342*q^6+1492*q^7+1350*q^8+1090*q^9+870*q^10+709*q^11+632*q^12 +697*q^13+960*q^14+1289*q^15+1464*q^16+1345*q^17+1066*q^18+831*q^19+751*q^20+ 801*q^21+1064*q^22+1387*q^23+1557*q^24+1379*q^25+1138*q^26+1056*q^27+1276*q^28+ 1620*q^29+1983*q^30+2146*q^31+2245*q^32+2445*q^33+2937*q^34+3462*q^35+3836*q^36 +3918*q^37+3940*q^38+4144*q^39+4322*q^40+4405*q^41+4471*q^42+4421*q^43+4243*q^ 44+3978*q^45+3574*q^46+2909*q^47+2251*q^48+1563*q^49+930*q^50+467*q^51+185*q^52 +54*q^53+10*q^54+q^55, 1+11*q+65*q^2+255*q^3+727*q^4+1563*q^5+2592*q^6+3366*q^7 +3526*q^8+3152*q^9+2627*q^10+2147*q^11+1768*q^12+1496*q^13+1493*q^14+1855*q^15+ 2560*q^16+3227*q^17+3440*q^18+3064*q^19+2491*q^20+2014*q^21+1729*q^22+1674*q^23 +2050*q^24+2768*q^25+3432*q^26+3553*q^27+3124*q^28+2518*q^29+2276*q^30+2498*q^ 31+3161*q^32+4002*q^33+4573*q^34+4618*q^35+4437*q^36+4599*q^37+5361*q^38+6632*q ^39+8129*q^40+9307*q^41+9702*q^42+9596*q^43+9975*q^44+10929*q^45+11832*q^46+ 12086*q^47+12003*q^48+12132*q^49+12851*q^50+13122*q^51+13124*q^52+12736*q^53+ 12079*q^54+11052*q^55+9786*q^56+8127*q^57+6236*q^58+4442*q^59+2806*q^60+1521*q^ 61+691*q^62+247*q^63+65*q^64+11*q^65+q^66, 1+12*q+77*q^2+330*q^3+1036*q^4+2482* q^5+4648*q^6+6906*q^7+8303*q^8+8360*q^9+7497*q^10+6386*q^11+5349*q^12+4425*q^13 +3718*q^14+3405*q^15+3805*q^16+5007*q^17+6715*q^18+7965*q^19+8060*q^20+7141*q^ 21+5964*q^22+4911*q^23+4141*q^24+3760*q^25+4158*q^26+5397*q^27+7143*q^28+8322*q ^29+8280*q^30+7130*q^31+5883*q^32+5113*q^33+5201*q^34+6194*q^35+8084*q^36+9870* q^37+10677*q^38+10229*q^39+9441*q^40+9318*q^41+10436*q^42+12339*q^43+14836*q^44 +17546*q^45+19960*q^46+21521*q^47+22702*q^48+24109*q^49+25935*q^50+27904*q^51+ 29890*q^52+31310*q^53+31496*q^54+31393*q^55+32658*q^56+34815*q^57+36557*q^58+ 37283*q^59+37719*q^60+38482*q^61+39152*q^62+38249*q^63+36626*q^64+34218*q^65+ 31039*q^66+27097*q^67+22539*q^68+17505*q^69+12594*q^70+8272*q^71+4792*q^72+2381 *q^73+986*q^74+321*q^75+77*q^76+12*q^77+q^78, 1+13*q+90*q^2+418*q^3+1431*q^4+ 3773*q^5+7859*q^6+13135*q^7+17888*q^8+20302*q^9+19994*q^10+18050*q^11+15684*q^ 12+13334*q^13+11194*q^14+9376*q^15+8262*q^16+8290*q^17+10082*q^18+13492*q^19+ 17263*q^20+19417*q^21+19071*q^22+16950*q^23+14426*q^24+12100*q^25+10192*q^26+ 8916*q^27+8915*q^28+10777*q^29+14321*q^30+18141*q^31+20125*q^32+19310*q^33+ 16765*q^34+14032*q^35+12051*q^36+11337*q^37+12624*q^38+16092*q^39+20660*q^40+ 24260*q^41+25122*q^42+23325*q^43+20854*q^44+19980*q^45+21545*q^46+25075*q^47+ 29686*q^48+34567*q^49+38721*q^50+41454*q^51+43224*q^52+45773*q^53+51131*q^54+ 58950*q^55+66655*q^56+71655*q^57+74026*q^58+75379*q^59+77256*q^60+79965*q^61+ 83784*q^62+87382*q^63+90679*q^64+93883*q^65+97312*q^66+99279*q^67+102585*q^68+ 108215*q^69+114089*q^70+116477*q^71+116387*q^72+115453*q^73+114535*q^74+111282* q^75+105135*q^76+97232*q^77+87441*q^78+75631*q^79+62757*q^80+49126*q^81+35845*q ^82+24122*q^83+14668*q^84+7842*q^85+3591*q^86+1365*q^87+408*q^88+90*q^89+13*q^ 90+q^91, 1+14*q+104*q^2+520*q^3+1926*q^4+5534*q^5+12670*q^6+23496*q^7+35778*q^8 +45472*q^9+49552*q^10+48220*q^11+43940*q^12+38716*q^13+33402*q^14+28351*q^15+ 23966*q^16+20657*q^17+19358*q^18+21247*q^19+27164*q^20+35759*q^21+43696*q^22+ 47199*q^23+45528*q^24+40727*q^25+35294*q^26+30057*q^27+25427*q^28+21821*q^29+ 20450*q^30+22414*q^31+28651*q^32+37608*q^33+45502*q^34+48345*q^35+45749*q^36+ 40029*q^37+34040*q^38+29046*q^39+26256*q^40+26860*q^41+32379*q^42+42021*q^43+ 52630*q^44+59388*q^45+59541*q^46+54258*q^47+47969*q^48+44442*q^49+45879*q^50+ 52216*q^51+62201*q^52+73184*q^53+82715*q^54+88598*q^55+90649*q^56+90773*q^57+ 92528*q^58+98948*q^59+112606*q^60+133092*q^61+156042*q^62+174327*q^63+183925*q^ 64+185874*q^65+185319*q^66+188667*q^67+200439*q^68+217512*q^69+231693*q^70+ 237134*q^71+237610*q^72+240287*q^73+248416*q^74+259804*q^75+272753*q^76+285358* q^77+298605*q^78+312155*q^79+324545*q^80+333299*q^81+342492*q^82+350920*q^83+ 355221*q^84+351811*q^85+343924*q^86+334596*q^87+322426*q^88+302901*q^89+276749* q^90+246632*q^91+212518*q^92+175463*q^93+138103*q^94+102017*q^95+70050*q^96+ 44107*q^97+24938*q^98+12369*q^99+5246*q^100+1842*q^101+509*q^102+104*q^103+14*q ^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+3*q^3+5*q^4+3*q^5+q^6, q^10+4*q^9+9*q^8+ 10*q^7+6*q^6+9*q^5+12*q^4+9*q^3+9*q^2+4*q+1, q^15+5*q^14+14*q^13+22*q^12+21*q^ 11+11*q^10+17*q^9+23*q^8+25*q^7+18*q^6+29*q^5+27*q^4+20*q^3+14*q^2+5*q+1, q^21+ 6*q^20+20*q^19+40*q^18+52*q^17+42*q^16+26*q^15+29*q^14+42*q^13+55*q^12+50*q^11+ 35*q^10+51*q^9+67*q^8+72*q^7+65*q^6+70*q^5+56*q^4+37*q^3+20*q^2+6*q+1, 1+65*q^ 25+91*q^22+106*q^24+59*q^21+116*q^23+7*q+27*q^2+27*q^26+61*q^3+106*q^4+150*q^5+ 175*q^6+185*q^7+203*q^8+195*q^9+160*q^10+139*q^11+146*q^12+119*q^13+93*q^14+70* q^15+103*q^16+q^28+122*q^17+107*q^18+72*q^19+59*q^20+7*q^27, 1+191*q^25+212*q^ 22+260*q^24+155*q^21+273*q^23+8*q+261*q^30+200*q^29+35*q^34+98*q^33+35*q^2+138* q^26+93*q^3+185*q^4+295*q^5+399*q^6+191*q^32+463*q^7+524*q^8+560*q^9+525*q^10+ 506*q^11+527*q^12+482*q^13+395*q^14+279*q^15+267*q^16+145*q^28+298*q^17+292*q^ 18+218*q^19+173*q^20+125*q^27+262*q^31+8*q^35+q^36, 1+563*q^25+140*q^42+525*q^ 22+663*q^24+531*q^21+647*q^23+9*q+9*q^44+597*q^30+q^45+463*q^37+470*q^29+631*q^ 39+316*q^41+279*q^34+587*q^38+358*q^33+44*q^2+410*q^26+134*q^3+302*q^4+540*q^5+ 820*q^6+501*q^32+1064*q^7+1273*q^8+1478*q^9+1496*q^10+1504*q^11+1539*q^12+1553* q^13+1461*q^14+1338*q^15+1229*q^16+44*q^43+353*q^28+1159*q^17+1050*q^18+872*q^ 19+672*q^20+344*q^27+518*q^40+625*q^31+295*q^35+351*q^36, 1+1983*q^25+697*q^42+ 2445*q^22+2146*q^24+2937*q^21+2245*q^23+10*q+491*q^51+709*q^44+54*q^53+1379*q^ 30+192*q^52+q^55+10*q^54+870*q^45+1066*q^37+1138*q^29+1090*q^46+1464*q^39+960*q ^41+801*q^34+932*q^50+1345*q^38+1064*q^33+54*q^2+1620*q^26+185*q^3+467*q^4+930* q^5+1563*q^6+1387*q^32+2251*q^7+2909*q^8+3574*q^9+3978*q^10+4243*q^11+4421*q^12 +4471*q^13+4405*q^14+4322*q^15+4144*q^16+632*q^43+1342*q^49+1056*q^28+3940*q^17 +3918*q^18+3836*q^19+3462*q^20+1276*q^27+1289*q^40+1557*q^31+751*q^35+1492*q^48 +831*q^36+1350*q^47, 1+9307*q^25+2050*q^42+9975*q^22+9702*q^24+10929*q^21+9596* q^23+11*q+1855*q^51+1729*q^44+1496*q^53+3152*q^57+4437*q^30+1493*q^52+3526*q^58 +2147*q^55+2592*q^60+1768*q^54+2014*q^45+2518*q^37+4599*q^29+2491*q^46+11*q^65+ 3553*q^39+1563*q^61+2768*q^41+3161*q^34+2560*q^50+65*q^64+3124*q^38+4002*q^33+ 2627*q^56+65*q^2+8129*q^26+247*q^3+691*q^4+1521*q^5+2806*q^6+4573*q^32+4442*q^7 +6236*q^8+8127*q^9+727*q^62+9786*q^10+11052*q^11+12079*q^12+12736*q^13+13124*q^ 14+13122*q^15+12851*q^16+q^66+1674*q^43+3227*q^49+5361*q^28+12132*q^17+12003*q^ 18+12086*q^19+11832*q^20+6632*q^27+3432*q^40+255*q^63+4618*q^31+2498*q^35+3440* q^48+3366*q^59+2276*q^36+3064*q^47, 1+31310*q^25+8084*q^42+32658*q^22+31496*q^ 24+34815*q^21+31393*q^23+12*q+5397*q^51+5201*q^44+3760*q^53+7141*q^57+22702*q^ 30+4158*q^52+8060*q^58+4911*q^55+6715*q^60+4141*q^54+5113*q^45+6906*q^71+1036*q ^74+9318*q^37+6386*q^67+24109*q^29+5883*q^46+4425*q^65+8360*q^69+10229*q^39+ 5007*q^61+9870*q^41+14836*q^34+7143*q^50+3718*q^64+9441*q^38+17546*q^33+5964*q^ 56+77*q^2+7497*q^68+29890*q^26+321*q^3+986*q^4+2381*q^5+4792*q^6+19960*q^32+ 8272*q^7+12594*q^8+17505*q^9+3805*q^62+22539*q^10+27097*q^11+31039*q^12+77*q^76 +34218*q^13+36626*q^14+38249*q^15+39152*q^16+5349*q^66+8303*q^70+12*q^77+q^78+ 6194*q^43+8322*q^49+25935*q^28+38482*q^17+37719*q^18+37283*q^19+36557*q^20+ 27904*q^27+4648*q^72+10677*q^40+2482*q^73+3405*q^63+21521*q^31+330*q^75+12339*q ^35+8280*q^48+7965*q^59+10436*q^36+7130*q^47, 1+97312*q^25+34567*q^42+108215*q^ 22+13*q^90+99279*q^24+114089*q^21+102585*q^23+13*q+20660*q^51+25075*q^44+12624* q^53+16765*q^57+79965*q^30+16092*q^52+19310*q^58+12051*q^55+18141*q^60+q^91+ 11337*q^54+21545*q^45+17263*q^71+8290*q^74+51131*q^37+14426*q^67+83784*q^29+ 19980*q^46+10192*q^65+19071*q^69+43224*q^39+14321*q^61+38721*q^41+71655*q^34+ 15684*q^79+24260*q^50+8916*q^64+45773*q^38+74026*q^33+14032*q^56+90*q^2+16950*q ^68+93883*q^26+408*q^3+1365*q^4+3591*q^5+7842*q^6+75379*q^32+14668*q^7+24122*q^ 8+35845*q^9+10777*q^62+49126*q^10+62757*q^11+75631*q^12+9376*q^76+87441*q^13+ 97232*q^14+105135*q^15+111282*q^16+12100*q^66+19417*q^70+11194*q^77+13334*q^78+ 29686*q^43+25122*q^49+87382*q^28+114535*q^17+115453*q^18+116387*q^19+116477*q^ 20+90679*q^27+13492*q^72+41454*q^40+10082*q^73+8915*q^63+77256*q^31+8262*q^75+ 66655*q^35+18050*q^80+23325*q^48+17888*q^83+19994*q^81+7859*q^85+3773*q^86+1431 *q^87+418*q^88+90*q^89+13135*q^84+20302*q^82+20125*q^59+58950*q^36+20854*q^47, 1+324545*q^25+174327*q^42+350920*q^22+28351*q^90+333299*q^24+355221*q^21+342492 *q^23+14*q+35778*q^97+82715*q^51+133092*q^44+62201*q^53+47969*q^57+259804*q^30+ 48220*q^94+73184*q^52+54258*q^58+45879*q^55+59388*q^60+33402*q^91+52216*q^54+ 112606*q^45+45502*q^71+22414*q^74+45472*q^96+23496*q^98+200439*q^37+34040*q^67+ 272753*q^29+98948*q^46+43940*q^93+12670*q^99+26256*q^65+45749*q^69+185319*q^39+ 52630*q^61+183925*q^41+237134*q^34+35294*q^79+88598*q^50+26860*q^64+5534*q^100+ 188667*q^38+237610*q^33+44442*q^56+104*q^2+40029*q^68+312155*q^26+509*q^3+1842* q^4+5246*q^5+12369*q^6+240287*q^32+24938*q^7+44107*q^8+70050*q^9+49552*q^95+ 42021*q^62+1926*q^101+102017*q^10+138103*q^11+175463*q^12+21821*q^76+212518*q^ 13+246632*q^14+276749*q^15+302901*q^16+29046*q^66+48345*q^70+25427*q^77+30057*q ^78+156042*q^43+520*q^102+90649*q^49+285358*q^28+322426*q^17+334596*q^18+343924 *q^19+351811*q^20+298605*q^27+37608*q^72+185874*q^40+28651*q^73+104*q^103+32379 *q^63+248416*q^31+20450*q^75+231693*q^35+14*q^104+40727*q^80+90773*q^48+q^105+ 43696*q^83+45528*q^81+27164*q^85+21247*q^86+19358*q^87+20657*q^88+23966*q^89+ 35759*q^84+47199*q^82+59541*q^59+217512*q^36+38716*q^92+92528*q^47] The number of permutations avoiding, {[1, 4, 3, 2], [2, 4, 1, 3], [3, 2, 1, 4]}, is given by [1, 2, 6, 21, 74, 253, 843, 2772, 9080, 29759, 97686, 321033, 1055596, 3471365, 11415280] The number of EVEN permutations avoiding, {[1, 4, 3, 2], [2, 4, 1, 3], [3, 2, 1, 4]}, is given by [1, 1, 3, 12, 38, 132, 427, 1405, 4566, 14939, 48943, 160713, 528176, 1736347, 5709088] The number of ODD permutations avoiding, {[1, 4, 3, 2], [2, 4, 1, 3], [3, 2, 1, 4]}, is given by [0, 1, 3, 9, 36, 121, 416, 1367, 4514, 14820, 48743, 160320, 527420, 1735018, 5706192] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 12, 38, 121, 416, 1405, 4566, 14820, 48743, 160713, 528176, 1735018, 5706192] ODD:, [0, 1, 3, 9, 36, 132, 427, 1367, 4514, 14939, 48943, 160320, 527420, 1736347, 5709088] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.054054054, 7.735177866, 11.13167260, 15.33225108, 20.40572687, 26.39167983, 33.30676863, 41.15751963, 49.94960383, 59.69089278, 70.39064990] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.573591585, 2.421144982, 3.527003070, 4.890301286, 6.482827707, 8.255149922, 10.15225763, 12.12802456, 14.15034358, 16.19850083, 18.25834742, 20.31879887] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.47619, 5.86194, 12.4398, 23.9150, 42.0271, 68.1475, 103.068, 147.089, 200.232, 262.391, 333.367, 412.854] Skewness: , [Float(undefined), 0., 0., 0., -0.06889504360, -0.1679099637, -0.2855544119, -0.4180916840, -0.5604017578, -0.7058546797, -0.8484504394, -0.9841036166, -1.110751832, -1.227833695, -1.335665584] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.081360971, 2.000143147, 1.958479450, 1.969918463, 2.045545362, 2.195180993, 2.419953823, 2.712009629, 3.058899238, 3.447412059, 3.866203987, 4.306329018] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 1, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 3, 1, 4], [4, 1, 3, 2]}, {[2, 4, 3, 1], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, {[1, 4, 2, 3], [3, 2, 4, 1], [4, 2, 1, 3]}, {[1, 4, 2, 3], [3, 2, 4, 1], [3, 1, 2, 4]}, {[1, 4, 2, 3], [2, 4, 3, 1], [3, 1, 2, 4]}, {[1, 3, 4, 2], [2, 3, 1, 4], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 0]}, {4}], [[2, 1], {}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {4}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 0]}, {2}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {2}], [[4, 5, 2, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [ [4, 5, 1, 3, 2], {[0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 1, 0]}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[3, 1, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 5, 1, 4, 3], {[0, 0, 1, 1, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]}, {1}], [ [2, 5, 1, 3, 4], {[0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[3, 5, 1, 4, 2], { [0, 1, 0, 1, 0, 0], [0, 1, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]}, {1}], [[3, 5, 1, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 1, 0, 0], [1, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[3, 4, 1, 2, 5], {[0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 5, 2, 4, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 1, 0]}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0], [1, 0, 1, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 1, 1, 0, 0], [1, 0, 0, 1, 0], [0, 1, 0, 1, 0]}, {}], [[4, 3, 1, 2], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 0]}, {4}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 1, 1, 0]}, {}], [[2, 3, 1], {[1, 0, 1, 0]}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [[4, 3, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[4, 2, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 2, 1, 4], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 0, 1, 1, 0], [0, 1, 0, 1, 0]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+3*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3 +12*q^4+11*q^5+9*q^6+7*q^7+6*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+25*q^4+28*q^5+ 26*q^6+25*q^7+22*q^8+20*q^9+15*q^10+14*q^11+14*q^12+10*q^13+5*q^14+q^15, 1+6*q+ 16*q^2+31*q^3+47*q^4+58*q^5+65*q^6+62*q^7+59*q^8+59*q^9+50*q^10+48*q^11+45*q^12 +38*q^13+35*q^14+31*q^15+28*q^16+31*q^17+25*q^18+15*q^19+6*q^20+q^21, 1+7*q+22* q^2+48*q^3+82*q^4+113*q^5+138*q^6+149*q^7+148*q^8+146*q^9+137*q^10+132*q^11+123 *q^12+115*q^13+105*q^14+97*q^15+90*q^16+90*q^17+83*q^18+71*q^19+66*q^20+65*q^21 +64*q^22+66*q^23+60*q^24+41*q^25+21*q^26+7*q^27+q^28, 1+8*q+29*q^2+71*q^3+135*q ^4+207*q^5+275*q^6+324*q^7+349*q^8+355*q^9+352*q^10+340*q^11+325*q^12+312*q^13+ 288*q^14+274*q^15+250*q^16+240*q^17+235*q^18+213*q^19+200*q^20+187*q^21+178*q^ 22+179*q^23+175*q^24+159*q^25+143*q^26+139*q^27+142*q^28+147*q^29+149*q^30+137* q^31+106*q^32+63*q^33+28*q^34+8*q^35+q^36, 1+9*q+37*q^2+101*q^3+212*q^4+359*q^5 +519*q^6+663*q^7+766*q^8+828*q^9+852*q^10+852*q^11+841*q^12+812*q^13+775*q^14+ 743*q^15+688*q^16+656*q^17+633*q^18+587*q^19+551*q^20+520*q^21+489*q^22+480*q^ 23+474*q^24+442*q^25+407*q^26+383*q^27+370*q^28+376*q^29+375*q^30+364*q^31+353* q^32+329*q^33+308*q^34+313*q^35+327*q^36+338*q^37+343*q^38+317*q^39+259*q^40+ 175*q^41+92*q^42+36*q^43+9*q^44+q^45, 1+10*q+46*q^2+139*q^3+320*q^4+594*q^5+933 *q^6+1287*q^7+1595*q^8+1828*q^9+1977*q^10+2051*q^11+2081*q^12+2064*q^13+2021*q^ 14+1957*q^15+1865*q^16+1782*q^17+1705*q^18+1625*q^19+1530*q^20+1439*q^21+1354*q ^22+1295*q^23+1257*q^24+1196*q^25+1132*q^26+1066*q^27+1011*q^28+983*q^29+970*q^ 30+952*q^31+925*q^32+870*q^33+801*q^34+771*q^35+774*q^36+794*q^37+818*q^38+816* q^39+780*q^40+744*q^41+721*q^42+710*q^43+732*q^44+770*q^45+793*q^46+793*q^47+ 742*q^48+624*q^49+456*q^50+274*q^51+129*q^52+45*q^53+10*q^54+q^55, 1+11*q+56*q^ 2+186*q^3+467*q^4+944*q^5+1606*q^6+2385*q^7+3166*q^8+3852*q^9+4384*q^10+4748*q^ 11+4971*q^12+5078*q^13+5086*q^14+5022*q^15+4901*q^16+4727*q^17+4565*q^18+4398*q ^19+4188*q^20+3990*q^21+3770*q^22+3583*q^23+3447*q^24+3291*q^25+3134*q^26+2963* q^27+2785*q^28+2662*q^29+2589*q^30+2519*q^31+2453*q^32+2364*q^33+2222*q^34+2099 *q^35+2032*q^36+2006*q^37+2005*q^38+1999*q^39+1930*q^40+1841*q^41+1750*q^42+ 1667*q^43+1657*q^44+1697*q^45+1750*q^46+1817*q^47+1844*q^48+1810*q^49+1736*q^50 +1662*q^51+1654*q^52+1694*q^53+1761*q^54+1835*q^55+1879*q^56+1859*q^57+1741*q^ 58+1501*q^59+1151*q^60+759*q^61+411*q^62+175*q^63+55*q^64+11*q^65+q^66, 1+12*q+ 67*q^2+243*q^3+662*q^4+1449*q^5+2660*q^6+4241*q^7+6018*q^8+7774*q^9+9326*q^10+ 10563*q^11+11474*q^12+12083*q^13+12423*q^14+12546*q^15+12483*q^16+12257*q^17+ 11977*q^18+11635*q^19+11240*q^20+10824*q^21+10320*q^22+9869*q^23+9475*q^24+9077 *q^25+8702*q^26+8298*q^27+7838*q^28+7437*q^29+7147*q^30+6883*q^31+6669*q^32+ 6438*q^33+6108*q^34+5778*q^35+5526*q^36+5353*q^37+5273*q^38+5227*q^39+5109*q^40 +4944*q^41+4724*q^42+4457*q^43+4264*q^44+4188*q^45+4187*q^46+4256*q^47+4281*q^ 48+4195*q^49+4027*q^50+3819*q^51+3672*q^52+3653*q^53+3731*q^54+3864*q^55+4035*q ^56+4162*q^57+4227*q^58+4227*q^59+4121*q^60+3971*q^61+3894*q^62+3947*q^63+4106* q^64+4289*q^65+4424*q^66+4479*q^67+4401*q^68+4116*q^69+3600*q^70+2864*q^71+2011 *q^72+1207*q^73+595*q^74+231*q^75+66*q^76+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+ 915*q^4+2158*q^5+4258*q^6+7268*q^7+11000*q^8+15077*q^9+19075*q^10+22636*q^11+ 25575*q^12+27844*q^13+29451*q^14+30474*q^15+30972*q^16+31014*q^17+30737*q^18+ 30225*q^19+29555*q^20+28730*q^21+27742*q^22+26712*q^23+25714*q^24+24769*q^25+ 23839*q^26+22901*q^27+21860*q^28+20823*q^29+19936*q^30+19140*q^31+18473*q^32+ 17840*q^33+17087*q^34+16251*q^35+15444*q^36+14783*q^37+14341*q^38+14027*q^39+ 13701*q^40+13303*q^41+12805*q^42+12220*q^43+11673*q^44+11307*q^45+11078*q^46+ 11010*q^47+10997*q^48+10863*q^49+10565*q^50+10101*q^51+9588*q^52+9205*q^53+8993 *q^54+8960*q^55+9094*q^56+9255*q^57+9338*q^58+9295*q^59+9053*q^60+8654*q^61+ 8318*q^62+8200*q^63+8333*q^64+8673*q^65+9100*q^66+9488*q^67+9800*q^68+9970*q^69 +9954*q^70+9805*q^71+9584*q^72+9386*q^73+9390*q^74+9657*q^75+10072*q^76+10485*q ^77+10738*q^78+10756*q^79+10484*q^80+9801*q^81+8645*q^82+7057*q^83+5195*q^84+ 3357*q^85+1848*q^86+836*q^87+298*q^88+78*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q ^3+1237*q^4+3130*q^5+6613*q^6+12050*q^7+19407*q^8+28190*q^9+37608*q^10+46802*q^ 11+55110*q^12+62159*q^13+67787*q^14+72000*q^15+74880*q^16+76532*q^17+77161*q^18 +77000*q^19+76213*q^20+74937*q^21+73210*q^22+71095*q^23+68899*q^24+66698*q^25+ 64453*q^26+62278*q^27+59908*q^28+57426*q^29+55100*q^30+52945*q^31+51060*q^32+ 49352*q^33+47549*q^34+45561*q^35+43528*q^36+41593*q^37+40023*q^38+38811*q^39+ 37715*q^40+36613*q^41+35397*q^42+33930*q^43+32410*q^44+31119*q^45+30094*q^46+ 29485*q^47+29158*q^48+28788*q^49+28217*q^50+27336*q^51+26203*q^52+25120*q^53+ 24273*q^54+23728*q^55+23524*q^56+23558*q^57+23599*q^58+23498*q^59+23051*q^60+ 22149*q^61+21073*q^62+20156*q^63+19621*q^64+19592*q^65+19933*q^66+20379*q^67+ 20825*q^68+21071*q^69+20974*q^70+20584*q^71+19977*q^72+19333*q^73+18990*q^74+ 19153*q^75+19787*q^76+20744*q^77+21768*q^78+22710*q^79+23463*q^80+23849*q^81+ 23871*q^82+23594*q^83+23123*q^84+22776*q^85+22743*q^86+23114*q^87+23913*q^88+ 24887*q^89+25719*q^90+26142*q^91+25982*q^92+25136*q^93+23456*q^94+20824*q^95+ 17303*q^96+13195*q^97+9019*q^98+5391*q^99+2740*q^100+1145*q^101+377*q^102+91*q^ 103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+3*q^2+6*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+ 11*q^7+12*q^6+11*q^5+9*q^4+7*q^3+6*q^2+4*q+1, q^15+5*q^14+11*q^13+19*q^12+25*q^ 11+28*q^10+26*q^9+25*q^8+22*q^7+20*q^6+15*q^5+14*q^4+14*q^3+10*q^2+5*q+1, q^21+ 6*q^20+16*q^19+31*q^18+47*q^17+58*q^16+65*q^15+62*q^14+59*q^13+59*q^12+50*q^11+ 48*q^10+45*q^9+38*q^8+35*q^7+31*q^6+28*q^5+31*q^4+25*q^3+15*q^2+6*q+1, 1+48*q^ 25+138*q^22+82*q^24+149*q^21+113*q^23+7*q+21*q^2+22*q^26+41*q^3+60*q^4+66*q^5+ 64*q^6+65*q^7+66*q^8+71*q^9+83*q^10+90*q^11+90*q^12+97*q^13+105*q^14+115*q^15+ 123*q^16+q^28+132*q^17+137*q^18+146*q^19+148*q^20+7*q^27, 1+340*q^25+288*q^22+ 325*q^24+274*q^21+312*q^23+8*q+275*q^30+324*q^29+29*q^34+71*q^33+28*q^2+352*q^ 26+63*q^3+106*q^4+137*q^5+149*q^6+135*q^32+147*q^7+142*q^8+139*q^9+143*q^10+159 *q^11+175*q^12+179*q^13+178*q^14+187*q^15+200*q^16+349*q^28+213*q^17+235*q^18+ 240*q^19+250*q^20+355*q^27+207*q^31+8*q^35+q^36, 1+551*q^25+101*q^42+480*q^22+ 520*q^24+474*q^21+489*q^23+9*q+9*q^44+743*q^30+q^45+766*q^37+688*q^29+519*q^39+ 212*q^41+852*q^34+663*q^38+841*q^33+36*q^2+587*q^26+92*q^3+175*q^4+259*q^5+317* q^6+812*q^32+343*q^7+338*q^8+327*q^9+313*q^10+308*q^11+329*q^12+353*q^13+364*q^ 14+375*q^15+376*q^16+37*q^43+656*q^28+370*q^17+383*q^18+407*q^19+442*q^20+633*q ^27+359*q^40+775*q^31+852*q^35+828*q^36, 1+970*q^25+2064*q^42+870*q^22+952*q^24 +801*q^21+925*q^23+10*q+320*q^51+2051*q^44+46*q^53+1196*q^30+139*q^52+q^55+10*q ^54+1977*q^45+1705*q^37+1132*q^29+1828*q^46+1865*q^39+2021*q^41+1439*q^34+594*q ^50+1782*q^38+1354*q^33+45*q^2+983*q^26+129*q^3+274*q^4+456*q^5+624*q^6+1295*q^ 32+742*q^7+793*q^8+793*q^9+770*q^10+732*q^11+710*q^12+721*q^13+744*q^14+780*q^ 15+816*q^16+2081*q^43+933*q^49+1066*q^28+818*q^17+794*q^18+774*q^19+771*q^20+ 1011*q^27+1957*q^40+1257*q^31+1530*q^35+1287*q^48+1625*q^36+1595*q^47, 1+1841*q ^25+3447*q^42+1657*q^22+1750*q^24+1697*q^21+1667*q^23+11*q+5022*q^51+3770*q^44+ 5078*q^53+3852*q^57+2032*q^30+5086*q^52+3166*q^58+4748*q^55+1606*q^60+4971*q^54 +3990*q^45+2662*q^37+2006*q^29+4188*q^46+11*q^65+2963*q^39+944*q^61+3291*q^41+ 2453*q^34+4901*q^50+56*q^64+2785*q^38+2364*q^33+4384*q^56+55*q^2+1930*q^26+175* q^3+411*q^4+759*q^5+1151*q^6+2222*q^32+1501*q^7+1741*q^8+1859*q^9+467*q^62+1879 *q^10+1835*q^11+1761*q^12+1694*q^13+1654*q^14+1662*q^15+1736*q^16+q^66+3583*q^ 43+4727*q^49+2005*q^28+1810*q^17+1844*q^18+1817*q^19+1750*q^20+1999*q^27+3134*q ^40+186*q^63+2099*q^31+2519*q^35+4565*q^48+2385*q^59+2589*q^36+4398*q^47, 1+ 3653*q^25+5526*q^42+4035*q^22+3731*q^24+4162*q^21+3864*q^23+12*q+8298*q^51+6108 *q^44+9077*q^53+10824*q^57+4281*q^30+8702*q^52+11240*q^58+9869*q^55+11977*q^60+ 9475*q^54+6438*q^45+4241*q^71+662*q^74+4944*q^37+10563*q^67+4195*q^29+6669*q^46 +12083*q^65+7774*q^69+5227*q^39+12257*q^61+5353*q^41+4264*q^34+7838*q^50+12423* q^64+5109*q^38+4188*q^33+10320*q^56+66*q^2+9326*q^68+3672*q^26+231*q^3+595*q^4+ 1207*q^5+2011*q^6+4187*q^32+2864*q^7+3600*q^8+4116*q^9+12483*q^62+4401*q^10+ 4479*q^11+4424*q^12+67*q^76+4289*q^13+4106*q^14+3947*q^15+3894*q^16+11474*q^66+ 6018*q^70+12*q^77+q^78+5778*q^43+7437*q^49+4027*q^28+3971*q^17+4121*q^18+4227*q ^19+4227*q^20+3819*q^27+2660*q^72+5273*q^40+1449*q^73+12546*q^63+4256*q^31+243* q^75+4457*q^35+7147*q^48+11635*q^59+4724*q^36+6883*q^47, 1+9100*q^25+10863*q^42 +9970*q^22+13*q^90+9488*q^24+9954*q^21+9800*q^23+13*q+13701*q^51+11010*q^44+ 14341*q^53+17087*q^57+8654*q^30+14027*q^52+17840*q^58+15444*q^55+19140*q^60+q^ 91+14783*q^54+11078*q^45+29555*q^71+31014*q^74+8993*q^37+25714*q^67+8318*q^29+ 11307*q^46+23839*q^65+27742*q^69+9588*q^39+19936*q^61+10565*q^41+9255*q^34+ 25575*q^79+13303*q^50+22901*q^64+9205*q^38+9338*q^33+16251*q^56+78*q^2+26712*q^ 68+8673*q^26+298*q^3+836*q^4+1848*q^5+3357*q^6+9295*q^32+5195*q^7+7057*q^8+8645 *q^9+20823*q^62+9801*q^10+10484*q^11+10756*q^12+30474*q^76+10738*q^13+10485*q^ 14+10072*q^15+9657*q^16+24769*q^66+28730*q^70+29451*q^77+27844*q^78+10997*q^43+ 12805*q^49+8200*q^28+9390*q^17+9386*q^18+9584*q^19+9805*q^20+8333*q^27+30225*q^ 72+10101*q^40+30737*q^73+21860*q^63+9053*q^31+30972*q^75+9094*q^35+22636*q^80+ 12220*q^48+11000*q^83+19075*q^81+4258*q^85+2158*q^86+915*q^87+311*q^88+79*q^89+ 7268*q^84+15077*q^82+18473*q^59+8960*q^36+11673*q^47, 1+23463*q^25+20156*q^42+ 23594*q^22+72000*q^90+23849*q^24+23123*q^21+23871*q^23+14*q+19407*q^97+24273*q^ 51+22149*q^44+26203*q^53+29158*q^57+19153*q^30+46802*q^94+25120*q^52+29485*q^58 +28217*q^55+31119*q^60+67787*q^91+27336*q^54+23051*q^45+47549*q^71+52945*q^74+ 28190*q^96+12050*q^98+20825*q^37+40023*q^67+19787*q^29+23498*q^46+55110*q^93+ 6613*q^99+37715*q^65+43528*q^69+19933*q^39+32410*q^61+19621*q^41+20584*q^34+ 64453*q^79+23728*q^50+36613*q^64+3130*q^100+20379*q^38+19977*q^33+28788*q^56+91 *q^2+41593*q^68+22710*q^26+377*q^3+1145*q^4+2740*q^5+5391*q^6+19333*q^32+9019*q ^7+13195*q^8+17303*q^9+37608*q^95+33930*q^62+1237*q^101+20824*q^10+23456*q^11+ 25136*q^12+57426*q^76+25982*q^13+26142*q^14+25719*q^15+24887*q^16+38811*q^66+ 45561*q^70+59908*q^77+62278*q^78+21073*q^43+391*q^102+23524*q^49+20744*q^28+ 23913*q^17+23114*q^18+22743*q^19+22776*q^20+21768*q^27+49352*q^72+19592*q^40+ 51060*q^73+92*q^103+35397*q^63+18990*q^31+55100*q^75+20974*q^35+14*q^104+66698* q^80+23558*q^48+q^105+73210*q^83+68899*q^81+76213*q^85+77000*q^86+77161*q^87+ 76532*q^88+74880*q^89+74937*q^84+71095*q^82+30094*q^59+21071*q^36+62159*q^92+ 23599*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 73, 241, 757, 2288, 6724, 19365, 54959, 154303, 429733, 1189430, 3276306] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 9, 36, 119, 377, 1141, 3361, 9681, 27479, 77148, 214865, 594711, 1638151] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 12, 37, 122, 380, 1147, 3363, 9684, 27480, 77155, 214868, 594719, 1638155] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 36, 122, 380, 1141, 3361, 9684, 27480, 77148, 214865, 594719, 1638155] ODD:, [0, 1, 3, 12, 37, 119, 377, 1147, 3363, 9681, 27479, 77155, 214868, 594711, 1638151] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.808219178, 7.045643154, 9.648612946, 12.59921329, 15.87968471, 19.47503227, 23.37429720, 27.57068236, 32.06102394, 36.84500055, 41.92431873] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.526782813, 2.297452635, 3.316310705, 4.602929835, 6.168293851, 8.020846435, 10.16772123, 12.61498386, 15.36775676, 18.43032773, 21.80623187, 25.49830805] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.33107, 5.27829, 10.9979, 21.1870, 38.0478, 64.3340, 103.383, 159.138, 236.168, 339.677, 475.512, 650.164] Skewness: , [Float(undefined), 0., 0., 0.08022105171, 0.1809737159, 0.2637170352, 0.3287543512, 0.3839143293, 0.4339171738, 0.4805646489, 0.5241978498, 0.5645961778, 0.6014527647, 0.6345500034, 0.6638281560] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.326456034, 2.309849331, 2.250933746, 2.206331784, 2.179610064, 2.166645476, 2.163782120, 2.168182043, 2.177414505, 2.189478102, 2.202792175, 2.216138366] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [3, 2, 1, 4], [4, 1, 3, 2]}, {[1, 4, 3, 2], [1, 2, 3, 4], [3, 2, 4, 1]}, {[1, 4, 3, 2], [1, 2, 3, 4], [4, 2, 1, 3]}, {[2, 3, 1, 4], [4, 1, 2, 3], [4, 3, 2, 1]}, {[2, 3, 4, 1], [3, 1, 2, 4], [4, 3, 2, 1]}, {[1, 3, 4, 2], [4, 1, 2, 3], [4, 3, 2, 1]}, {[1, 2, 3, 4], [2, 4, 3, 1], [3, 2, 1, 4]}, {[1, 4, 2, 3], [2, 3, 4, 1], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [3, 2, 1, 4], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {2}], [[4, 3, 2, 1], %3, {1}], [[4, 2, 1, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}], [[4, 3, 1, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {1}], [[3, 2, 4, 1], %3, {1}], [[3, 4, 2, 1], %3, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 2, 0], [0, 3, 0, 0, 0], [0, 2, 0, 1, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 5, 3], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}], [[2, 3, 1, 5, 4], {[0, 0, 3, 0, 0, 0], [0, 1, 2, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 2, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[3, 4, 1, 5, 2], %2, {2}], [[3, 4, 2, 5, 1], %1, {1}], [[3, 1, 5, 4, 2], %2, {2}], [[3, 2, 5, 4, 1], %1, {1}], [[3, 1, 4, 5, 2], %2, {2}], [[3, 2, 4, 5, 1], %1, {1}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0], [0, 3, 0, 0, 0]}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[3, 2, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {2}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[2, 1, 4, 5, 3], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[2, 1, 3], {[0, 3, 0, 0]}, {}], [[2, 3, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 2, 0], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {2}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 2, 0, 0]}, {4}], [[2, 3, 1, 4], { [0, 0, 0, 0, 1], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0]}, {}], [[2, 1, 5, 4, 3], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {2}], [ [2, 1, 4, 3, 5], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 1]}, {5}], [ [1, 3, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [[2, 3, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {}], [[3, 1, 2], {[0, 0, 2, 0], [0, 1, 0, 0]}, {}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1]}, {}] , [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {4}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1], [0, 3, 0, 0, 0]}, {}], [[2, 4, 3, 1], { [0, 0, 0, 0, 1], [0, 0, 0, 2, 0], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {3}], [[2, 1], {[0, 3, 0]}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[2, 3, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1, 3, 5, 4], {[0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[2, 1, 3, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 3, 4], { [0, 3, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}]} %1 := {[0, 0, 3, 0, 0, 0], [0, 1, 2, 0, 0, 0], [0, 3, 0, 0, 0, 0], [0, 2, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} %2 := {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]} %3 := {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, 3*q^2+9*q^3+12*q^4+ 13*q^5+12*q^6+9*q^7+7*q^8+4*q^9+q^10, q^3+6*q^4+12*q^5+18*q^6+22*q^7+28*q^8+30* q^9+26*q^10+23*q^11+16*q^12+11*q^13+5*q^14+q^15, q^5+3*q^6+8*q^7+15*q^8+21*q^9+ 28*q^10+41*q^11+55*q^12+62*q^13+63*q^14+62*q^15+52*q^16+41*q^17+27*q^18+16*q^19 +6*q^20+q^21, q^8+3*q^9+5*q^10+9*q^11+18*q^12+31*q^13+42*q^14+54*q^15+71*q^16+ 94*q^17+121*q^18+141*q^19+144*q^20+136*q^21+121*q^22+98*q^23+70*q^24+43*q^25+22 *q^26+7*q^27+q^28, q^12+3*q^13+5*q^14+6*q^15+11*q^16+24*q^17+44*q^18+64*q^19+84 *q^20+102*q^21+125*q^22+163*q^23+221*q^24+279*q^25+311*q^26+318*q^27+302*q^28+ 275*q^29+231*q^30+175*q^31+115*q^32+65*q^33+29*q^34+8*q^35+q^36, q^17+3*q^18+5* q^19+6*q^20+8*q^21+16*q^22+32*q^23+58*q^24+94*q^25+134*q^26+165*q^27+189*q^28+ 227*q^29+290*q^30+388*q^31+506*q^32+607*q^33+673*q^34+691*q^35+673*q^36+621*q^ 37+536*q^38+425*q^39+299*q^40+182*q^41+94*q^42+37*q^43+9*q^44+q^45, q^23+3*q^24 +5*q^25+6*q^26+8*q^27+13*q^28+24*q^29+42*q^30+76*q^31+132*q^32+205*q^33+268*q^ 34+317*q^35+361*q^36+420*q^37+511*q^38+666*q^39+881*q^40+1113*q^41+1309*q^42+ 1438*q^43+1501*q^44+1490*q^45+1393*q^46+1235*q^47+1010*q^48+752*q^49+492*q^50+ 278*q^51+131*q^52+46*q^53+10*q^54+q^55, q^30+3*q^31+5*q^32+6*q^33+8*q^34+13*q^ 35+21*q^36+34*q^37+58*q^38+101*q^39+179*q^40+296*q^41+427*q^42+529*q^43+608*q^ 44+682*q^45+782*q^46+929*q^47+1161*q^48+1511*q^49+1954*q^50+2410*q^51+2802*q^52 +3086*q^53+3246*q^54+3259*q^55+3121*q^56+2813*q^57+2376*q^58+1839*q^59+1283*q^ 60+783*q^61+411*q^62+177*q^63+56*q^64+11*q^65+q^66, q^38+3*q^39+5*q^40+6*q^41+8 *q^42+13*q^43+21*q^44+31*q^45+50*q^46+82*q^47+137*q^48+237*q^49+416*q^50+643*q^ 51+860*q^52+1024*q^53+1162*q^54+1299*q^55+1472*q^56+1713*q^57+2074*q^58+2617*q^ 59+3374*q^60+4279*q^61+5188*q^62+5973*q^63+6586*q^64+6984*q^65+7113*q^66+6922*q ^67+6378*q^68+5517*q^69+4427*q^70+3238*q^71+2118*q^72+1209*q^73+590*q^74+233*q^ 75+67*q^76+12*q^77+q^78, q^47+3*q^48+5*q^49+6*q^50+8*q^51+13*q^52+21*q^53+31*q^ 54+47*q^55+74*q^56+118*q^57+188*q^58+321*q^59+563*q^60+928*q^61+1338*q^62+1704* q^63+1981*q^64+2216*q^65+2468*q^66+2791*q^67+3214*q^68+3774*q^69+4602*q^70+5837 *q^71+7458*q^72+9278*q^73+11053*q^74+12654*q^75+14010*q^76+14992*q^77+15465*q^ 78+15264*q^79+14343*q^80+12700*q^81+10500*q^82+7997*q^83+5524*q^84+3394*q^85+ 1816*q^86+825*q^87+300*q^88+79*q^89+13*q^90+q^91, q^57+3*q^58+5*q^59+6*q^60+8*q ^61+13*q^62+21*q^63+31*q^64+47*q^65+71*q^66+110*q^67+169*q^68+269*q^69+437*q^70 +755*q^71+1286*q^72+2004*q^73+2735*q^74+3338*q^75+3802*q^76+4229*q^77+4711*q^78 +5313*q^79+6053*q^80+6984*q^81+8280*q^82+10203*q^83+12912*q^84+16275*q^85+19876 *q^86+23409*q^87+26749*q^88+29740*q^89+32119*q^90+33477*q^91+33515*q^92+32025*q ^93+28981*q^94+24610*q^95+19405*q^96+14025*q^97+9153*q^98+5294*q^99+2660*q^100+ 1127*q^101+379*q^102+92*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+5*q^4+5*q^3+4*q^2+3*q+1, 3*q^8+9*q^7+12*q^6+13* q^5+12*q^4+9*q^3+7*q^2+4*q+1, q^12+6*q^11+12*q^10+18*q^9+22*q^8+28*q^7+30*q^6+ 26*q^5+23*q^4+16*q^3+11*q^2+5*q+1, q^16+3*q^15+8*q^14+15*q^13+21*q^12+28*q^11+ 41*q^10+55*q^9+62*q^8+63*q^7+62*q^6+52*q^5+41*q^4+27*q^3+16*q^2+6*q+1, q^20+3*q ^19+5*q^18+9*q^17+18*q^16+31*q^15+42*q^14+54*q^13+71*q^12+94*q^11+121*q^10+141* q^9+144*q^8+136*q^7+121*q^6+98*q^5+70*q^4+43*q^3+22*q^2+7*q+1, 1+5*q^22+q^24+6* q^21+3*q^23+8*q+29*q^2+65*q^3+115*q^4+175*q^5+231*q^6+275*q^7+302*q^8+318*q^9+ 311*q^10+279*q^11+221*q^12+163*q^13+125*q^14+102*q^15+84*q^16+64*q^17+44*q^18+ 24*q^19+11*q^20, 1+6*q^25+32*q^22+8*q^24+58*q^21+16*q^23+9*q+37*q^2+5*q^26+94*q ^3+182*q^4+299*q^5+425*q^6+536*q^7+621*q^8+673*q^9+691*q^10+673*q^11+607*q^12+ 506*q^13+388*q^14+290*q^15+227*q^16+q^28+189*q^17+165*q^18+134*q^19+94*q^20+3*q ^27, 1+42*q^25+205*q^22+76*q^24+268*q^21+132*q^23+10*q+5*q^30+6*q^29+46*q^2+24* q^26+131*q^3+278*q^4+492*q^5+752*q^6+q^32+1010*q^7+1235*q^8+1393*q^9+1490*q^10+ 1501*q^11+1438*q^12+1309*q^13+1113*q^14+881*q^15+666*q^16+8*q^28+511*q^17+420*q ^18+361*q^19+317*q^20+13*q^27+3*q^31, 1+296*q^25+608*q^22+427*q^24+682*q^21+529 *q^23+11*q+21*q^30+34*q^29+5*q^34+6*q^33+56*q^2+179*q^26+177*q^3+411*q^4+783*q^ 5+1283*q^6+8*q^32+1839*q^7+2376*q^8+2813*q^9+3121*q^10+3259*q^11+3246*q^12+3086 *q^13+2802*q^14+2410*q^15+1954*q^16+58*q^28+1511*q^17+1161*q^18+929*q^19+782*q^ 20+101*q^27+13*q^31+3*q^35+q^36, 1+1024*q^25+1472*q^22+1162*q^24+1713*q^21+1299 *q^23+12*q+137*q^30+6*q^37+237*q^29+3*q^39+21*q^34+5*q^38+31*q^33+67*q^2+860*q^ 26+233*q^3+590*q^4+1209*q^5+2118*q^6+50*q^32+3238*q^7+4427*q^8+5517*q^9+6378*q^ 10+6922*q^11+7113*q^12+6984*q^13+6586*q^14+5973*q^15+5188*q^16+416*q^28+4279*q^ 17+3374*q^18+2617*q^19+2074*q^20+643*q^27+q^40+82*q^31+13*q^35+8*q^36, 1+2468*q ^25+5*q^42+3774*q^22+2791*q^24+4602*q^21+3214*q^23+13*q+q^44+928*q^30+31*q^37+ 1338*q^29+13*q^39+6*q^41+118*q^34+21*q^38+188*q^33+79*q^2+2216*q^26+300*q^3+825 *q^4+1816*q^5+3394*q^6+321*q^32+5524*q^7+7997*q^8+10500*q^9+12700*q^10+14343*q^ 11+15264*q^12+15465*q^13+14992*q^14+14010*q^15+12654*q^16+3*q^43+1704*q^28+ 11053*q^17+9278*q^18+7458*q^19+5837*q^20+1981*q^27+8*q^40+563*q^31+74*q^35+47*q ^36, 1+6053*q^25+21*q^42+10203*q^22+6984*q^24+12912*q^21+8280*q^23+14*q+8*q^44+ 3338*q^30+6*q^45+169*q^37+3802*q^29+5*q^46+71*q^39+31*q^41+755*q^34+110*q^38+ 1286*q^33+92*q^2+5313*q^26+379*q^3+1127*q^4+2660*q^5+5294*q^6+2004*q^32+9153*q^ 7+14025*q^8+19405*q^9+24610*q^10+28981*q^11+32025*q^12+33515*q^13+33477*q^14+ 32119*q^15+29740*q^16+13*q^43+4229*q^28+26749*q^17+23409*q^18+19876*q^19+16275* q^20+4711*q^27+47*q^40+2735*q^31+437*q^35+q^48+269*q^36+3*q^47] The number of permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 70, 199, 502, 1232, 2962, 6970, 16138, 36982, 84083, 189918, 426722] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [1, 1, 3, 10, 35, 99, 249, 616, 1480, 3486, 8063, 18500, 42048, 94963, 213360] The number of ODD permutations avoiding, {[1, 2, 3, 4], [3, 2, 1, 4], [4, 1, 3, 2]}, is given by [0, 1, 3, 11, 35, 100, 253, 616, 1482, 3484, 8075, 18482, 42035, 94955, 213362] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 35, 100, 253, 616, 1480, 3484, 8075, 18500, 42048, 94955, 213362] ODD:, [0, 1, 3, 11, 35, 99, 249, 616, 1482, 3486, 8063, 18482, 42035, 94963, 213360] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.095238095, 5.471428571, 8.924623116, 13.65936255, 19.47077922, 26.32950709, 34.24189383, 43.20516793, 53.20842572, 64.24557877, 76.31391969, 89.41085765] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.411003085, 1.925182208, 2.510149595, 2.996262943, 3.422014772, 3.811404804, 4.178309897, 4.527628854, 4.857243475, 5.168488359, 5.463486018, 5.742650228] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.99093, 3.70633, 6.30085, 8.97759, 11.7102, 14.5268, 17.4583, 20.4994, 23.5928, 26.7133, 29.8497, 32.9780] Skewness: , [Float(undefined), 0., 0., 0.2379317809, 0.2297233919, 0.01481404220, -0.1997753258, -0.3400137007, -0.4346773611, -0.5087458729, -0.5769066908, -0.6392687242, -0.6964791350, -0.7505236214, -0.8024035001] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.142917187, 2.293329726, 2.386481206, 2.633146125, 2.854579386, 3.018595859, 3.137127125, 3.241141537, 3.344560547, 3.450152229, 3.560453038, 3.677702704] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [2, 3, 1, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, {[3, 2, 4, 1], [3, 1, 2, 4], [4, 1, 3, 2]}, {[1, 3, 4, 2], [3, 2, 4, 1], [3, 1, 2, 4]}, {[1, 4, 2, 3], [2, 4, 3, 1], [2, 3, 1, 4]}, {[1, 4, 2, 3], [2, 3, 1, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [3, 2, 4, 1], [4, 1, 3, 2]}, {[1, 4, 2, 3], [2, 4, 3, 1], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 1, 1]}, {4}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 0, 1, 1], [0, 1, 0, 1, 0]}, {}], [[1, 3, 2], {[0, 0, 1, 1]}, {}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]}, {3}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [[2, 4, 3, 1], { [0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [ [3, 1, 5, 4, 2], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 3, 4], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 1]}, {}], [[3, 2, 1], {[0, 1, 1, 0]}, {2}], [[2, 3, 1], {[0, 1, 1, 0], [0, 0, 1, 1]}, {1}], [[3, 2, 5, 4, 1], { [0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 1, 1, 0], [0, 1, 1, 0, 0, 0], [0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 0, 0, 0, 1, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+3*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+5*q^2+7*q^3+ 9*q^4+12*q^5+12*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+8*q^2+11*q^3+14*q^4+16*q^5+ 18*q^6+23*q^7+29*q^8+28*q^9+29*q^10+25*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*q+ 12*q^2+17*q^3+24*q^4+25*q^5+33*q^6+37*q^7+37*q^8+44*q^9+54*q^10+55*q^11+67*q^12 +68*q^13+69*q^14+68*q^15+59*q^16+47*q^17+31*q^18+16*q^19+6*q^20+q^21, 1+7*q+17* q^2+26*q^3+39*q^4+44*q^5+53*q^6+65*q^7+72*q^8+78*q^9+86*q^10+92*q^11+99*q^12+ 108*q^13+120*q^14+131*q^15+145*q^16+158*q^17+159*q^18+167*q^19+166*q^20+159*q^ 21+142*q^22+114*q^23+82*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+23*q^2+39*q^3+ 61*q^4+76*q^5+90*q^6+111*q^7+124*q^8+144*q^9+161*q^10+178*q^11+188*q^12+197*q^ 13+209*q^14+220*q^15+233*q^16+252*q^17+266*q^18+289*q^19+313*q^20+338*q^21+352* q^22+380*q^23+387*q^24+399*q^25+404*q^26+397*q^27+378*q^28+338*q^29+280*q^30+ 208*q^31+135*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+9*q+30*q^2+57*q^3+93*q^4+127*q ^5+152*q^6+193*q^7+216*q^8+249*q^9+290*q^10+327*q^11+361*q^12+397*q^13+405*q^14 +433*q^15+460*q^16+474*q^17+497*q^18+526*q^19+534*q^20+567*q^21+598*q^22+619*q^ 23+663*q^24+712*q^25+755*q^26+807*q^27+844*q^28+875*q^29+919*q^30+943*q^31+969* q^32+986*q^33+979*q^34+955*q^35+902*q^36+810*q^37+682*q^38+525*q^39+360*q^40+ 212*q^41+101*q^42+37*q^43+9*q^44+q^45, 1+10*q+38*q^2+81*q^3+139*q^4+205*q^5+255 *q^6+328*q^7+383*q^8+436*q^9+507*q^10+580*q^11+660*q^12+739*q^13+809*q^14+855*q ^15+916*q^16+958*q^17+992*q^18+1049*q^19+1095*q^20+1144*q^21+1180*q^22+1208*q^ 23+1241*q^24+1281*q^25+1327*q^26+1377*q^27+1441*q^28+1504*q^29+1575*q^30+1673*q ^31+1763*q^32+1849*q^33+1943*q^34+2033*q^35+2111*q^36+2178*q^37+2262*q^38+2311* q^39+2375*q^40+2413*q^41+2416*q^42+2389*q^43+2304*q^44+2165*q^45+1950*q^46+1659 *q^47+1312*q^48+940*q^49+595*q^50+320*q^51+139*q^52+46*q^53+10*q^54+q^55, 1+11* q+47*q^2+112*q^3+204*q^4+321*q^5+422*q^6+548*q^7+669*q^8+768*q^9+899*q^10+1020* q^11+1172*q^12+1335*q^13+1493*q^14+1643*q^15+1796*q^16+1912*q^17+2025*q^18+2107 *q^19+2201*q^20+2307*q^21+2433*q^22+2532*q^23+2629*q^24+2681*q^25+2754*q^26+ 2804*q^27+2868*q^28+2939*q^29+3040*q^30+3110*q^31+3214*q^32+3299*q^33+3387*q^34 +3517*q^35+3674*q^36+3837*q^37+4019*q^38+4203*q^39+4389*q^40+4592*q^41+4778*q^ 42+4934*q^43+5108*q^44+5276*q^45+5436*q^46+5605*q^47+5737*q^48+5854*q^49+5946*q ^50+5976*q^51+5949*q^52+5826*q^53+5590*q^54+5223*q^55+4709*q^56+4043*q^57+3256* q^58+2417*q^59+1614*q^60+945*q^61+467*q^62+186*q^63+56*q^64+11*q^65+q^66, 1+12* q+57*q^2+151*q^3+294*q^4+490*q^5+685*q^6+904*q^7+1147*q^8+1342*q^9+1590*q^10+ 1817*q^11+2076*q^12+2378*q^13+2694*q^14+2999*q^15+3371*q^16+3687*q^17+3963*q^18 +4261*q^19+4453*q^20+4672*q^21+4913*q^22+5129*q^23+5376*q^24+5637*q^25+5842*q^ 26+6013*q^27+6209*q^28+6298*q^29+6418*q^30+6558*q^31+6686*q^32+6859*q^33+7036*q ^34+7158*q^35+7316*q^36+7458*q^37+7588*q^38+7757*q^39+7965*q^40+8176*q^41+8478* q^42+8804*q^43+9106*q^44+9446*q^45+9819*q^46+10187*q^47+10615*q^48+11012*q^49+ 11393*q^50+11789*q^51+12161*q^52+12517*q^53+12887*q^54+13237*q^55+13594*q^56+ 13989*q^57+14279*q^58+14554*q^59+14758*q^60+14843*q^61+14831*q^62+14648*q^63+ 14252*q^64+13602*q^65+12662*q^66+11413*q^67+9862*q^68+8061*q^69+6141*q^70+4281* q^71+2669*q^72+1450*q^73+662*q^74+243*q^75+67*q^76+12*q^77+q^78, 1+13*q+68*q^2+ 199*q^3+416*q^4+732*q^5+1088*q^6+1474*q^7+1930*q^8+2323*q^9+2780*q^10+3234*q^11 +3697*q^12+4248*q^13+4815*q^14+5415*q^15+6105*q^16+6846*q^17+7519*q^18+8192*q^ 19+8817*q^20+9361*q^21+9901*q^22+10423*q^23+10890*q^24+11435*q^25+11960*q^26+ 12487*q^27+13030*q^28+13532*q^29+13918*q^30+14287*q^31+14536*q^32+14797*q^33+ 15096*q^34+15426*q^35+15733*q^36+16113*q^37+16398*q^38+16713*q^39+16946*q^40+ 17196*q^41+17401*q^42+17677*q^43+17976*q^44+18338*q^45+18751*q^46+19207*q^47+ 19641*q^48+20157*q^49+20670*q^50+21262*q^51+21926*q^52+22665*q^53+23472*q^54+ 24337*q^55+25181*q^56+25974*q^57+26772*q^58+27605*q^59+28431*q^60+29274*q^61+ 30087*q^62+30880*q^63+31717*q^64+32569*q^65+33394*q^66+34200*q^67+35006*q^68+ 35717*q^69+36345*q^70+36818*q^71+37071*q^72+37106*q^73+36803*q^74+36119*q^75+ 34948*q^76+33192*q^77+30808*q^78+27761*q^79+24093*q^80+19924*q^81+15494*q^82+ 11164*q^83+7317*q^84+4268*q^85+2159*q^86+915*q^87+311*q^88+79*q^89+13*q^90+q^91 , 1+14*q+80*q^2+257*q^3+578*q^4+1073*q^5+1691*q^6+2373*q^7+3193*q^8+3973*q^9+ 4810*q^10+5707*q^11+6567*q^12+7597*q^13+8638*q^14+9739*q^15+10987*q^16+12412*q^ 17+13829*q^18+15358*q^19+16759*q^20+18163*q^21+19525*q^22+20754*q^23+21954*q^24 +23139*q^25+24256*q^26+25403*q^27+26606*q^28+27808*q^29+29093*q^30+30246*q^31+ 31279*q^32+32210*q^33+32955*q^34+33674*q^35+34310*q^36+35011*q^37+35690*q^38+ 36462*q^39+37170*q^40+37923*q^41+38655*q^42+39212*q^43+39683*q^44+40155*q^45+ 40551*q^46+41059*q^47+41624*q^48+42244*q^49+42909*q^50+43698*q^51+44381*q^52+ 45118*q^53+45838*q^54+46572*q^55+47431*q^56+48491*q^57+49656*q^58+51019*q^59+ 52476*q^60+53965*q^61+55511*q^62+57120*q^63+58718*q^64+60450*q^65+62256*q^66+ 64049*q^67+65915*q^68+67771*q^69+69476*q^70+71222*q^71+72981*q^72+74795*q^73+ 76677*q^74+78596*q^75+80512*q^76+82493*q^77+84461*q^78+86313*q^79+88114*q^80+ 89684*q^81+91112*q^82+92256*q^83+92948*q^84+93143*q^85+92678*q^86+91409*q^87+ 89196*q^88+85858*q^89+81221*q^90+75180*q^91+67730*q^92+58977*q^93+49206*q^94+ 38907*q^95+28779*q^96+19621*q^97+12109*q^98+6624*q^99+3131*q^100+1237*q^101+391 *q^102+92*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+5*q^8+7 *q^7+9*q^6+12*q^5+12*q^4+11*q^3+7*q^2+4*q+1, q^15+5*q^14+8*q^13+11*q^12+14*q^11 +16*q^10+18*q^9+23*q^8+29*q^7+28*q^6+29*q^5+25*q^4+19*q^3+11*q^2+5*q+1, q^21+6* q^20+12*q^19+17*q^18+24*q^17+25*q^16+33*q^15+37*q^14+37*q^13+44*q^12+54*q^11+55 *q^10+67*q^9+68*q^8+69*q^7+68*q^6+59*q^5+47*q^4+31*q^3+16*q^2+6*q+1, 1+26*q^25+ 53*q^22+39*q^24+65*q^21+44*q^23+7*q+22*q^2+17*q^26+48*q^3+82*q^4+114*q^5+142*q^ 6+159*q^7+166*q^8+167*q^9+159*q^10+158*q^11+145*q^12+131*q^13+120*q^14+108*q^15 +99*q^16+q^28+92*q^17+86*q^18+78*q^19+72*q^20+7*q^27, 1+178*q^25+209*q^22+188*q ^24+220*q^21+197*q^23+8*q+90*q^30+111*q^29+23*q^34+39*q^33+29*q^2+161*q^26+71*q ^3+135*q^4+208*q^5+280*q^6+61*q^32+338*q^7+378*q^8+397*q^9+404*q^10+399*q^11+ 387*q^12+380*q^13+352*q^14+338*q^15+313*q^16+124*q^28+289*q^17+266*q^18+252*q^ 19+233*q^20+144*q^27+76*q^31+8*q^35+q^36, 1+534*q^25+57*q^42+619*q^22+567*q^24+ 663*q^21+598*q^23+9*q+9*q^44+433*q^30+q^45+216*q^37+460*q^29+152*q^39+93*q^41+ 327*q^34+193*q^38+361*q^33+37*q^2+526*q^26+101*q^3+212*q^4+360*q^5+525*q^6+397* q^32+682*q^7+810*q^8+902*q^9+955*q^10+979*q^11+986*q^12+969*q^13+943*q^14+919*q ^15+875*q^16+30*q^43+474*q^28+844*q^17+807*q^18+755*q^19+712*q^20+497*q^27+127* q^40+405*q^31+290*q^35+249*q^36, 1+1575*q^25+739*q^42+1849*q^22+1673*q^24+1943* q^21+1763*q^23+10*q+139*q^51+580*q^44+38*q^53+1281*q^30+81*q^52+q^55+10*q^54+ 507*q^45+992*q^37+1327*q^29+436*q^46+916*q^39+809*q^41+1144*q^34+205*q^50+958*q ^38+1180*q^33+46*q^2+1504*q^26+139*q^3+320*q^4+595*q^5+940*q^6+1208*q^32+1312*q ^7+1659*q^8+1950*q^9+2165*q^10+2304*q^11+2389*q^12+2416*q^13+2413*q^14+2375*q^ 15+2311*q^16+660*q^43+255*q^49+1377*q^28+2262*q^17+2178*q^18+2111*q^19+2033*q^ 20+1441*q^27+855*q^40+1241*q^31+1095*q^35+328*q^48+1049*q^36+383*q^47, 1+4592*q ^25+2629*q^42+5108*q^22+4778*q^24+5276*q^21+4934*q^23+11*q+1643*q^51+2433*q^44+ 1335*q^53+768*q^57+3674*q^30+1493*q^52+669*q^58+1020*q^55+422*q^60+1172*q^54+ 2307*q^45+2939*q^37+3837*q^29+2201*q^46+11*q^65+2804*q^39+321*q^61+2681*q^41+ 3214*q^34+1796*q^50+47*q^64+2868*q^38+3299*q^33+899*q^56+56*q^2+4389*q^26+186*q ^3+467*q^4+945*q^5+1614*q^6+3387*q^32+2417*q^7+3256*q^8+4043*q^9+204*q^62+4709* q^10+5223*q^11+5590*q^12+5826*q^13+5949*q^14+5976*q^15+5946*q^16+q^66+2532*q^43 +1912*q^49+4019*q^28+5854*q^17+5737*q^18+5605*q^19+5436*q^20+4203*q^27+2754*q^ 40+112*q^63+3517*q^31+3110*q^35+2025*q^48+548*q^59+3040*q^36+2107*q^47, 1+12517 *q^25+7316*q^42+13594*q^22+12887*q^24+13989*q^21+13237*q^23+12*q+6013*q^51+7036 *q^44+5637*q^53+4672*q^57+10615*q^30+5842*q^52+4453*q^58+5129*q^55+3963*q^60+ 5376*q^54+6859*q^45+904*q^71+294*q^74+8176*q^37+1817*q^67+11012*q^29+6686*q^46+ 2378*q^65+1342*q^69+7757*q^39+3687*q^61+7458*q^41+9106*q^34+6209*q^50+2694*q^64 +7965*q^38+9446*q^33+4913*q^56+67*q^2+1590*q^68+12161*q^26+243*q^3+662*q^4+1450 *q^5+2669*q^6+9819*q^32+4281*q^7+6141*q^8+8061*q^9+3371*q^62+9862*q^10+11413*q^ 11+12662*q^12+57*q^76+13602*q^13+14252*q^14+14648*q^15+14831*q^16+2076*q^66+ 1147*q^70+12*q^77+q^78+7158*q^43+6298*q^49+11393*q^28+14843*q^17+14758*q^18+ 14554*q^19+14279*q^20+11789*q^27+685*q^72+7588*q^40+490*q^73+2999*q^63+10187*q^ 31+151*q^75+8804*q^35+6418*q^48+4261*q^59+8478*q^36+6558*q^47, 1+33394*q^25+ 20157*q^42+35717*q^22+13*q^90+34200*q^24+36345*q^21+35006*q^23+13*q+16946*q^51+ 19207*q^44+16398*q^53+15096*q^57+29274*q^30+16713*q^52+14797*q^58+15733*q^55+ 14287*q^60+q^91+16113*q^54+18751*q^45+8817*q^71+6846*q^74+23472*q^37+10890*q^67 +30087*q^29+18338*q^46+11960*q^65+9901*q^69+21926*q^39+13918*q^61+20670*q^41+ 25974*q^34+3697*q^79+17196*q^50+12487*q^64+22665*q^38+26772*q^33+15426*q^56+79* q^2+10423*q^68+32569*q^26+311*q^3+915*q^4+2159*q^5+4268*q^6+27605*q^32+7317*q^7 +11164*q^8+15494*q^9+13532*q^62+19924*q^10+24093*q^11+27761*q^12+5415*q^76+ 30808*q^13+33192*q^14+34948*q^15+36119*q^16+11435*q^66+9361*q^70+4815*q^77+4248 *q^78+19641*q^43+17401*q^49+30880*q^28+36803*q^17+37106*q^18+37071*q^19+36818*q ^20+31717*q^27+8192*q^72+21262*q^40+7519*q^73+13030*q^63+28431*q^31+6105*q^75+ 25181*q^35+3234*q^80+17677*q^48+1930*q^83+2780*q^81+1088*q^85+732*q^86+416*q^87 +199*q^88+68*q^89+1474*q^84+2323*q^82+14536*q^59+24337*q^36+17976*q^47, 1+88114 *q^25+57120*q^42+92256*q^22+9739*q^90+89684*q^24+92948*q^21+91112*q^23+14*q+ 3193*q^97+45838*q^51+53965*q^44+44381*q^53+41624*q^57+78596*q^30+5707*q^94+ 45118*q^52+41059*q^58+42909*q^55+40155*q^60+8638*q^91+43698*q^54+52476*q^45+ 32955*q^71+30246*q^74+3973*q^96+2373*q^98+65915*q^37+35690*q^67+80512*q^29+ 51019*q^46+6567*q^93+1691*q^99+37170*q^65+34310*q^69+62256*q^39+39683*q^61+ 58718*q^41+71222*q^34+24256*q^79+46572*q^50+37923*q^64+1073*q^100+64049*q^38+ 72981*q^33+42244*q^56+92*q^2+35011*q^68+86313*q^26+391*q^3+1237*q^4+3131*q^5+ 6624*q^6+74795*q^32+12109*q^7+19621*q^8+28779*q^9+4810*q^95+39212*q^62+578*q^ 101+38907*q^10+49206*q^11+58977*q^12+27808*q^76+67730*q^13+75180*q^14+81221*q^ 15+85858*q^16+36462*q^66+33674*q^70+26606*q^77+25403*q^78+55511*q^43+257*q^102+ 47431*q^49+82493*q^28+89196*q^17+91409*q^18+92678*q^19+93143*q^20+84461*q^27+ 32210*q^72+60450*q^40+31279*q^73+80*q^103+38655*q^63+76677*q^31+29093*q^75+ 69476*q^35+14*q^104+23139*q^80+48491*q^48+q^105+19525*q^83+21954*q^81+16759*q^ 85+15358*q^86+13829*q^87+12412*q^88+10987*q^89+18163*q^84+20754*q^82+40551*q^59 +67771*q^36+7597*q^92+49656*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 73, 243, 777, 2408, 7288, 21661, 63471, 183877, 527761, 1503086, 4252938] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 35, 123, 393, 1204, 3635, 10820, 31739, 91983, 263917, 751431, 2126311] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 1, 2, 4], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 38, 120, 384, 1204, 3653, 10841, 31732, 91894, 263844, 751655, 2126627] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 35, 120, 384, 1204, 3635, 10841, 31732, 91983, 263917, 751655, 2126627] ODD:, [0, 1, 3, 12, 38, 123, 393, 1204, 3653, 10820, 31739, 91894, 263844, 751431, 2126311] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.232876712, 8.082304527, 11.60360360, 15.80481728, 20.69552689, 26.28544389, 32.58347907, 39.59739391, 47.33377419, 55.79813996, 64.99509586] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.526782813, 2.266616257, 3.214136668, 4.378770288, 5.760886852, 7.359210701, 9.172482750, 11.19970542, 13.44014094, 15.89328380, 18.55883201, 21.43665545] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.33107, 5.13755, 10.3307, 19.1736, 33.1878, 54.1580, 84.1344, 125.433, 180.637, 252.596, 344.430, 459.530] Skewness: , [Float(undefined), 0., 0., -0.08021543222, -0.2046139656, -0.3093392422, -0.3818419503, -0.4318589899, -0.4685214444, -0.4972327115, -0.5208966266, -0.5410571656, -0.5585844924, -0.5739949759, -0.5876232692] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.326456034, 2.397127878, 2.405808756, 2.400262090, 2.393669897, 2.388833099, 2.386037710, 2.384996923, 2.385272881, 2.386468495, 2.388174696, 2.390113218] end of this data For the equivalence class of patterns, { {[2, 3, 4, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 4, 3, 2], [3, 4, 2, 1], [3, 2, 1, 4]}, {[1, 2, 4, 3], [2, 3, 4, 1], [4, 1, 2, 3]}} the member , {[2, 3, 4, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 1, 0]}, {}], [[2, 3, 1], {[0, 1, 1, 1]}, {}], [ [4, 2, 5, 3, 1], {[0, 1, 0, 0, 1, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 2, 1], {}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 0, 0, 1]}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[1, 4, 3, 2, 5], {[0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {1}], [ [2, 5, 4, 3, 1], {[0, 1, 1, 0, 0, 1], [0, 1, 0, 0, 1, 0], [0, 1, 0, 1, 0, 0]}, {3}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {}], [[1, 5, 4, 3, 2], {}, {3}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0], [0, 1, 1, 0, 1]}, {1}], [[4, 3, 2, 1], {}, {2}], [[3, 4, 2, 1], {[0, 0, 1, 0, 1], [0, 1, 0, 1, 1]}, {3}], [[4, 3, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[4, 2, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1]}, {1}], [[1, 5, 3, 2, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {3}], [ [2, 5, 4, 1, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [3, 5, 4, 1, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {2}], [ [1, 5, 4, 2, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0]}, {1}], [[4, 3, 2, 5, 1], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 5, 3, 1, 4], { [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1, 5], {[0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {2}], [ [4, 2, 1, 5, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[3, 2, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 1, 5, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0]}, {3}], [[4, 3, 1, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [4, 1, 5, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[3, 1, 4, 2, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[1, 3, 2], {}, {}], [[1, 4, 2, 3], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 1, 0, 1, 0], [0, 1, 1, 0, 1]}, {}], [[3, 2, 1, 4], {[0, 0, 0, 0, 1]}, {}], [[3, 4, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[3, 5, 4, 2, 1], {[0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 0, 1]}, {4}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+4*q^3+5*q^4+3*q^5+q^6, 1+2*q+5*q^2+6*q^3+ 12*q^4+12*q^5+10*q^6+12*q^7+9*q^8+4*q^9+q^10, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+ 18*q^6+28*q^7+30*q^8+28*q^9+26*q^10+31*q^11+25*q^12+14*q^13+5*q^14+q^15, 1+2*q+ 5*q^2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^7+41*q^8+46*q^9+50*q^10+64*q^11+74*q^12+ 70*q^13+70*q^14+74*q^15+82*q^16+70*q^17+44*q^18+20*q^19+6*q^20+q^21, 1+2*q+5*q^ 2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+108*q ^13+121*q^14+130*q^15+158*q^16+182*q^17+181*q^18+184*q^19+199*q^20+215*q^21+227 *q^22+196*q^23+134*q^24+70*q^25+27*q^26+7*q^27+q^28, 1+2*q+5*q^2+6*q^3+12*q^4+ 17*q^5+18*q^6+31*q^7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+ 157*q^15+194*q^16+237*q^17+258*q^18+289*q^19+321*q^20+361*q^21+418*q^22+478*q^ 23+487*q^24+509*q^25+548*q^26+600*q^27+641*q^28+644*q^29+558*q^30+400*q^31+231* q^32+104*q^33+35*q^34+8*q^35+q^36, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^ 7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+157*q^15+194*q^16+246 *q^17+278*q^18+324*q^19+373*q^20+437*q^21+512*q^22+597*q^23+650*q^24+728*q^25+ 828*q^26+933*q^27+1066*q^28+1211*q^29+1344*q^30+1397*q^31+1476*q^32+1588*q^33+ 1728*q^34+1867*q^35+1931*q^36+1870*q^37+1609*q^38+1190*q^39+735*q^40+370*q^41+ 147*q^42+44*q^43+9*q^44+q^45, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^7+41* q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+157*q^15+194*q^16+246*q^17 +278*q^18+324*q^19+373*q^20+442*q^21+534*q^22+638*q^23+703*q^24+801*q^25+938*q^ 26+1084*q^27+1259*q^28+1435*q^29+1624*q^30+1763*q^31+1992*q^32+2262*q^33+2581*q ^34+2956*q^35+3352*q^36+3725*q^37+4052*q^38+4238*q^39+4481*q^40+4802*q^41+5187* q^42+5584*q^43+5860*q^44+5872*q^45+5514*q^46+4704*q^47+3542*q^48+2296*q^49+1252 *q^50+561*q^51+200*q^52+54*q^53+10*q^54+q^55, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+ 18*q^6+31*q^7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+157*q^15+ 194*q^16+246*q^17+278*q^18+324*q^19+373*q^20+442*q^21+534*q^22+638*q^23+703*q^ 24+801*q^25+949*q^26+1113*q^27+1312*q^28+1510*q^29+1726*q^30+1894*q^31+2165*q^ 32+2486*q^33+2891*q^34+3365*q^35+3829*q^36+4306*q^37+4786*q^38+5230*q^39+5834*q ^40+6661*q^41+7580*q^42+8680*q^43+9845*q^44+10981*q^45+11978*q^46+12835*q^47+ 13430*q^48+14135*q^49+15039*q^50+16116*q^51+17228*q^52+18112*q^53+18456*q^54+ 17958*q^55+16461*q^56+13907*q^57+10586*q^58+7099*q^59+4110*q^60+2013*q^61+815*q ^62+264*q^63+65*q^64+11*q^65+q^66, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^ 7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+157*q^15+194*q^16+246 *q^17+278*q^18+324*q^19+373*q^20+442*q^21+534*q^22+638*q^23+703*q^24+801*q^25+ 949*q^26+1113*q^27+1312*q^28+1510*q^29+1726*q^30+1900*q^31+2194*q^32+2547*q^33+ 2975*q^34+3469*q^35+3970*q^36+4511*q^37+5054*q^38+5556*q^39+6222*q^40+7166*q^41 +8261*q^42+9568*q^43+10961*q^44+12379*q^45+13718*q^46+15141*q^47+16536*q^48+ 18361*q^49+20723*q^50+23542*q^51+26767*q^52+30235*q^53+33696*q^54+36905*q^55+ 39686*q^56+42009*q^57+43871*q^58+45915*q^59+48461*q^60+51471*q^61+54605*q^62+ 57229*q^63+58651*q^64+58144*q^65+55188*q^66+49588*q^67+41516*q^68+31792*q^69+ 21849*q^70+13232*q^71+6939*q^72+3092*q^73+1144*q^74+340*q^75+77*q^76+12*q^77+q^ 78, 1+2*q+5*q^2+6*q^3+12*q^4+17*q^5+18*q^6+31*q^7+41*q^8+46*q^9+57*q^10+77*q^11 +98*q^12+112*q^13+137*q^14+157*q^15+194*q^16+246*q^17+278*q^18+324*q^19+373*q^ 20+442*q^21+534*q^22+638*q^23+703*q^24+801*q^25+949*q^26+1113*q^27+1312*q^28+ 1510*q^29+1726*q^30+1900*q^31+2194*q^32+2547*q^33+2975*q^34+3469*q^35+3970*q^36 +4524*q^37+5094*q^38+5636*q^39+6338*q^40+7315*q^41+8440*q^42+9797*q^43+11262*q^ 44+12791*q^45+14266*q^46+15827*q^47+17345*q^48+19316*q^49+21965*q^50+25186*q^51 +28984*q^52+33125*q^53+37422*q^54+41666*q^55+45975*q^56+50313*q^57+54933*q^58+ 60558*q^59+67701*q^60+76188*q^61+85960*q^62+96340*q^63+106923*q^64+117016*q^65+ 126159*q^66+133993*q^67+140642*q^68+146325*q^69+152410*q^70+159654*q^71+168117* q^72+176930*q^73+184548*q^74+189226*q^75+189241*q^76+183185*q^77+170232*q^78+ 150466*q^79+124920*q^80+95973*q^81+67137*q^82+42085*q^83+23274*q^84+11176*q^85+ 4576*q^86+1561*q^87+429*q^88+90*q^89+13*q^90+q^91, 1+2*q+5*q^2+6*q^3+12*q^4+17* q^5+18*q^6+31*q^7+41*q^8+46*q^9+57*q^10+77*q^11+98*q^12+112*q^13+137*q^14+157*q ^15+194*q^16+246*q^17+278*q^18+324*q^19+373*q^20+442*q^21+534*q^22+638*q^23+703 *q^24+801*q^25+949*q^26+1113*q^27+1312*q^28+1510*q^29+1726*q^30+1900*q^31+2194* q^32+2547*q^33+2975*q^34+3469*q^35+3970*q^36+4524*q^37+5094*q^38+5636*q^39+6338 *q^40+7315*q^41+8440*q^42+9804*q^43+11299*q^44+12879*q^45+14401*q^46+15993*q^47 +17544*q^48+19582*q^49+22328*q^50+25630*q^51+29516*q^52+33784*q^53+38299*q^54+ 42852*q^55+47492*q^56+52107*q^57+57091*q^58+63287*q^59+71225*q^60+80907*q^61+ 92351*q^62+104851*q^63+118130*q^64+131626*q^65+145201*q^66+158910*q^67+173186*q ^68+188478*q^69+206604*q^70+228726*q^71+254892*q^72+284603*q^73+316428*q^74+ 348858*q^75+380464*q^76+409930*q^77+436091*q^78+458683*q^79+478147*q^80+495687* q^81+513976*q^82+535048*q^83+559002*q^84+583826*q^85+605567*q^86+619754*q^87+ 621965*q^88+608578*q^89+577076*q^90+526758*q^91+459178*q^92+378374*q^93+291134* q^94+206340*q^95+132836*q^96+76612*q^97+39038*q^98+17314*q^99+6566*q^100+2080*q ^101+532*q^102+104*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+5*q^4+4*q^3+5*q^2+3*q+1, q^10+2*q^9+5*q^8+6 *q^7+12*q^6+12*q^5+10*q^4+12*q^3+9*q^2+4*q+1, q^15+2*q^14+5*q^13+6*q^12+12*q^11 +17*q^10+18*q^9+28*q^8+30*q^7+28*q^6+26*q^5+31*q^4+25*q^3+14*q^2+5*q+1, q^21+2* q^20+5*q^19+6*q^18+12*q^17+17*q^16+18*q^15+31*q^14+41*q^13+46*q^12+50*q^11+64*q ^10+74*q^9+70*q^8+70*q^7+74*q^6+82*q^5+70*q^4+44*q^3+20*q^2+6*q+1, 1+6*q^25+18* q^22+12*q^24+31*q^21+17*q^23+7*q+27*q^2+5*q^26+70*q^3+134*q^4+196*q^5+227*q^6+ 215*q^7+199*q^8+184*q^9+181*q^10+182*q^11+158*q^12+130*q^13+121*q^14+108*q^15+ 98*q^16+q^28+77*q^17+57*q^18+46*q^19+41*q^20+2*q^27, 1+77*q^25+137*q^22+98*q^24 +157*q^21+112*q^23+8*q+18*q^30+31*q^29+5*q^34+6*q^33+35*q^2+57*q^26+104*q^3+231 *q^4+400*q^5+558*q^6+12*q^32+644*q^7+641*q^8+600*q^9+548*q^10+509*q^11+487*q^12 +478*q^13+418*q^14+361*q^15+321*q^16+41*q^28+289*q^17+258*q^18+237*q^19+194*q^ 20+46*q^27+17*q^31+2*q^35+q^36, 1+373*q^25+6*q^42+597*q^22+437*q^24+650*q^21+ 512*q^23+9*q+2*q^44+157*q^30+q^45+41*q^37+194*q^29+18*q^39+12*q^41+77*q^34+31*q ^38+98*q^33+44*q^2+324*q^26+147*q^3+370*q^4+735*q^5+1190*q^6+112*q^32+1609*q^7+ 1870*q^8+1931*q^9+1867*q^10+1728*q^11+1588*q^12+1476*q^13+1397*q^14+1344*q^15+ 1211*q^16+5*q^43+246*q^28+1066*q^17+933*q^18+828*q^19+728*q^20+278*q^27+17*q^40 +137*q^31+57*q^35+46*q^36, 1+1624*q^25+112*q^42+2262*q^22+1763*q^24+2581*q^21+ 1992*q^23+10*q+12*q^51+77*q^44+5*q^53+801*q^30+6*q^52+q^55+2*q^54+57*q^45+278*q ^37+938*q^29+46*q^46+194*q^39+137*q^41+442*q^34+17*q^50+246*q^38+534*q^33+54*q^ 2+1435*q^26+200*q^3+561*q^4+1252*q^5+2296*q^6+638*q^32+3542*q^7+4704*q^8+5514*q ^9+5872*q^10+5860*q^11+5584*q^12+5187*q^13+4802*q^14+4481*q^15+4238*q^16+98*q^ 43+18*q^49+1084*q^28+4052*q^17+3725*q^18+3352*q^19+2956*q^20+1259*q^27+157*q^40 +703*q^31+373*q^35+31*q^48+324*q^36+41*q^47, 1+6661*q^25+703*q^42+9845*q^22+ 7580*q^24+10981*q^21+8680*q^23+11*q+157*q^51+534*q^44+112*q^53+46*q^57+3829*q^ 30+137*q^52+41*q^58+77*q^55+18*q^60+98*q^54+442*q^45+1510*q^37+4306*q^29+373*q^ 46+2*q^65+1113*q^39+17*q^61+801*q^41+2165*q^34+194*q^50+5*q^64+1312*q^38+2486*q ^33+57*q^56+65*q^2+5834*q^26+264*q^3+815*q^4+2013*q^5+4110*q^6+2891*q^32+7099*q ^7+10586*q^8+13907*q^9+12*q^62+16461*q^10+17958*q^11+18456*q^12+18112*q^13+ 17228*q^14+16116*q^15+15039*q^16+q^66+638*q^43+246*q^49+4786*q^28+14135*q^17+ 13430*q^18+12835*q^19+11978*q^20+5230*q^27+949*q^40+6*q^63+3365*q^31+1894*q^35+ 278*q^48+31*q^59+1726*q^36+324*q^47, 1+30235*q^25+3970*q^42+39686*q^22+33696*q^ 24+42009*q^21+36905*q^23+12*q+1113*q^51+2975*q^44+801*q^53+442*q^57+16536*q^30+ 949*q^52+373*q^58+638*q^55+278*q^60+703*q^54+2547*q^45+31*q^71+12*q^74+7166*q^ 37+77*q^67+18361*q^29+2194*q^46+112*q^65+46*q^69+5556*q^39+246*q^61+4511*q^41+ 10961*q^34+1312*q^50+137*q^64+6222*q^38+12379*q^33+534*q^56+77*q^2+57*q^68+ 26767*q^26+340*q^3+1144*q^4+3092*q^5+6939*q^6+13718*q^32+13232*q^7+21849*q^8+ 31792*q^9+194*q^62+41516*q^10+49588*q^11+55188*q^12+5*q^76+58144*q^13+58651*q^ 14+57229*q^15+54605*q^16+98*q^66+41*q^70+2*q^77+q^78+3469*q^43+1510*q^49+20723* q^28+51471*q^17+48461*q^18+45915*q^19+43871*q^20+23542*q^27+18*q^72+5054*q^40+ 17*q^73+157*q^63+15141*q^31+6*q^75+9568*q^35+1726*q^48+324*q^59+8261*q^36+1900* q^47, 1+126159*q^25+19316*q^42+146325*q^22+2*q^90+133993*q^24+152410*q^21+ 140642*q^23+13*q+6338*q^51+15827*q^44+5094*q^53+2975*q^57+76188*q^30+5636*q^52+ 2547*q^58+3970*q^55+1900*q^60+q^91+4524*q^54+14266*q^45+373*q^71+246*q^74+37422 *q^37+703*q^67+85960*q^29+12791*q^46+949*q^65+534*q^69+28984*q^39+1726*q^61+ 21965*q^41+50313*q^34+98*q^79+7315*q^50+1113*q^64+33125*q^38+54933*q^33+3469*q^ 56+90*q^2+638*q^68+117016*q^26+429*q^3+1561*q^4+4576*q^5+11176*q^6+60558*q^32+ 23274*q^7+42085*q^8+67137*q^9+1510*q^62+95973*q^10+124920*q^11+150466*q^12+157* q^76+170232*q^13+183185*q^14+189241*q^15+189226*q^16+801*q^66+442*q^70+137*q^77 +112*q^78+17345*q^43+8440*q^49+96340*q^28+184548*q^17+176930*q^18+168117*q^19+ 159654*q^20+106923*q^27+324*q^72+25186*q^40+278*q^73+1312*q^63+67701*q^31+194*q ^75+45975*q^35+77*q^80+9797*q^48+41*q^83+57*q^81+18*q^85+17*q^86+12*q^87+6*q^88 +5*q^89+31*q^84+46*q^82+2194*q^59+41666*q^36+11262*q^47, 1+478147*q^25+104851*q ^42+535048*q^22+157*q^90+495687*q^24+559002*q^21+513976*q^23+14*q+41*q^97+38299 *q^51+80907*q^44+29516*q^53+17544*q^57+348858*q^30+77*q^94+33784*q^52+15993*q^ 58+22328*q^55+12879*q^60+137*q^91+25630*q^54+71225*q^45+2975*q^71+1900*q^74+46* q^96+31*q^98+173186*q^37+5094*q^67+380464*q^29+63287*q^46+98*q^93+18*q^99+6338* q^65+3970*q^69+145201*q^39+11299*q^61+118130*q^41+228726*q^34+949*q^79+42852*q^ 50+7315*q^64+17*q^100+158910*q^38+254892*q^33+19582*q^56+104*q^2+4524*q^68+ 458683*q^26+532*q^3+2080*q^4+6566*q^5+17314*q^6+284603*q^32+39038*q^7+76612*q^8 +132836*q^9+57*q^95+9804*q^62+12*q^101+206340*q^10+291134*q^11+378374*q^12+1510 *q^76+459178*q^13+526758*q^14+577076*q^15+608578*q^16+5636*q^66+3469*q^70+1312* q^77+1113*q^78+92351*q^43+6*q^102+47492*q^49+409930*q^28+621965*q^17+619754*q^ 18+605567*q^19+583826*q^20+436091*q^27+2547*q^72+131626*q^40+2194*q^73+5*q^103+ 8440*q^63+316428*q^31+1726*q^75+206604*q^35+2*q^104+801*q^80+52107*q^48+q^105+ 534*q^83+703*q^81+373*q^85+324*q^86+278*q^87+246*q^88+194*q^89+442*q^84+638*q^ 82+14401*q^59+188478*q^36+112*q^92+57091*q^47] The number of permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, is given by [1, 2, 6, 21, 74, 249, 804, 2551, 8139, 26500, 88531, 303112, 1059129, 3759584, 13505901] The number of EVEN permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, is given by [1, 1, 3, 12, 38, 122, 403, 1280, 4061, 13249, 44295, 151537, 529503, 1879883, 6753022] The number of ODD permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, is given by [0, 1, 3, 9, 36, 127, 401, 1271, 4078, 13251, 44236, 151575, 529626, 1879701, 6752879] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 12, 38, 127, 401, 1280, 4061, 13251, 44236, 151537, 529503, 1879701, 6752879] ODD:, [0, 1, 3, 9, 36, 122, 403, 1271, 4078, 13249, 44295, 151575, 529626, 1879883, 6753022] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.095238095, 5.405405405, 8.542168675, 12.60820896, 17.69070953, 23.83167465, 31.01173585, 39.17048266, 48.24580023, 58.19962441, 69.02027751, 80.71267256] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.508855192, 2.186685255, 3.008560230, 3.949240405, 4.943703843, 5.910744094, 6.790841087, 7.573817270, 8.292812772, 8.994992966, 9.716788529, 10.47626641] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.27664, 4.78159, 9.05143, 15.5965, 24.4402, 34.9369, 46.1155, 57.3627, 68.7707, 80.9099, 94.4160, 109.752] Skewness: , [Float(undefined), 0., 0., -0.07871347439, -0.1980904487, -0.3372091340, -0.4866941614, -0.6459949577, -0.8071646855, -0.9529703148, -1.064578495, -1.132025829, -1.159080185, -1.158770824, -1.144939150] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.311169730, 2.439646112, 2.541207176, 2.689253097, 2.935796707, 3.299691099, 3.750151636, 4.206378015, 4.574968920, 4.803547408, 4.901069376, 4.912281983] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [2, 4, 1, 3], [2, 3, 1, 4]}, {[2, 4, 1, 3], [4, 3, 2, 1], [4, 2, 1, 3]}, {[3, 1, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, {[2, 4, 3, 1], [2, 4, 1, 3], [4, 3, 2, 1]}, {[1, 3, 4, 2], [1, 2, 3, 4], [3, 1, 4, 2]}, {[1, 4, 2, 3], [1, 2, 3, 4], [2, 4, 1, 3]}, {[1, 2, 3, 4], [3, 1, 4, 2], [3, 1, 2, 4]}, {[3, 2, 4, 1], [3, 1, 4, 2], [4, 3, 2, 1]}} the member , {[3, 1, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 2], {}, {1}], [[2, 1, 3], {[0, 1, 0, 0]}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0]}, {1, 2}], [[3, 2, 1], {[1, 0, 0, 0]}, {2}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1, 2}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+3*q^5, 1+4*q+9*q^2+13*q^3+14* q^4+14*q^5+11*q^6+7*q^7+3*q^8, 1+5*q+14*q^2+26*q^3+36*q^4+42*q^5+42*q^6+38*q^7+ 31*q^8+19*q^9+12*q^10+7*q^11+q^12, 1+6*q+20*q^2+45*q^3+76*q^4+105*q^5+124*q^6+ 130*q^7+122*q^8+104*q^9+84*q^10+62*q^11+42*q^12+29*q^13+19*q^14+6*q^15+3*q^16, 1+7*q+27*q^2+71*q^3+141*q^4+227*q^5+310*q^6+373*q^7+401*q^8+388*q^9+348*q^10+ 296*q^11+239*q^12+186*q^13+149*q^14+112*q^15+80*q^16+50*q^17+32*q^18+16*q^19+6* q^20+3*q^21, 1+8*q+35*q^2+105*q^3+239*q^4+440*q^5+684*q^6+930*q^7+1129*q^8+1238 *q^9+1243*q^10+1156*q^11+1014*q^12+855*q^13+704*q^14+580*q^15+478*q^16+386*q^17 +311*q^18+235*q^19+160*q^20+108*q^21+78*q^22+44*q^23+24*q^24+8*q^25+7*q^26+q^27 , 1+9*q+44*q^2+148*q^3+379*q^4+785*q^5+1370*q^6+2081*q^7+2814*q^8+3438*q^9+3841 *q^10+3965*q^11+3820*q^12+3469*q^13+3017*q^14+2563*q^15+2142*q^16+1785*q^17+ 1526*q^18+1304*q^19+1072*q^20+872*q^21+698*q^22+527*q^23+396*q^24+283*q^25+203* q^26+134*q^27+88*q^28+47*q^29+24*q^30+15*q^31+6*q^32+3*q^33, 1+10*q+54*q^2+201* q^3+571*q^4+1313*q^5+2542*q^6+4271*q^7+6370*q^8+8572*q^9+10536*q^10+11947*q^11+ 12617*q^12+12514*q^13+11754*q^14+10574*q^15+9222*q^16+7864*q^17+6657*q^18+5677* q^19+4864*q^20+4198*q^21+3617*q^22+3084*q^23+2574*q^24+2094*q^25+1690*q^26+1343 *q^27+1062*q^28+814*q^29+591*q^30+415*q^31+300*q^32+206*q^33+133*q^34+69*q^35+ 48*q^36+22*q^37+15*q^38+6*q^39+3*q^40, 1+11*q+65*q^2+265*q^3+826*q^4+2086*q^5+ 4435*q^6+8170*q^7+13331*q^8+19592*q^9+26266*q^10+32446*q^11+37257*q^12+40084*q^ 13+40714*q^14+39358*q^15+36503*q^16+32747*q^17+28692*q^18+24773*q^19+21230*q^20 +18212*q^21+15716*q^22+13680*q^23+11968*q^24+10414*q^25+8968*q^26+7647*q^27+ 6458*q^28+5422*q^29+4489*q^30+3659*q^31+2976*q^32+2375*q^33+1851*q^34+1374*q^35 +1034*q^36+762*q^37+546*q^38+379*q^39+259*q^40+157*q^41+95*q^42+58*q^43+38*q^44 +21*q^45+8*q^46+7*q^47+q^48, 1+12*q+77*q^2+341*q^3+1156*q^4+3178*q^5+7357*q^6+ 14745*q^7+26137*q^8+41664*q^9+60505*q^10+80870*q^11+100340*q^12+116442*q^13+ 127257*q^14+131877*q^15+130505*q^16+124188*q^17+114438*q^18+102892*q^19+90909*q ^20+79336*q^21+68727*q^22+59490*q^23+51683*q^24+45089*q^25+39561*q^26+34820*q^ 27+30550*q^28+26741*q^29+23203*q^30+19946*q^31+17131*q^32+14720*q^33+12530*q^34 +10476*q^35+8699*q^36+7132*q^37+5743*q^38+4573*q^39+3618*q^40+2813*q^41+2144*q^ 42+1576*q^43+1162*q^44+848*q^45+610*q^46+397*q^47+265*q^48+162*q^49+113*q^50+66 *q^51+45*q^52+21*q^53+15*q^54+6*q^55+3*q^56, 1+13*q+90*q^2+430*q^3+1574*q^4+ 4676*q^5+11702*q^6+25345*q^7+48504*q^8+83372*q^9+130395*q^10+187501*q^11+250012 *q^12+311393*q^13+364650*q^14+403938*q^15+425827*q^16+429816*q^17+417990*q^18+ 394104*q^19+362563*q^20+327409*q^21+291587*q^22+257128*q^23+225368*q^24+196919* q^25+172022*q^26+150596*q^27+132431*q^28+117061*q^29+103658*q^30+91595*q^31+ 80658*q^32+70924*q^33+62307*q^34+54596*q^35+47668*q^36+41471*q^37+35856*q^38+ 30715*q^39+26119*q^40+21959*q^41+18265*q^42+15021*q^43+12295*q^44+9998*q^45+ 8076*q^46+6376*q^47+4916*q^48+3737*q^49+2858*q^50+2147*q^51+1584*q^52+1132*q^53 +806*q^54+536*q^55+365*q^56+244*q^57+161*q^58+105*q^59+69*q^60+43*q^61+21*q^62+ 15*q^63+6*q^64+3*q^65, 1+14*q+104*q^2+533*q^3+2094*q^4+6681*q^5+17964*q^6+41800 *q^7+85892*q^8+158377*q^9+265452*q^10+408655*q^11+582810*q^12+775607*q^13+ 969276*q^14+1144023*q^15+1282197*q^16+1371873*q^17+1408739*q^18+1395969*q^19+ 1342429*q^20+1259787*q^21+1159719*q^22+1052210*q^23+944500*q^24+841132*q^25+ 744813*q^26+656831*q^27+578001*q^28+508862*q^29+449085*q^30+397528*q^31+352870* q^32+313678*q^33+278892*q^34+247763*q^35+219948*q^36+195364*q^37+173494*q^38+ 153935*q^39+136316*q^40+120258*q^41+105502*q^42+91855*q^43+79413*q^44+68175*q^ 45+58260*q^46+49534*q^47+41727*q^48+34854*q^49+28982*q^50+23924*q^51+19492*q^52 +15633*q^53+12484*q^54+9843*q^55+7711*q^56+5936*q^57+4516*q^58+3382*q^59+2544*q ^60+1862*q^61+1321*q^62+905*q^63+649*q^64+439*q^65+312*q^66+199*q^67+147*q^68+ 86*q^69+58*q^70+35*q^71+21*q^72+8*q^73+7*q^74+q^75] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+5*q^2+4*q+3), q^2*(q^8+4*q^7+9*q^6 +13*q^5+14*q^4+14*q^3+11*q^2+7*q+3), q^3*(q^12+5*q^11+14*q^10+26*q^9+36*q^8+42* q^7+42*q^6+38*q^5+31*q^4+19*q^3+12*q^2+7*q+1), q^5*(q^16+6*q^15+20*q^14+45*q^13 +76*q^12+105*q^11+124*q^10+130*q^9+122*q^8+104*q^7+84*q^6+62*q^5+42*q^4+29*q^3+ 19*q^2+6*q+3), q^7*(q^21+7*q^20+27*q^19+71*q^18+141*q^17+227*q^16+310*q^15+373* q^14+401*q^13+388*q^12+348*q^11+296*q^10+239*q^9+186*q^8+149*q^7+112*q^6+80*q^5 +50*q^4+32*q^3+16*q^2+6*q+3), q^9*(1+35*q^25+440*q^22+105*q^24+684*q^21+239*q^ 23+7*q+8*q^2+8*q^26+24*q^3+44*q^4+78*q^5+108*q^6+160*q^7+235*q^8+311*q^9+386*q^ 10+478*q^11+580*q^12+704*q^13+855*q^14+1014*q^15+1156*q^16+1243*q^17+1238*q^18+ 1129*q^19+930*q^20+q^27), q^12*(3+2814*q^25+3965*q^22+3438*q^24+3820*q^21+3841* q^23+6*q+148*q^30+379*q^29+q^33+15*q^2+2081*q^26+24*q^3+47*q^4+88*q^5+134*q^6+9 *q^32+203*q^7+283*q^8+396*q^9+527*q^10+698*q^11+872*q^12+1072*q^13+1304*q^14+ 1526*q^15+1785*q^16+785*q^28+2142*q^17+2563*q^18+3017*q^19+3469*q^20+1370*q^27+ 44*q^31), q^15*(3+10574*q^25+6657*q^22+9222*q^24+5677*q^21+7864*q^23+6*q+10536* q^30+201*q^37+11947*q^29+10*q^39+2542*q^34+54*q^38+4271*q^33+15*q^2+11754*q^26+ 22*q^3+48*q^4+69*q^5+133*q^6+6370*q^32+206*q^7+300*q^8+415*q^9+591*q^10+814*q^ 11+1062*q^12+1343*q^13+1690*q^14+2094*q^15+2574*q^16+12617*q^28+3084*q^17+3617* q^18+4198*q^19+4864*q^20+12514*q^27+q^40+8572*q^31+1313*q^35+571*q^36), q^18*(1 +13680*q^25+4435*q^42+8968*q^22+11968*q^24+7647*q^21+10414*q^23+7*q+826*q^44+ 28692*q^30+265*q^45+32446*q^37+24773*q^29+65*q^46+19592*q^39+8170*q^41+40714*q^ 34+26266*q^38+39358*q^33+8*q^2+15716*q^26+21*q^3+38*q^4+58*q^5+95*q^6+36503*q^ 32+157*q^7+259*q^8+379*q^9+546*q^10+762*q^11+1034*q^12+1374*q^13+1851*q^14+2375 *q^15+2976*q^16+2086*q^43+21230*q^28+3659*q^17+4489*q^18+5422*q^19+6458*q^20+ 18212*q^27+13331*q^40+32747*q^31+40084*q^35+q^48+37257*q^36+11*q^47), q^22*(3+ 19946*q^25+127257*q^42+12530*q^22+17131*q^24+10476*q^21+14720*q^23+6*q+3178*q^ 51+100340*q^44+341*q^53+39561*q^30+1156*q^52+12*q^55+77*q^54+80870*q^45+102892* q^37+34820*q^29+60505*q^46+124188*q^39+131877*q^41+68727*q^34+7357*q^50+114438* q^38+59490*q^33+q^56+15*q^2+23203*q^26+21*q^3+45*q^4+66*q^5+113*q^6+51683*q^32+ 162*q^7+265*q^8+397*q^9+610*q^10+848*q^11+1162*q^12+1576*q^13+2144*q^14+2813*q^ 15+3618*q^16+116442*q^43+14745*q^49+30550*q^28+4573*q^17+5743*q^18+7132*q^19+ 8699*q^20+26741*q^27+130505*q^40+45089*q^31+79336*q^35+26137*q^48+90909*q^36+ 41664*q^47), q^26*(3+26119*q^25+257128*q^42+15021*q^22+21959*q^24+12295*q^21+ 18265*q^23+6*q+364650*q^51+327409*q^44+250012*q^53+48504*q^57+54596*q^30+311393 *q^52+25345*q^58+130395*q^55+4676*q^60+187501*q^54+362563*q^45+132431*q^37+ 47668*q^29+394104*q^46+q^65+172022*q^39+1574*q^61+225368*q^41+91595*q^34+403938 *q^50+13*q^64+150596*q^38+80658*q^33+83372*q^56+15*q^2+30715*q^26+21*q^3+43*q^4 +69*q^5+105*q^6+70924*q^32+161*q^7+244*q^8+365*q^9+430*q^62+536*q^10+806*q^11+ 1132*q^12+1584*q^13+2147*q^14+2858*q^15+3737*q^16+291587*q^43+425827*q^49+41471 *q^28+4916*q^17+6376*q^18+8076*q^19+9998*q^20+35856*q^27+196919*q^40+90*q^63+ 62307*q^31+103658*q^35+429816*q^48+11702*q^59+117061*q^36+417990*q^47), q^30*(1 +28982*q^25+313678*q^42+15633*q^22+23924*q^24+12484*q^21+19492*q^23+7*q+944500* q^51+397528*q^44+1159719*q^53+1408739*q^57+68175*q^30+1052210*q^52+1371873*q^58 +1342429*q^55+1144023*q^60+1259787*q^54+449085*q^45+2094*q^71+14*q^74+173494*q^ 37+85892*q^67+58260*q^29+508862*q^46+265452*q^65+17964*q^69+219948*q^39+969276* q^61+278892*q^41+120258*q^34+841132*q^50+408655*q^64+195364*q^38+105502*q^33+ 1395969*q^56+8*q^2+41800*q^68+34854*q^26+21*q^3+35*q^4+58*q^5+86*q^6+91855*q^32 +147*q^7+199*q^8+312*q^9+775607*q^62+439*q^10+649*q^11+905*q^12+1321*q^13+1862* q^14+2544*q^15+3382*q^16+158377*q^66+6681*q^70+352870*q^43+744813*q^49+49534*q^ 28+4516*q^17+5936*q^18+7711*q^19+9843*q^20+41727*q^27+533*q^72+247763*q^40+104* q^73+582810*q^63+79413*q^31+q^75+136316*q^35+656831*q^48+1282197*q^59+153935*q^ 36+578001*q^47)] The number of permutations avoiding, {[3, 1, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, is given by [1, 2, 6, 21, 76, 274, 978, 3463, 12201, 42869, 150415, 527426, 1848905, 6480722, 22715293] The number of EVEN permutations avoiding, {[3, 1, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, is given by [1, 1, 3, 10, 38, 137, 491, 1734, 6107, 21441, 75221, 263727, 924484, 3240404, 11357742] The number of ODD permutations avoiding, {[3, 1, 4, 2], [4, 1, 3, 2], [4, 3, 2, 1]}, is given by [0, 1, 3, 11, 38, 137, 487, 1729, 6094, 21428, 75194, 263699, 924421, 3240318, 11357551] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 38, 137, 487, 1734, 6107, 21428, 75194, 263727, 924484, 3240318, 11357551] ODD:, [0, 1, 3, 11, 38, 137, 491, 1729, 6094, 21441, 75221, 263699, 924421, 3240404, 11357742] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.809523810, 4.289473684, 5.879562044, 7.549079755, 9.276638753, 11.04630768, 12.84695234, 14.67136921, 16.51509216, 18.37528645, 20.24996135, 22.13751436] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.401327521, 1.862481292, 2.368455676, 2.923031764, 3.518400954, 4.143765467, 4.789011871, 5.446030889, 6.108930913, 6.773720585, 7.437804091, 8.099495168] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.96372, 3.46884, 5.60958, 8.54411, 12.3791, 17.1708, 22.9346, 29.6593, 37.3190, 45.8833, 55.3209, 65.6018] Skewness: , [Float(undefined), 0., 0., -0.07423825638, -0.009751336573, 0.1221884125, 0.2668740973, 0.4041551784, 0.5294343696, 0.6427709758, 0.7452069820, 0.8378619651, 0.9217772456, 0.9979465954, 1.067293135] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.145300573, 2.334027083, 2.508161801, 2.672563664, 2.845144723, 3.033947244, 3.240120453, 3.461122797, 3.693201219, 3.932284986, 4.174941755, 4.418537260] end of this data For the equivalence class of patterns, { {[2, 4, 3, 1], [3, 4, 1, 2], [4, 1, 3, 2]}, {[3, 4, 1, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, {[1, 3, 4, 2], [2, 3, 1, 4], [2, 1, 4, 3]}, {[1, 4, 2, 3], [2, 1, 4, 3], [3, 1, 2, 4]}} the member , {[2, 4, 3, 1], [3, 4, 1, 2], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 0, 2, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 0, 2, 0, 0]}, {2}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {4}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 3, 1], {[0, 1, 0, 0]}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[1, 3, 2], {[1, 0, 0, 0]}, {1}], [[3, 2, 1], {}, {2}], [[1, 2, 3], {[1, 1, 0, 0]}, {1}], [[2, 1, 3], {[1, 1, 0, 0], [0, 2, 0, 0]}, {2}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+2*q^4+3*q^5+q^6, 1+4*q+9*q^2+15*q^3 +14*q^4+10*q^5+8*q^6+5*q^7+3*q^8+4*q^9+q^10, 1+5*q+14*q^2+29*q^3+40*q^4+41*q^5+ 30*q^6+24*q^7+16*q^8+17*q^9+13*q^10+8*q^11+7*q^12+4*q^13+5*q^14+q^15, 1+6*q+20* q^2+49*q^3+86*q^4+115*q^5+115*q^6+89*q^7+73*q^8+58*q^9+51*q^10+43*q^11+33*q^12+ 28*q^13+27*q^14+22*q^15+15*q^16+11*q^17+9*q^18+5*q^19+6*q^20+q^21, 1+7*q+27*q^2 +76*q^3+159*q^4+259*q^5+329*q^6+324*q^7+269*q^8+227*q^9+190*q^10+166*q^11+138*q ^12+109*q^13+100*q^14+93*q^15+81*q^16+64*q^17+59*q^18+47*q^19+45*q^20+35*q^21+ 27*q^22+20*q^23+14*q^24+11*q^25+6*q^26+7*q^27+q^28, 1+8*q+35*q^2+111*q^3+267*q^ 4+508*q^5+774*q^6+941*q^7+935*q^8+810*q^9+709*q^10+613*q^11+545*q^12+450*q^13+ 381*q^14+341*q^15+302*q^16+265*q^17+226*q^18+201*q^19+189*q^20+172*q^21+153*q^ 22+129*q^23+113*q^24+96*q^25+83*q^26+72*q^27+58*q^28+44*q^29+35*q^30+25*q^31+17 *q^32+13*q^33+7*q^34+8*q^35+q^36, 1+9*q+44*q^2+155*q^3+419*q^4+906*q^5+1596*q^6 +2296*q^7+2724*q^8+2721*q^9+2455*q^10+2205*q^11+1973*q^12+1750*q^13+1514*q^14+ 1315*q^15+1167*q^16+1042*q^17+900*q^18+771*q^19+689*q^20+629*q^21+584*q^22+529* q^23+469*q^24+422*q^25+389*q^26+357*q^27+324*q^28+290*q^29+251*q^30+221*q^31+ 187*q^32+164*q^33+133*q^34+120*q^35+92*q^36+73*q^37+56*q^38+43*q^39+30*q^40+20* q^41+15*q^42+8*q^43+9*q^44+q^45, 1+10*q+54*q^2+209*q^3+625*q^4+1507*q^5+2995*q^ 6+4942*q^7+6819*q^8+7934*q^9+8007*q^10+7451*q^11+6867*q^12+6257*q^13+5658*q^14+ 5004*q^15+4451*q^16+4011*q^17+3599*q^18+3151*q^19+2736*q^20+2443*q^21+2194*q^22 +2010*q^23+1780*q^24+1589*q^25+1444*q^26+1333*q^27+1249*q^28+1160*q^29+1072*q^ 30+969*q^31+888*q^32+814*q^33+739*q^34+679*q^35+614*q^36+547*q^37+476*q^38+424* q^39+363*q^40+312*q^41+263*q^42+221*q^43+185*q^44+151*q^45+115*q^46+91*q^47+68* q^48+51*q^49+35*q^50+23*q^51+17*q^52+9*q^53+10*q^54+q^55, 1+11*q+65*q^2+274*q^3 +896*q^4+2376*q^5+5236*q^6+9694*q^7+15191*q^8+20260*q^9+23281*q^10+23748*q^11+ 22675*q^12+21319*q^13+19800*q^14+18167*q^15+16390*q^16+14861*q^17+13550*q^18+ 12277*q^19+10931*q^20+9717*q^21+8718*q^22+7932*q^23+7148*q^24+6342*q^25+5640*q^ 26+5077*q^27+4636*q^28+4276*q^29+3950*q^30+3621*q^31+3340*q^32+3094*q^33+2866*q ^34+2689*q^35+2520*q^36+2356*q^37+2174*q^38+2003*q^39+1840*q^40+1695*q^41+1542* q^42+1407*q^43+1275*q^44+1153*q^45+1022*q^46+900*q^47+791*q^48+684*q^49+594*q^ 50+498*q^51+424*q^52+346*q^53+295*q^54+230*q^55+186*q^56+141*q^57+109*q^58+80*q ^59+59*q^60+40*q^61+26*q^62+19*q^63+10*q^64+11*q^65+q^66, 1+12*q+77*q^2+351*q^3 +1244*q^4+3590*q^5+8661*q^6+17706*q^7+30935*q^8+46440*q^9+60313*q^10+68764*q^11 +70891*q^12+69116*q^13+66154*q^14+62427*q^15+58060*q^16+53345*q^17+49070*q^18+ 45186*q^19+41345*q^20+37346*q^21+33717*q^22+30755*q^23+28126*q^24+25579*q^25+ 22920*q^26+20580*q^27+18550*q^28+16899*q^29+15412*q^30+14030*q^31+12782*q^32+ 11635*q^33+10699*q^34+9870*q^35+9226*q^36+8630*q^37+8113*q^38+7567*q^39+7083*q^ 40+6636*q^41+6210*q^42+5823*q^43+5458*q^44+5127*q^45+4779*q^46+4448*q^47+4093*q ^48+3782*q^49+3483*q^50+3188*q^51+2899*q^52+2628*q^53+2376*q^54+2128*q^55+1896* q^56+1664*q^57+1461*q^58+1268*q^59+1098*q^60+930*q^61+791*q^62+657*q^63+544*q^ 64+448*q^65+361*q^66+280*q^67+224*q^68+167*q^69+127*q^70+92*q^71+67*q^72+45*q^ 73+29*q^74+21*q^75+11*q^76+12*q^77+q^78, 1+13*q+90*q^2+441*q^3+1682*q^4+5239*q^ 5+13702*q^6+30561*q^7+58696*q^8+97662*q^9+141507*q^10+180015*q^11+204272*q^12+ 212730*q^13+211051*q^14+205169*q^15+196252*q^16+185036*q^17+172549*q^18+160747* q^19+149435*q^20+137913*q^21+126055*q^22+115341*q^23+106165*q^24+97870*q^25+ 89693*q^26+81499*q^27+74081*q^28+67592*q^29+61884*q^30+56602*q^31+51645*q^32+ 46927*q^33+42684*q^34+38921*q^35+35687*q^36+32888*q^37+30468*q^38+28269*q^39+ 26232*q^40+24423*q^41+22827*q^42+21397*q^43+20149*q^44+19017*q^45+17967*q^46+ 17010*q^47+16019*q^48+15088*q^49+14217*q^50+13394*q^51+12601*q^52+11824*q^53+ 11098*q^54+10388*q^55+9702*q^56+9007*q^57+8331*q^58+7681*q^59+7087*q^60+6474*q^ 61+5915*q^62+5357*q^63+4843*q^64+4346*q^65+3894*q^66+3443*q^67+3036*q^68+2658*q ^69+2308*q^70+1993*q^71+1705*q^72+1449*q^73+1207*q^74+1015*q^75+825*q^76+685*q^ 77+541*q^78+433*q^79+333*q^80+262*q^81+193*q^82+145*q^83+104*q^84+75*q^85+50*q^ 86+32*q^87+23*q^88+12*q^89+13*q^90+q^91, 1+14*q+104*q^2+545*q^3+2224*q^4+7427*q ^5+20895*q^6+50375*q^7+105199*q^8+191605*q^9+305892*q^10+430444*q^11+538775*q^ 12+609915*q^13+641137*q^14+645530*q^15+635952*q^16+615969*q^17+587938*q^18+ 555571*q^19+523323*q^20+490847*q^21+456900*q^22+422211*q^23+390332*q^24+361938* q^25+335688*q^26+309655*q^27+284155*q^28+261095*q^29+240476*q^30+222037*q^31+ 204721*q^32+188209*q^33+172461*q^34+157913*q^35+144694*q^36+132712*q^37+122204* q^38+112515*q^39+103733*q^40+95493*q^41+88189*q^42+81582*q^43+75885*q^44+70804* q^45+66299*q^46+62387*q^47+58716*q^48+55318*q^49+52172*q^50+49364*q^51+46751*q^ 52+44342*q^53+42016*q^54+39900*q^55+37880*q^56+35972*q^57+34028*q^58+32191*q^59 +30420*q^60+28714*q^61+27079*q^62+25470*q^63+23912*q^64+22390*q^65+20947*q^66+ 19503*q^67+18121*q^68+16745*q^69+15457*q^70+14201*q^71+13019*q^72+11866*q^73+ 10779*q^74+9735*q^75+8757*q^76+7847*q^77+6989*q^78+6183*q^79+5434*q^80+4759*q^ 81+4130*q^82+3572*q^83+3055*q^84+2604*q^85+2192*q^86+1835*q^87+1514*q^88+1249*q ^89+1016*q^90+816*q^91+641*q^92+508*q^93+386*q^94+300*q^95+219*q^96+163*q^97+ 116*q^98+83*q^99+55*q^100+35*q^101+25*q^102+13*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+2*q^2+3*q+1, q^10+4*q^9+9*q^8+ 15*q^7+14*q^6+10*q^5+8*q^4+5*q^3+3*q^2+4*q+1, q^15+5*q^14+14*q^13+29*q^12+40*q^ 11+41*q^10+30*q^9+24*q^8+16*q^7+17*q^6+13*q^5+8*q^4+7*q^3+4*q^2+5*q+1, q^21+6*q ^20+20*q^19+49*q^18+86*q^17+115*q^16+115*q^15+89*q^14+73*q^13+58*q^12+51*q^11+ 43*q^10+33*q^9+28*q^8+27*q^7+22*q^6+15*q^5+11*q^4+9*q^3+5*q^2+6*q+1, 1+76*q^25+ 329*q^22+159*q^24+324*q^21+259*q^23+7*q+6*q^2+27*q^26+11*q^3+14*q^4+20*q^5+27*q ^6+35*q^7+45*q^8+47*q^9+59*q^10+64*q^11+81*q^12+93*q^13+100*q^14+109*q^15+138*q ^16+q^28+166*q^17+190*q^18+227*q^19+269*q^20+7*q^27, 1+613*q^25+381*q^22+545*q^ 24+341*q^21+450*q^23+8*q+774*q^30+941*q^29+35*q^34+111*q^33+7*q^2+709*q^26+13*q ^3+17*q^4+25*q^5+35*q^6+267*q^32+44*q^7+58*q^8+72*q^9+83*q^10+96*q^11+113*q^12+ 129*q^13+153*q^14+172*q^15+189*q^16+935*q^28+201*q^17+226*q^18+265*q^19+302*q^ 20+810*q^27+508*q^31+8*q^35+q^36, 1+689*q^25+155*q^42+529*q^22+629*q^24+469*q^ 21+584*q^23+9*q+9*q^44+1315*q^30+q^45+2724*q^37+1167*q^29+1596*q^39+419*q^41+ 2205*q^34+2296*q^38+1973*q^33+8*q^2+771*q^26+15*q^3+20*q^4+30*q^5+43*q^6+1750*q ^32+56*q^7+73*q^8+92*q^9+120*q^10+133*q^11+164*q^12+187*q^13+221*q^14+251*q^15+ 290*q^16+44*q^43+1042*q^28+324*q^17+357*q^18+389*q^19+422*q^20+900*q^27+906*q^ 40+1514*q^31+2455*q^35+2721*q^36, 1+1072*q^25+6257*q^42+814*q^22+969*q^24+739*q ^21+888*q^23+10*q+625*q^51+7451*q^44+54*q^53+1589*q^30+209*q^52+q^55+10*q^54+ 8007*q^45+3599*q^37+1444*q^29+7934*q^46+4451*q^39+5658*q^41+2443*q^34+1507*q^50 +4011*q^38+2194*q^33+9*q^2+1160*q^26+17*q^3+23*q^4+35*q^5+51*q^6+2010*q^32+68*q ^7+91*q^8+115*q^9+151*q^10+185*q^11+221*q^12+263*q^13+312*q^14+363*q^15+424*q^ 16+6867*q^43+2995*q^49+1333*q^28+476*q^17+547*q^18+614*q^19+679*q^20+1249*q^27+ 5004*q^40+1780*q^31+2736*q^35+4942*q^48+3151*q^36+6819*q^47, 1+1695*q^25+7148*q ^42+1275*q^22+1542*q^24+1153*q^21+1407*q^23+11*q+18167*q^51+8718*q^44+21319*q^ 53+20260*q^57+2520*q^30+19800*q^52+15191*q^58+23748*q^55+5236*q^60+22675*q^54+ 9717*q^45+4276*q^37+2356*q^29+10931*q^46+11*q^65+5077*q^39+2376*q^61+6342*q^41+ 3340*q^34+16390*q^50+65*q^64+4636*q^38+3094*q^33+23281*q^56+10*q^2+1840*q^26+19 *q^3+26*q^4+40*q^5+59*q^6+2866*q^32+80*q^7+109*q^8+141*q^9+896*q^62+186*q^10+ 230*q^11+295*q^12+346*q^13+424*q^14+498*q^15+594*q^16+q^66+7932*q^43+14861*q^49 +2174*q^28+684*q^17+791*q^18+900*q^19+1022*q^20+2003*q^27+5640*q^40+274*q^63+ 2689*q^31+3621*q^35+13550*q^48+9694*q^59+3950*q^36+12277*q^47, 1+2628*q^25+9226 *q^42+1896*q^22+2376*q^24+1664*q^21+2128*q^23+12*q+20580*q^51+10699*q^44+25579* q^53+37346*q^57+4093*q^30+22920*q^52+41345*q^58+30755*q^55+49070*q^60+28126*q^ 54+11635*q^45+17706*q^71+1244*q^74+6636*q^37+68764*q^67+3782*q^29+12782*q^46+ 69116*q^65+46440*q^69+7567*q^39+53345*q^61+8630*q^41+5458*q^34+18550*q^50+66154 *q^64+7083*q^38+5127*q^33+33717*q^56+11*q^2+60313*q^68+2899*q^26+21*q^3+29*q^4+ 45*q^5+67*q^6+4779*q^32+92*q^7+127*q^8+167*q^9+58060*q^62+224*q^10+280*q^11+361 *q^12+77*q^76+448*q^13+544*q^14+657*q^15+791*q^16+70891*q^66+30935*q^70+12*q^77 +q^78+9870*q^43+16899*q^49+3483*q^28+930*q^17+1098*q^18+1268*q^19+1461*q^20+ 3188*q^27+8661*q^72+8113*q^40+3590*q^73+62427*q^63+4448*q^31+351*q^75+5823*q^35 +15412*q^48+45186*q^59+6210*q^36+14030*q^47, 1+3894*q^25+15088*q^42+2658*q^22+ 13*q^90+3443*q^24+2308*q^21+3036*q^23+13*q+26232*q^51+17010*q^44+30468*q^53+ 42684*q^57+6474*q^30+28269*q^52+46927*q^58+35687*q^55+56602*q^60+q^91+32888*q^ 54+17967*q^45+149435*q^71+185036*q^74+11098*q^37+106165*q^67+5915*q^29+19017*q^ 46+89693*q^65+126055*q^69+12601*q^39+61884*q^61+14217*q^41+9007*q^34+204272*q^ 79+24423*q^50+81499*q^64+11824*q^38+8331*q^33+38921*q^56+12*q^2+115341*q^68+ 4346*q^26+23*q^3+32*q^4+50*q^5+75*q^6+7681*q^32+104*q^7+145*q^8+193*q^9+67592*q ^62+262*q^10+333*q^11+433*q^12+205169*q^76+541*q^13+685*q^14+825*q^15+1015*q^16 +97870*q^66+137913*q^70+211051*q^77+212730*q^78+16019*q^43+22827*q^49+5357*q^28 +1207*q^17+1449*q^18+1705*q^19+1993*q^20+4843*q^27+160747*q^72+13394*q^40+ 172549*q^73+74081*q^63+7087*q^31+196252*q^75+9702*q^35+180015*q^80+21397*q^48+ 58696*q^83+141507*q^81+13702*q^85+5239*q^86+1682*q^87+441*q^88+90*q^89+30561*q^ 84+97662*q^82+51645*q^59+10388*q^36+20149*q^47, 1+5434*q^25+25470*q^42+3572*q^ 22+645530*q^90+4759*q^24+3055*q^21+4130*q^23+14*q+105199*q^97+42016*q^51+28714* q^44+46751*q^53+58716*q^57+9735*q^30+430444*q^94+44342*q^52+62387*q^58+52172*q^ 55+70804*q^60+641137*q^91+49364*q^54+30420*q^45+172461*q^71+222037*q^74+191605* q^96+50375*q^98+18121*q^37+122204*q^67+8757*q^29+32191*q^46+538775*q^93+20895*q ^99+103733*q^65+144694*q^69+20947*q^39+75885*q^61+23912*q^41+14201*q^34+335688* q^79+39900*q^50+95493*q^64+7427*q^100+19503*q^38+13019*q^33+55318*q^56+13*q^2+ 132712*q^68+6183*q^26+25*q^3+35*q^4+55*q^5+83*q^6+11866*q^32+116*q^7+163*q^8+ 219*q^9+305892*q^95+81582*q^62+2224*q^101+300*q^10+386*q^11+508*q^12+261095*q^ 76+641*q^13+816*q^14+1016*q^15+1249*q^16+112515*q^66+157913*q^70+284155*q^77+ 309655*q^78+27079*q^43+545*q^102+37880*q^49+7847*q^28+1514*q^17+1835*q^18+2192* q^19+2604*q^20+6989*q^27+188209*q^72+22390*q^40+204721*q^73+104*q^103+88189*q^ 63+10779*q^31+240476*q^75+15457*q^35+14*q^104+361938*q^80+35972*q^48+q^105+ 456900*q^83+390332*q^81+523323*q^85+555571*q^86+587938*q^87+615969*q^88+635952* q^89+490847*q^84+422211*q^82+66299*q^59+16745*q^36+609915*q^92+34028*q^47] The number of permutations avoiding, {[2, 4, 3, 1], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 74, 255, 863, 2891, 9638, 32068, 106627, 354480, 1178459, 3917863, 13025489] The number of EVEN permutations avoiding, {[2, 4, 3, 1], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 36, 126, 436, 1446, 4831, 16021, 53324, 177182, 589287, 1958819, 6513017] The number of ODD permutations avoiding, {[2, 4, 3, 1], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 38, 129, 427, 1445, 4807, 16047, 53303, 177298, 589172, 1959044, 6512472] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 36, 129, 427, 1446, 4831, 16047, 53303, 177182, 589287, 1959044, 6512472] ODD:, [0, 1, 3, 12, 38, 126, 436, 1445, 4807, 16021, 53324, 177298, 589172, 1958819, 6513017] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.857142857, 4.405405405, 6.090196078, 7.887601390, 9.776202006, 11.73552604, 13.74831608, 15.80141052, 17.88525164, 19.99292466, 22.11930330, 24.26046531] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.520830421, 2.211266721, 3.031162004, 3.965099386, 4.991178751, 6.086470152, 7.229897136, 8.403601433, 9.593216203, 10.78753395, 11.97796313, 13.15799295] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.31293, 4.88970, 9.18794, 15.7220, 24.9119, 37.0451, 52.2714, 70.6205, 92.0298, 116.371, 143.472, 173.133] Skewness: , [Float(undefined), 0., 0., 0.2420216426, 0.5161679509, 0.7182479952, 0.8639466427, 0.9796940977, 1.081180869, 1.175674118, 1.265941795, 1.352703248, 1.435894658, 1.515212598, 1.590353823] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.396775778, 2.722838586, 3.009396970, 3.263151590, 3.511505483, 3.775152030, 4.064122906, 4.381007170, 4.724301927, 5.090373125, 5.474874859, 5.873187336] end of this data For the equivalence class of patterns, { {[1, 2, 4, 3], [2, 4, 1, 3], [2, 3, 1, 4]}, {[2, 4, 1, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, {[1, 4, 2, 3], [2, 4, 1, 3], [2, 1, 3, 4]}, {[3, 4, 2, 1], [3, 1, 4, 2], [4, 1, 3, 2]}, {[1, 2, 4, 3], [3, 1, 4, 2], [3, 1, 2, 4]}, {[3, 2, 4, 1], [3, 1, 4, 2], [4, 3, 1, 2]}, {[1, 3, 4, 2], [2, 1, 3, 4], [3, 1, 4, 2]}, {[2, 4, 3, 1], [2, 4, 1, 3], [4, 3, 1, 2]}} the member , {[2, 4, 1, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[1, 4, 2, 3], {[1, 1, 0, 0, 0], [1, 0, 1, 1, 0], [0, 1, 1, 1, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [1, 1, 1, 0, 0]}, {2}], [[1, 3, 2, 4], {[1, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 1, 1, 0, 0]}, {1}], [[3, 1, 2, 4], {[1, 1, 0, 0, 0], [1, 0, 1, 0, 0], [0, 1, 1, 0, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0], [1, 1, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[1, 1, 0, 0, 0], [1, 0, 1, 1, 0], [0, 1, 1, 1, 0]}, {2}], [[1, 3, 2], {[1, 1, 1, 0]}, {}], [[2, 1, 3], {[1, 1, 0, 0]}, {2}], [[3, 1, 2], {[1, 1, 1, 0]}, {}], [[1, 2, 3], {[1, 1, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+2*q^5+q^6, 1+4*q+9*q^2+13*q^3 +14*q^4+12*q^5+10*q^6+7*q^7+3*q^8+2*q^9+q^10, 1+5*q+14*q^2+26*q^3+36*q^4+39*q^5 +37*q^6+34*q^7+27*q^8+20*q^9+14*q^10+10*q^11+5*q^12+3*q^13+2*q^14+q^15, 1+6*q+ 20*q^2+45*q^3+76*q^4+101*q^5+113*q^6+114*q^7+106*q^8+92*q^9+78*q^10+64*q^11+49* q^12+37*q^13+26*q^14+19*q^15+12*q^16+8*q^17+5*q^18+3*q^19+2*q^20+q^21, 1+7*q+27 *q^2+71*q^3+141*q^4+222*q^5+291*q^6+334*q^7+348*q^8+334*q^9+303*q^10+269*q^11+ 233*q^12+196*q^13+162*q^14+133*q^15+105*q^16+80*q^17+60*q^18+45*q^19+30*q^20+24 *q^21+18*q^22+10*q^23+8*q^24+5*q^25+3*q^26+2*q^27+q^28, 1+8*q+35*q^2+105*q^3+ 239*q^4+434*q^5+655*q^6+854*q^7+996*q^8+1064*q^9+1057*q^10+996*q^11+906*q^12+ 808*q^13+704*q^14+610*q^15+529*q^16+450*q^17+374*q^18+308*q^19+245*q^20+195*q^ 21+156*q^22+120*q^23+92*q^24+69*q^25+52*q^26+40*q^27+31*q^28+23*q^29+16*q^30+10 *q^31+8*q^32+5*q^33+3*q^34+2*q^35+q^36, 1+9*q+44*q^2+148*q^3+379*q^4+778*q^5+ 1329*q^6+1951*q^7+2536*q^8+2992*q^9+3263*q^10+3339*q^11+3249*q^12+3047*q^13+ 2775*q^14+2478*q^15+2195*q^16+1938*q^17+1700*q^18+1485*q^19+1280*q^20+1090*q^21 +925*q^22+773*q^23+638*q^24+522*q^25+421*q^26+341*q^27+271*q^28+218*q^29+174*q^ 30+134*q^31+108*q^32+83*q^33+67*q^34+51*q^35+41*q^36+30*q^37+21*q^38+16*q^39+10 *q^40+8*q^41+5*q^42+3*q^43+2*q^44+q^45, 1+10*q+54*q^2+201*q^3+571*q^4+1305*q^5+ 2487*q^6+4067*q^7+5856*q^8+7596*q^9+9045*q^10+10037*q^11+10512*q^12+10516*q^13+ 10137*q^14+9476*q^15+8667*q^16+7822*q^17+6991*q^18+6217*q^19+5516*q^20+4875*q^ 21+4295*q^22+3770*q^23+3282*q^24+2828*q^25+2416*q^26+2061*q^27+1743*q^28+1458*q ^29+1218*q^30+1008*q^31+828*q^32+680*q^33+557*q^34+449*q^35+373*q^36+306*q^37+ 245*q^38+204*q^39+164*q^40+128*q^41+106*q^42+85*q^43+66*q^44+52*q^45+40*q^46+28 *q^47+21*q^48+16*q^49+10*q^50+8*q^51+5*q^52+3*q^53+2*q^54+q^55, 1+11*q+65*q^2+ 265*q^3+826*q^4+2077*q^5+4364*q^6+7869*q^7+12460*q^8+17685*q^9+22908*q^10+27491 *q^11+30946*q^12+33043*q^13+33795*q^14+33357*q^15+31971*q^16+29958*q^17+27597*q ^18+25092*q^19+22603*q^20+20248*q^21+18094*q^22+16148*q^23+14402*q^24+12812*q^ 25+11329*q^26+9980*q^27+8764*q^28+7649*q^29+6656*q^30+5764*q^31+4956*q^32+4233* q^33+3597*q^34+3038*q^35+2562*q^36+2161*q^37+1809*q^38+1516*q^39+1267*q^40+1047 *q^41+880*q^42+734*q^43+614*q^44+515*q^45+432*q^46+358*q^47+297*q^48+245*q^49+ 199*q^50+164*q^51+130*q^52+108*q^53+84*q^54+67*q^55+51*q^56+38*q^57+28*q^58+21* q^59+16*q^60+10*q^61+8*q^62+5*q^63+3*q^64+2*q^65+q^66, 1+12*q+77*q^2+341*q^3+ 1156*q^4+3168*q^5+7268*q^6+14321*q^7+24754*q^8+38240*q^9+53680*q^10+69474*q^11+ 83922*q^12+95629*q^13+103768*q^14+108104*q^15+108856*q^16+106595*q^17+102060*q^ 18+95971*q^19+88933*q^20+81453*q^21+73959*q^22+66770*q^23+60090*q^24+54006*q^25 +48471*q^26+43426*q^27+38868*q^28+34695*q^29+30894*q^30+27477*q^31+24360*q^32+ 21519*q^33+18934*q^34+16581*q^35+14468*q^36+12572*q^37+10889*q^38+9397*q^39+ 8071*q^40+6900*q^41+5885*q^42+5015*q^43+4272*q^44+3631*q^45+3091*q^46+2624*q^47 +2230*q^48+1894*q^49+1610*q^50+1378*q^51+1168*q^52+998*q^53+850*q^54+720*q^55+ 614*q^56+519*q^57+432*q^58+359*q^59+299*q^60+246*q^61+203*q^62+166*q^63+132*q^ 64+107*q^65+85*q^66+66*q^67+49*q^68+38*q^69+28*q^70+21*q^71+16*q^72+10*q^73+8*q ^74+5*q^75+3*q^76+2*q^77+q^78, 1+13*q+90*q^2+430*q^3+1574*q^4+4665*q^5+11593*q^ 6+24769*q^7+46416*q^8+77613*q^9+117588*q^10+163637*q^11+211671*q^12+257153*q^13 +296090*q^14+325727*q^15+344748*q^16+353195*q^17+352203*q^18+343521*q^19+329008 *q^20+310391*q^21+289225*q^22+266796*q^23+244148*q^24+222151*q^25+201339*q^26+ 181955*q^27+164204*q^28+148059*q^29+133359*q^30+120026*q^31+107955*q^32+96989*q ^33+86989*q^34+77880*q^35+69599*q^36+62058*q^37+55210*q^38+48990*q^39+43320*q^ 40+38141*q^41+33461*q^42+29265*q^43+25526*q^44+22238*q^45+19321*q^46+16718*q^47 +14447*q^48+12458*q^49+10719*q^50+9246*q^51+7970*q^52+6852*q^53+5912*q^54+5094* q^55+4393*q^56+3813*q^57+3297*q^58+2844*q^59+2469*q^60+2137*q^61+1838*q^62+1596 *q^63+1373*q^64+1178*q^65+1015*q^66+872*q^67+734*q^68+625*q^69+525*q^70+435*q^ 71+367*q^72+304*q^73+250*q^74+205*q^75+168*q^76+131*q^77+108*q^78+84*q^79+64*q^ 80+49*q^81+38*q^82+28*q^83+21*q^84+16*q^85+10*q^86+8*q^87+5*q^88+3*q^89+2*q^90+ q^91, 1+14*q+104*q^2+533*q^3+2094*q^4+6669*q^5+17833*q^6+41040*q^7+82864*q^8+ 149181*q^9+242902*q^10+362293*q^11+500649*q^12+647439*q^13+790478*q^14+918359*q ^15+1022297*q^16+1097055*q^17+1141097*q^18+1156114*q^19+1145926*q^20+1115254*q^ 21+1068999*q^22+1011798*q^23+947625*q^24+879783*q^25+811009*q^26+743288*q^27+ 678138*q^28+616738*q^29+559713*q^30+507293*q^31+459443*q^32+415908*q^33+376310* q^34+340216*q^35+307431*q^36+277661*q^37+250605*q^38+226080*q^39+203726*q^40+ 183243*q^41+164451*q^42+147221*q^43+131454*q^44+117116*q^45+104121*q^46+92306*q ^47+81640*q^48+72037*q^49+63368*q^50+55631*q^51+48744*q^52+42612*q^53+37220*q^ 54+32453*q^55+28267*q^56+24642*q^57+21457*q^58+18670*q^59+16275*q^60+14191*q^61 +12366*q^62+10805*q^63+9445*q^64+8265*q^65+7252*q^66+6371*q^67+5577*q^68+4898*q ^69+4297*q^70+3757*q^71+3289*q^72+2869*q^73+2505*q^74+2182*q^75+1894*q^76+1638* q^77+1417*q^78+1219*q^79+1042*q^80+889*q^81+751*q^82+633*q^83+534*q^84+447*q^85 +372*q^86+308*q^87+252*q^88+207*q^89+167*q^90+132*q^91+107*q^92+82*q^93+64*q^94 +49*q^95+38*q^96+28*q^97+21*q^98+16*q^99+10*q^100+8*q^101+5*q^102+3*q^103+2*q^ 104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+5*q^3+4*q^2+2*q+1, q^10+4*q^9+9*q^8+ 13*q^7+14*q^6+12*q^5+10*q^4+7*q^3+3*q^2+2*q+1, q^15+5*q^14+14*q^13+26*q^12+36*q ^11+39*q^10+37*q^9+34*q^8+27*q^7+20*q^6+14*q^5+10*q^4+5*q^3+3*q^2+2*q+1, q^21+6 *q^20+20*q^19+45*q^18+76*q^17+101*q^16+113*q^15+114*q^14+106*q^13+92*q^12+78*q^ 11+64*q^10+49*q^9+37*q^8+26*q^7+19*q^6+12*q^5+8*q^4+5*q^3+3*q^2+2*q+1, 1+71*q^ 25+291*q^22+141*q^24+334*q^21+222*q^23+2*q+3*q^2+27*q^26+5*q^3+8*q^4+10*q^5+18* q^6+24*q^7+30*q^8+45*q^9+60*q^10+80*q^11+105*q^12+133*q^13+162*q^14+196*q^15+ 233*q^16+q^28+269*q^17+303*q^18+334*q^19+348*q^20+7*q^27, 1+996*q^25+704*q^22+ 906*q^24+610*q^21+808*q^23+2*q+655*q^30+854*q^29+35*q^34+105*q^33+3*q^2+1057*q^ 26+5*q^3+8*q^4+10*q^5+16*q^6+239*q^32+23*q^7+31*q^8+40*q^9+52*q^10+69*q^11+92*q ^12+120*q^13+156*q^14+195*q^15+245*q^16+996*q^28+308*q^17+374*q^18+450*q^19+529 *q^20+1064*q^27+434*q^31+8*q^35+q^36, 1+1280*q^25+148*q^42+773*q^22+1090*q^24+ 638*q^21+925*q^23+2*q+9*q^44+2478*q^30+q^45+2536*q^37+2195*q^29+1329*q^39+379*q ^41+3339*q^34+1951*q^38+3249*q^33+3*q^2+1485*q^26+5*q^3+8*q^4+10*q^5+16*q^6+ 3047*q^32+21*q^7+30*q^8+41*q^9+51*q^10+67*q^11+83*q^12+108*q^13+134*q^14+174*q^ 15+218*q^16+44*q^43+1938*q^28+271*q^17+341*q^18+421*q^19+522*q^20+1700*q^27+778 *q^40+2775*q^31+3263*q^35+2992*q^36, 1+1218*q^25+10516*q^42+680*q^22+1008*q^24+ 557*q^21+828*q^23+2*q+571*q^51+10037*q^44+54*q^53+2828*q^30+201*q^52+q^55+10*q^ 54+9045*q^45+6991*q^37+2416*q^29+7596*q^46+8667*q^39+10137*q^41+4875*q^34+1305* q^50+7822*q^38+4295*q^33+3*q^2+1458*q^26+5*q^3+8*q^4+10*q^5+16*q^6+3770*q^32+21 *q^7+28*q^8+40*q^9+52*q^10+66*q^11+85*q^12+106*q^13+128*q^14+164*q^15+204*q^16+ 10512*q^43+2487*q^49+2061*q^28+245*q^17+306*q^18+373*q^19+449*q^20+1743*q^27+ 9476*q^40+3282*q^31+5516*q^35+4067*q^48+6217*q^36+5856*q^47, 1+1047*q^25+14402* q^42+614*q^22+880*q^24+515*q^21+734*q^23+2*q+33357*q^51+18094*q^44+33043*q^53+ 17685*q^57+2562*q^30+33795*q^52+12460*q^58+27491*q^55+4364*q^60+30946*q^54+ 20248*q^45+7649*q^37+2161*q^29+22603*q^46+11*q^65+9980*q^39+2077*q^61+12812*q^ 41+4956*q^34+31971*q^50+65*q^64+8764*q^38+4233*q^33+22908*q^56+3*q^2+1267*q^26+ 5*q^3+8*q^4+10*q^5+16*q^6+3597*q^32+21*q^7+28*q^8+38*q^9+826*q^62+51*q^10+67*q^ 11+84*q^12+108*q^13+130*q^14+164*q^15+199*q^16+q^66+16148*q^43+29958*q^49+1809* q^28+245*q^17+297*q^18+358*q^19+432*q^20+1516*q^27+11329*q^40+265*q^63+3038*q^ 31+5764*q^35+27597*q^48+7869*q^59+6656*q^36+25092*q^47, 1+998*q^25+14468*q^42+ 614*q^22+850*q^24+519*q^21+720*q^23+2*q+43426*q^51+18934*q^44+54006*q^53+81453* q^57+2230*q^30+48471*q^52+88933*q^58+66770*q^55+102060*q^60+60090*q^54+21519*q^ 45+14321*q^71+1156*q^74+6900*q^37+69474*q^67+1894*q^29+24360*q^46+95629*q^65+ 38240*q^69+9397*q^39+106595*q^61+12572*q^41+4272*q^34+38868*q^50+103768*q^64+ 8071*q^38+3631*q^33+73959*q^56+3*q^2+53680*q^68+1168*q^26+5*q^3+8*q^4+10*q^5+16 *q^6+3091*q^32+21*q^7+28*q^8+38*q^9+108856*q^62+49*q^10+66*q^11+85*q^12+77*q^76 +107*q^13+132*q^14+166*q^15+203*q^16+83922*q^66+24754*q^70+12*q^77+q^78+16581*q ^43+34695*q^49+1610*q^28+246*q^17+299*q^18+359*q^19+432*q^20+1378*q^27+7268*q^ 72+10889*q^40+3168*q^73+108104*q^63+2624*q^31+341*q^75+5015*q^35+30894*q^48+ 95971*q^59+5885*q^36+27477*q^47, 1+1015*q^25+12458*q^42+625*q^22+13*q^90+872*q^ 24+525*q^21+734*q^23+2*q+43320*q^51+16718*q^44+55210*q^53+86989*q^57+2137*q^30+ 48990*q^52+96989*q^58+69599*q^55+120026*q^60+q^91+62058*q^54+19321*q^45+329008* q^71+353195*q^74+5912*q^37+244148*q^67+1838*q^29+22238*q^46+201339*q^65+289225* q^69+7970*q^39+133359*q^61+10719*q^41+3813*q^34+211671*q^79+38141*q^50+181955*q ^64+6852*q^38+3297*q^33+77880*q^56+3*q^2+266796*q^68+1178*q^26+5*q^3+8*q^4+10*q ^5+16*q^6+2844*q^32+21*q^7+28*q^8+38*q^9+148059*q^62+49*q^10+64*q^11+84*q^12+ 325727*q^76+108*q^13+131*q^14+168*q^15+205*q^16+222151*q^66+310391*q^70+296090* q^77+257153*q^78+14447*q^43+33461*q^49+1596*q^28+250*q^17+304*q^18+367*q^19+435 *q^20+1373*q^27+343521*q^72+9246*q^40+352203*q^73+164204*q^63+2469*q^31+344748* q^75+4393*q^35+163637*q^80+29265*q^48+46416*q^83+117588*q^81+11593*q^85+4665*q^ 86+1574*q^87+430*q^88+90*q^89+24769*q^84+77613*q^82+107955*q^59+5094*q^36+25526 *q^47, 1+1042*q^25+10805*q^42+633*q^22+918359*q^90+889*q^24+534*q^21+751*q^23+2 *q+82864*q^97+37220*q^51+14191*q^44+48744*q^53+81640*q^57+2182*q^30+362293*q^94 +42612*q^52+92306*q^58+63368*q^55+117116*q^60+790478*q^91+55631*q^54+16275*q^45 +376310*q^71+507293*q^74+149181*q^96+41040*q^98+5577*q^37+250605*q^67+1894*q^29 +18670*q^46+500649*q^93+17833*q^99+203726*q^65+307431*q^69+7252*q^39+131454*q^ 61+9445*q^41+3757*q^34+811009*q^79+32453*q^50+183243*q^64+6669*q^100+6371*q^38+ 3289*q^33+72037*q^56+3*q^2+277661*q^68+1219*q^26+5*q^3+8*q^4+10*q^5+16*q^6+2869 *q^32+21*q^7+28*q^8+38*q^9+242902*q^95+147221*q^62+2094*q^101+49*q^10+64*q^11+ 82*q^12+616738*q^76+107*q^13+132*q^14+167*q^15+207*q^16+226080*q^66+340216*q^70 +678138*q^77+743288*q^78+12366*q^43+533*q^102+28267*q^49+1638*q^28+252*q^17+308 *q^18+372*q^19+447*q^20+1417*q^27+415908*q^72+8265*q^40+459443*q^73+104*q^103+ 164451*q^63+2505*q^31+559713*q^75+4297*q^35+14*q^104+879783*q^80+24642*q^48+q^ 105+1068999*q^83+947625*q^81+1145926*q^85+1156114*q^86+1141097*q^87+1097055*q^ 88+1022297*q^89+1115254*q^84+1011798*q^82+104121*q^59+4898*q^36+647439*q^92+ 21457*q^47] The number of permutations avoiding, {[2, 4, 1, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [1, 2, 6, 21, 76, 274, 978, 3463, 12201, 42869, 150415, 527426, 1848905, 6480722, 22715293] The number of EVEN permutations avoiding, {[2, 4, 1, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [1, 1, 3, 11, 38, 136, 488, 1731, 6100, 21434, 75208, 263715, 924455, 3240363, 11357648] The number of ODD permutations avoiding, {[2, 4, 1, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, is given by [0, 1, 3, 10, 38, 138, 490, 1732, 6101, 21435, 75207, 263711, 924450, 3240359, 11357645] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 38, 138, 490, 1731, 6100, 21435, 75207, 263715, 924455, 3240359, 11357645] ODD:, [0, 1, 3, 10, 38, 136, 488, 1732, 6101, 21434, 75208, 263711, 924450, 3240363, 11357648] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.857142857, 4.447368421, 6.197080292, 8.057259714, 9.991336991, 11.97303500, 13.98532739, 16.01840907, 18.06722649, 20.12935007, 22.20352933, 24.28889299] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.489190073, 2.098838882, 2.783788879, 3.529425452, 4.315848615, 5.123538401, 5.936427483, 6.743127894, 7.536823828, 8.314362402, 9.075069247, 9.819664237] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.21769, 4.40512, 7.74948, 12.4568, 18.6265, 26.2506, 35.2412, 45.4698, 56.8037, 69.1286, 82.3569, 96.4258] Skewness: , [Float(undefined), 0., 0., 0.1589038037, 0.3214490818, 0.4693755925, 0.5986998469, 0.7122587157, 0.8128652902, 0.9019109207, 0.9799132779, 1.047247496, 1.104523357, 1.152706935, 1.193058446] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.435936804, 2.699179495, 2.941347870, 3.183180122, 3.433901170, 3.695127223, 3.962841500, 4.229580909, 4.486744343, 4.726850526, 4.944727445, 5.138183961] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, {[1, 4, 2, 3], [2, 1, 4, 3], [4, 3, 2, 1]}, {[2, 3, 1, 4], [2, 1, 4, 3], [4, 3, 2, 1]}, {[2, 1, 4, 3], [3, 1, 2, 4], [4, 3, 2, 1]}, {[1, 2, 3, 4], [3, 4, 1, 2], [4, 2, 1, 3]}, {[1, 2, 3, 4], [3, 4, 1, 2], [3, 2, 4, 1]}, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 3, 2, 1]}, {[1, 2, 3, 4], [2, 4, 3, 1], [3, 4, 1, 2]}} the member , {[1, 2, 3, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 2, 0, 0]}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {3}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 2, 0]}, {1}], [ [2, 3, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {1}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 2, 0, 0]}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {}], [[4, 2, 3, 1], %1, {4}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1], [0, 2, 0, 0, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 3, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 3, 0]}, {3}], [[2, 4, 1, 3], %1, {1}], [[2, 4, 3, 1, 5], {[0, 0, 1, 1, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {4}], [[1, 4, 3, 5, 2], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [2, 4, 3, 5, 1], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[1, 3, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 3, 2, 5, 4], {[0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [ [1, 4, 2, 5, 3], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 5, 4, 1, 2], {[0, 0, 0, 0, 0, 0]}, {2}], [[3, 5, 4, 2, 1], { [0, 0, 0, 2, 1, 0], [0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [[3, 1, 2], {[0, 0, 2, 0], [0, 1, 0, 0]}, {}], [[2, 5, 3, 1, 4], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}], [[2, 5, 4, 1, 3], { [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[2, 1, 5, 3, 4], { [0, 2, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[4, 1, 2, 5, 3], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 1, 2, 5, 4], {[0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {4}], [ [4, 2, 3, 5, 1], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 1, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 5, 2], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 3, 0, 0], [0, 0, 2, 1, 0]}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 2, 0, 0, 0, 0]}, {4}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0], [0, 2, 0, 0, 0]}, {}], [[3, 1, 4, 2], %1, {1}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {2}], [[2, 1], {[0, 3, 0]}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 2, 1, 0], [0, 0, 0, 3, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 1, 5, 4, 2], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 0, 0, 2, 0, 0], [0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 3, 0], [0, 1, 0, 0]}, {}]} %1 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 3*q+5*q^2+6*q^3+3*q^4+3*q^5+q^6, 3*q^2+11*q^3+14*q^4+ 14*q^5+12*q^6+8*q^7+4*q^8+4*q^9+q^10, q^3+10*q^4+20*q^5+31*q^6+32*q^7+31*q^8+28 *q^9+24*q^10+15*q^11+10*q^12+5*q^13+5*q^14+q^15, 3*q^5+13*q^6+30*q^7+49*q^8+54* q^9+64*q^10+67*q^11+63*q^12+55*q^13+50*q^14+41*q^15+29*q^16+18*q^17+12*q^18+6*q ^19+6*q^20+q^21, 3*q^7+11*q^8+33*q^9+51*q^10+78*q^11+97*q^12+104*q^13+110*q^14+ 123*q^15+122*q^16+112*q^17+103*q^18+92*q^19+79*q^20+67*q^21+48*q^22+34*q^23+21* q^24+14*q^25+7*q^26+7*q^27+q^28, q^9+8*q^10+21*q^11+42*q^12+78*q^13+104*q^14+ 132*q^15+151*q^16+175*q^17+189*q^18+196*q^19+194*q^20+207*q^21+201*q^22+192*q^ 23+175*q^24+155*q^25+138*q^26+122*q^27+100*q^28+77*q^29+55*q^30+39*q^31+24*q^32 +16*q^33+8*q^34+8*q^35+q^36, 3*q^12+9*q^13+27*q^14+47*q^15+85*q^16+118*q^17+159 *q^18+197*q^19+231*q^20+241*q^21+269*q^22+305*q^23+311*q^24+314*q^25+316*q^26+ 320*q^27+325*q^28+318*q^29+298*q^30+276*q^31+253*q^32+225*q^33+197*q^34+174*q^ 35+148*q^36+113*q^37+87*q^38+62*q^39+44*q^40+27*q^41+18*q^42+9*q^43+9*q^44+q^45 , 3*q^15+9*q^16+23*q^17+47*q^18+71*q^19+116*q^20+167*q^21+209*q^22+243*q^23+304 *q^24+338*q^25+378*q^26+395*q^27+415*q^28+445*q^29+475*q^30+476*q^31+491*q^32+ 492*q^33+491*q^34+477*q^35+485*q^36+472*q^37+448*q^38+416*q^39+382*q^40+345*q^ 41+311*q^42+274*q^43+240*q^44+205*q^45+165*q^46+126*q^47+97*q^48+69*q^49+49*q^ 50+30*q^51+20*q^52+10*q^53+10*q^54+q^55, q^18+8*q^19+15*q^20+34*q^21+59*q^22+96 *q^23+129*q^24+190*q^25+240*q^26+304*q^27+357*q^28+408*q^29+459*q^30+503*q^31+ 541*q^32+591*q^33+615*q^34+624*q^35+650*q^36+676*q^37+709*q^38+720*q^39+723*q^ 40+719*q^41+720*q^42+704*q^43+698*q^44+696*q^45+671*q^46+635*q^47+603*q^48+556* q^49+509*q^50+460*q^51+411*q^52+368*q^53+324*q^54+277*q^55+226*q^56+182*q^57+ 139*q^58+107*q^59+76*q^60+54*q^61+33*q^62+22*q^63+11*q^64+11*q^65+q^66, 3*q^22+ 9*q^23+21*q^24+39*q^25+63*q^26+98*q^27+139*q^28+195*q^29+243*q^30+322*q^31+385* q^32+442*q^33+517*q^34+605*q^35+648*q^36+698*q^37+726*q^38+779*q^39+827*q^40+ 888*q^41+894*q^42+917*q^43+940*q^44+966*q^45+986*q^46+1019*q^47+1019*q^48+1029* q^49+1027*q^50+1021*q^51+999*q^52+993*q^53+961*q^54+956*q^55+930*q^56+887*q^57+ 830*q^58+777*q^59+721*q^60+659*q^61+596*q^62+535*q^63+476*q^64+426*q^65+367*q^ 66+303*q^67+247*q^68+199*q^69+152*q^70+117*q^71+83*q^72+59*q^73+36*q^74+24*q^75 +12*q^76+12*q^77+q^78, 3*q^26+9*q^27+21*q^28+35*q^29+65*q^30+86*q^31+131*q^32+ 182*q^33+233*q^34+286*q^35+375*q^36+445*q^37+540*q^38+610*q^39+680*q^40+766*q^ 41+861*q^42+916*q^43+965*q^44+1026*q^45+1056*q^46+1099*q^47+1159*q^48+1213*q^49 +1252*q^50+1276*q^51+1294*q^52+1336*q^53+1347*q^54+1348*q^55+1379*q^56+1418*q^ 57+1419*q^58+1422*q^59+1402*q^60+1401*q^61+1381*q^62+1364*q^63+1329*q^64+1303*q ^65+1284*q^66+1242*q^67+1184*q^68+1131*q^69+1058*q^70+983*q^71+906*q^72+834*q^ 73+756*q^74+680*q^75+609*q^76+542*q^77+476*q^78+398*q^79+329*q^80+268*q^81+216* q^82+165*q^83+127*q^84+90*q^85+64*q^86+39*q^87+26*q^88+13*q^89+13*q^90+q^91, q^ 30+8*q^31+15*q^32+28*q^33+53*q^34+78*q^35+109*q^36+146*q^37+201*q^38+261*q^39+ 329*q^40+383*q^41+473*q^42+574*q^43+671*q^44+753*q^45+866*q^46+955*q^47+1059*q^ 48+1113*q^49+1192*q^50+1266*q^51+1335*q^52+1387*q^53+1471*q^54+1509*q^55+1541*q ^56+1571*q^57+1643*q^58+1681*q^59+1749*q^60+1756*q^61+1794*q^62+1820*q^63+1829* q^64+1831*q^65+1862*q^66+1881*q^67+1908*q^68+1901*q^69+1904*q^70+1885*q^71+1887 *q^72+1846*q^73+1822*q^74+1791*q^75+1752*q^76+1704*q^77+1687*q^78+1625*q^79+ 1560*q^80+1481*q^81+1398*q^82+1314*q^83+1223*q^84+1123*q^85+1030*q^86+941*q^87+ 850*q^88+763*q^89+684*q^90+599*q^91+513*q^92+429*q^93+355*q^94+289*q^95+233*q^ 96+178*q^97+137*q^98+97*q^99+69*q^100+42*q^101+28*q^102+14*q^103+14*q^104+q^105 ] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+5*q^4+6*q^3+3*q^2+3*q+1, 3*q^8+11*q^7+14*q^6+14 *q^5+12*q^4+8*q^3+4*q^2+4*q+1, q^12+10*q^11+20*q^10+31*q^9+32*q^8+31*q^7+28*q^6 +24*q^5+15*q^4+10*q^3+5*q^2+5*q+1, 3*q^16+13*q^15+30*q^14+49*q^13+54*q^12+64*q^ 11+67*q^10+63*q^9+55*q^8+50*q^7+41*q^6+29*q^5+18*q^4+12*q^3+6*q^2+6*q+1, 3*q^21 +11*q^20+33*q^19+51*q^18+78*q^17+97*q^16+104*q^15+110*q^14+123*q^13+122*q^12+ 112*q^11+103*q^10+92*q^9+79*q^8+67*q^7+48*q^6+34*q^5+21*q^4+14*q^3+7*q^2+7*q+1, 1+21*q^25+104*q^22+42*q^24+132*q^21+78*q^23+8*q+8*q^2+8*q^26+16*q^3+24*q^4+39*q ^5+55*q^6+77*q^7+100*q^8+122*q^9+138*q^10+155*q^11+175*q^12+192*q^13+201*q^14+ 207*q^15+194*q^16+196*q^17+189*q^18+175*q^19+151*q^20+q^27, 1+231*q^25+305*q^22 +241*q^24+311*q^21+269*q^23+9*q+47*q^30+85*q^29+3*q^33+9*q^2+197*q^26+18*q^3+27 *q^4+44*q^5+62*q^6+9*q^32+87*q^7+113*q^8+148*q^9+174*q^10+197*q^11+225*q^12+253 *q^13+276*q^14+298*q^15+318*q^16+118*q^28+325*q^17+320*q^18+316*q^19+314*q^20+ 159*q^27+27*q^31, 1+475*q^25+492*q^22+476*q^24+491*q^21+491*q^23+10*q+338*q^30+ 47*q^37+378*q^29+9*q^39+167*q^34+23*q^38+209*q^33+10*q^2+445*q^26+20*q^3+30*q^4 +49*q^5+69*q^6+243*q^32+97*q^7+126*q^8+165*q^9+205*q^10+240*q^11+274*q^12+311*q ^13+345*q^14+382*q^15+416*q^16+395*q^28+448*q^17+472*q^18+485*q^19+477*q^20+415 *q^27+3*q^40+304*q^31+116*q^35+71*q^36, 1+719*q^25+129*q^42+698*q^22+720*q^24+ 696*q^21+704*q^23+11*q+59*q^44+650*q^30+34*q^45+408*q^37+676*q^29+15*q^46+304*q ^39+190*q^41+541*q^34+357*q^38+591*q^33+11*q^2+723*q^26+22*q^3+33*q^4+54*q^5+76 *q^6+615*q^32+107*q^7+139*q^8+182*q^9+226*q^10+277*q^11+324*q^12+368*q^13+411*q ^14+460*q^15+509*q^16+96*q^43+709*q^28+556*q^17+603*q^18+635*q^19+671*q^20+720* q^27+240*q^40+624*q^31+503*q^35+q^48+459*q^36+8*q^47, 1+993*q^25+648*q^42+930*q ^22+961*q^24+887*q^21+956*q^23+12*q+98*q^51+517*q^44+39*q^53+1019*q^30+63*q^52+ 9*q^55+21*q^54+442*q^45+888*q^37+1029*q^29+385*q^46+779*q^39+698*q^41+940*q^34+ 139*q^50+827*q^38+966*q^33+3*q^56+12*q^2+999*q^26+24*q^3+36*q^4+59*q^5+83*q^6+ 986*q^32+117*q^7+152*q^8+199*q^9+247*q^10+303*q^11+367*q^12+426*q^13+476*q^14+ 535*q^15+596*q^16+605*q^43+195*q^49+1027*q^28+659*q^17+721*q^18+777*q^19+830*q^ 20+1021*q^27+726*q^40+1019*q^31+917*q^35+243*q^48+894*q^36+322*q^47, 1+1284*q^ 25+1213*q^42+1131*q^22+1242*q^24+1058*q^21+1184*q^23+13*q+680*q^51+1099*q^44+ 540*q^53+233*q^57+1401*q^30+610*q^52+182*q^58+375*q^55+86*q^60+445*q^54+1056*q^ 45+1347*q^37+1381*q^29+1026*q^46+3*q^65+1294*q^39+65*q^61+1252*q^41+1418*q^34+ 766*q^50+9*q^64+1336*q^38+1419*q^33+286*q^56+13*q^2+1303*q^26+26*q^3+39*q^4+64* q^5+90*q^6+1422*q^32+127*q^7+165*q^8+216*q^9+35*q^62+268*q^10+329*q^11+398*q^12 +476*q^13+542*q^14+609*q^15+680*q^16+1159*q^43+861*q^49+1364*q^28+756*q^17+834* q^18+906*q^19+983*q^20+1329*q^27+1276*q^40+21*q^63+1402*q^31+1379*q^35+916*q^48 +131*q^59+1348*q^36+965*q^47, 1+1560*q^25+1820*q^42+1314*q^22+1481*q^24+1223*q^ 21+1398*q^23+14*q+1471*q^51+1756*q^44+1335*q^53+1059*q^57+1791*q^30+1387*q^52+ 955*q^58+1192*q^55+753*q^60+1266*q^54+1749*q^45+53*q^71+8*q^74+1908*q^37+201*q^ 67+1752*q^29+1681*q^46+329*q^65+109*q^69+1862*q^39+671*q^61+1829*q^41+1885*q^34 +1509*q^50+383*q^64+1881*q^38+1887*q^33+1113*q^56+14*q^2+146*q^68+1625*q^26+28* q^3+42*q^4+69*q^5+97*q^6+1846*q^32+137*q^7+178*q^8+233*q^9+574*q^62+289*q^10+ 355*q^11+429*q^12+513*q^13+599*q^14+684*q^15+763*q^16+261*q^66+78*q^70+1794*q^ 43+1541*q^49+1704*q^28+850*q^17+941*q^18+1030*q^19+1123*q^20+1687*q^27+28*q^72+ 1831*q^40+15*q^73+473*q^63+1822*q^31+q^75+1904*q^35+1571*q^48+866*q^59+1901*q^ 36+1643*q^47] The number of permutations avoiding, {[1, 2, 3, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 71, 213, 561, 1317, 2809, 5536, 10220, 17865, 29823, 47867, 74271 ] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 1, 3, 9, 34, 111, 286, 650, 1390, 2780, 5142, 8920, 14849, 23940, 37247] The number of ODD permutations avoiding, {[1, 2, 3, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [0, 1, 3, 12, 37, 102, 275, 667, 1419, 2756, 5078, 8945, 14974, 23927, 37024] For the reverse patterns and complement patterns, we get EVEN:, [ 1, 1, 3, 9, 34, 102, 275, 650, 1390, 2756, 5078, 8920, 14849, 23927, 37024] ODD:, [ 0, 1, 3, 12, 37, 111, 286, 667, 1419, 2780, 5142, 8945, 14974, 23940, 37247 ] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.047619048, 5.225352113, 8.112676056, 11.73796791, 16.10554290, 21.21359915, 27.05960983, 33.64148728, 40.95768262, 49.00707508, 57.78885245, 67.30241952] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.396464600, 1.877946010, 2.462198505, 3.173104802, 4.015944830, 4.991336050, 6.099374684, 7.340318445, 8.714548374, 10.22247168, 11.86446616, 13.64085901] The centralized moments are Second: , [0., 0.250000, 0.916667, 1.95011, 3.52668, 6.06242, 10.0686, 16.1278, 24.9134, 37.2024, 53.8803, 75.9434, 104.499, 140.766, 186.073] Skewness: , [Float(undefined), 0., 0., 0.3348086229, 0.4506415497, 0.4061130085, 0.3241255356, 0.2489349379, 0.1871507930, 0.1368829037, 0.09559577458, 0.06133004686, 0.03265843905, 0.008508989435, -0.01194746509] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.263860296, 2.580243032, 2.662837584, 2.611369125, 2.539176992, 2.480048014, 2.435942756, 2.402735658, 2.376824911, 2.355806173, 2.338296437, 2.323433127] end of this data For the equivalence class of patterns, { {[2, 4, 1, 3], [2, 3, 1, 4], [3, 4, 2, 1]}, {[3, 1, 4, 2], [3, 1, 2, 4], [4, 3, 1, 2]}, {[2, 4, 1, 3], [2, 1, 3, 4], [4, 2, 1, 3]}, {[1, 4, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}, {[1, 2, 4, 3], [3, 1, 4, 2], [4, 1, 3, 2]}, {[1, 3, 4, 2], [3, 4, 2, 1], [3, 1, 4, 2]}, {[1, 2, 4, 3], [2, 4, 3, 1], [2, 4, 1, 3]}, {[2, 1, 3, 4], [3, 2, 4, 1], [3, 1, 4, 2]}} the member , {[1, 4, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 1, 3, 4], {}, {3}], [[2, 1, 3], {}, {}], [[1, 2, 3], {}, {2}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 1, 0]}, {1}], [[3, 1, 2], {}, {2}], [[1, 3, 2], {[0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+7*q^2+10*q^3 +13*q^4+13*q^5+12*q^6+9*q^7+5*q^8+2*q^9+q^10, 1+5*q+11*q^2+18*q^3+26*q^4+32*q^5 +37*q^6+38*q^7+34*q^8+29*q^9+23*q^10+16*q^11+9*q^12+5*q^13+2*q^14+q^15, 1+6*q+ 16*q^2+30*q^3+48*q^4+66*q^5+84*q^6+101*q^7+109*q^8+111*q^9+107*q^10+95*q^11+84* q^12+70*q^13+53*q^14+38*q^15+27*q^16+16*q^17+9*q^18+5*q^19+2*q^20+q^21, 1+7*q+ 22*q^2+47*q^3+83*q^4+126*q^5+173*q^6+224*q^7+270*q^8+308*q^9+334*q^10+342*q^11+ 336*q^12+320*q^13+294*q^14+259*q^15+228*q^16+188*q^17+152*q^18+118*q^19+86*q^20 +62*q^21+42*q^22+27*q^23+16*q^24+9*q^25+5*q^26+2*q^27+q^28, 1+8*q+29*q^2+70*q^3 +136*q^4+226*q^5+335*q^6+462*q^7+596*q^8+730*q^9+858*q^10+962*q^11+1037*q^12+ 1078*q^13+1083*q^14+1057*q^15+1013*q^16+944*q^17+861*q^18+772*q^19+669*q^20+578 *q^21+484*q^22+396*q^23+315*q^24+243*q^25+182*q^26+134*q^27+95*q^28+66*q^29+42* q^30+27*q^31+16*q^32+9*q^33+5*q^34+2*q^35+q^36, 1+9*q+37*q^2+100*q^3+213*q^4+ 385*q^5+615*q^6+903*q^7+1237*q^8+1601*q^9+1987*q^10+2370*q^11+2728*q^12+3040*q^ 13+3279*q^14+3435*q^15+3516*q^16+3514*q^17+3444*q^18+3323*q^19+3139*q^20+2929*q ^21+2687*q^22+2427*q^23+2160*q^24+1900*q^25+1641*q^26+1400*q^27+1173*q^28+963*q ^29+778*q^30+619*q^31+480*q^32+366*q^33+273*q^34+198*q^35+143*q^36+99*q^37+66*q ^38+42*q^39+27*q^40+16*q^41+9*q^42+5*q^43+2*q^44+q^45, 1+10*q+46*q^2+138*q^3+ 321*q^4+628*q^5+1078*q^6+1684*q^7+2443*q^8+3333*q^9+4337*q^10+5421*q^11+6547*q^ 12+7674*q^13+8739*q^14+9686*q^15+10476*q^16+11077*q^17+11465*q^18+11659*q^19+ 11654*q^20+11480*q^21+11165*q^22+10705*q^23+10139*q^24+9494*q^25+8793*q^26+8049 *q^27+7301*q^28+6537*q^29+5803*q^30+5095*q^31+4427*q^32+3793*q^33+3224*q^34+ 2693*q^35+2232*q^36+1819*q^37+1465*q^38+1162*q^39+911*q^40+699*q^41+531*q^42+ 396*q^43+289*q^44+207*q^45+147*q^46+99*q^47+66*q^48+42*q^49+27*q^50+16*q^51+9*q ^52+5*q^53+2*q^54+q^55, 1+11*q+56*q^2+185*q^3+468*q^4+987*q^5+1815*q^6+3013*q^7 +4620*q^8+6635*q^9+9043*q^10+11801*q^11+14845*q^12+18111*q^13+21495*q^14+24877* q^15+28134*q^16+31132*q^17+33749*q^18+35913*q^19+37551*q^20+38649*q^21+39244*q^ 22+39327*q^23+38955*q^24+38166*q^25+37010*q^26+35539*q^27+33823*q^28+31879*q^29 +29810*q^30+27646*q^31+25441*q^32+23210*q^33+21023*q^34+18869*q^35+16823*q^36+ 14875*q^37+13042*q^38+11350*q^39+9784*q^40+8356*q^41+7072*q^42+5932*q^43+4921*q ^44+4048*q^45+3294*q^46+2653*q^47+2113*q^48+1662*q^49+1291*q^50+991*q^51+750*q^ 52+561*q^53+412*q^54+298*q^55+211*q^56+147*q^57+99*q^58+66*q^59+42*q^60+27*q^61 +16*q^62+9*q^63+5*q^64+2*q^65+q^66, 1+12*q+67*q^2+242*q^3+663*q^4+1502*q^5+2950 *q^6+5196*q^7+8408*q^8+12693*q^9+18107*q^10+24647*q^11+32242*q^12+40797*q^13+ 50159*q^14+60126*q^15+70465*q^16+80885*q^17+91051*q^18+100652*q^19+109381*q^20+ 116969*q^21+123289*q^22+128193*q^23+131665*q^24+133728*q^25+134420*q^26+133843* q^27+132098*q^28+129272*q^29+125497*q^30+120929*q^31+115671*q^32+109895*q^33+ 103713*q^34+97251*q^35+90600*q^36+83920*q^37+77215*q^38+70664*q^39+64246*q^40+ 58081*q^41+52164*q^42+46604*q^43+41347*q^44+36482*q^45+31966*q^46+27833*q^47+ 24060*q^48+20668*q^49+17603*q^50+14914*q^51+12521*q^52+10445*q^53+8633*q^54+ 7090*q^55+5763*q^56+4655*q^57+3717*q^58+2945*q^59+2306*q^60+1791*q^61+1371*q^62 +1042*q^63+780*q^64+577*q^65+421*q^66+302*q^67+211*q^68+147*q^69+99*q^70+66*q^ 71+42*q^72+27*q^73+16*q^74+9*q^75+5*q^76+2*q^77+q^78, 1+13*q+79*q^2+310*q^3+916 *q^4+2222*q^5+4648*q^6+8671*q^7+14786*q^8+23432*q^9+34957*q^10+49596*q^11+67429 *q^12+88424*q^13+112419*q^14+139124*q^15+168168*q^16+199054*q^17+231149*q^18+ 263721*q^19+295956*q^20+326986*q^21+356058*q^22+382455*q^23+405602*q^24+425129* q^25+440762*q^26+452425*q^27+460150*q^28+464042*q^29+464254*q^30+461061*q^31+ 454667*q^32+445399*q^33+433557*q^34+419501*q^35+403522*q^36+386077*q^37+367365* q^38+347801*q^39+327601*q^40+307070*q^41+286394*q^42+265876*q^43+245630*q^44+ 225886*q^45+206759*q^46+188361*q^47+170789*q^48+154144*q^49+138440*q^50+123761* q^51+110106*q^52+97454*q^53+85826*q^54+75191*q^55+65528*q^56+56796*q^57+48966*q ^58+41962*q^59+35765*q^60+30304*q^61+25519*q^62+21355*q^63+17768*q^64+14670*q^ 65+12045*q^66+9813*q^67+7938*q^68+6372*q^69+5076*q^70+4005*q^71+3138*q^72+2435* q^73+1871*q^74+1422*q^75+1072*q^76+796*q^77+586*q^78+425*q^79+302*q^80+211*q^81 +147*q^82+99*q^83+66*q^84+42*q^85+27*q^86+16*q^87+9*q^88+5*q^89+2*q^90+q^91, 1+ 14*q+92*q^2+390*q^3+1238*q^4+3206*q^5+7124*q^6+14050*q^7+25212*q^8+41889*q^9+ 65293*q^10+96484*q^11+136253*q^12+185107*q^13+243242*q^14+310507*q^15+386488*q^ 16+470490*q^17+561515*q^18+658273*q^19+759154*q^20+862217*q^21+965395*q^22+ 1066499*q^23+1163334*q^24+1253978*q^25+1336654*q^26+1410018*q^27+1473071*q^28+ 1525161*q^29+1565967*q^30+1595529*q^31+1613991*q^32+1621721*q^33+1619234*q^34+ 1607092*q^35+1586013*q^36+1556794*q^37+1520247*q^38+1477244*q^39+1428755*q^40+ 1375537*q^41+1318572*q^42+1258579*q^43+1196428*q^44+1132689*q^45+1068206*q^46+ 1003362*q^47+938955*q^48+875291*q^49+812952*q^50+752180*q^51+693485*q^52+636882 *q^53+582831*q^54+531292*q^55+482585*q^56+436629*q^57+393646*q^58+353434*q^59+ 316199*q^60+281698*q^61+249998*q^62+220909*q^63+194445*q^64+170366*q^65+148690* q^66+129151*q^67+111716*q^68+96162*q^69+82423*q^70+70271*q^71+59659*q^72+50361* q^73+42319*q^74+35351*q^75+29391*q^76+24277*q^77+19960*q^78+16292*q^79+13231*q^ 80+10663*q^81+8545*q^82+6789*q^83+5364*q^84+4198*q^85+3267*q^86+2515*q^87+1922* q^88+1452*q^89+1088*q^90+805*q^91+590*q^92+425*q^93+302*q^94+211*q^95+147*q^96+ 99*q^97+66*q^98+42*q^99+27*q^100+16*q^101+9*q^102+5*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+ 10*q^7+13*q^6+13*q^5+12*q^4+9*q^3+5*q^2+2*q+1, q^15+5*q^14+11*q^13+18*q^12+26*q ^11+32*q^10+37*q^9+38*q^8+34*q^7+29*q^6+23*q^5+16*q^4+9*q^3+5*q^2+2*q+1, q^21+6 *q^20+16*q^19+30*q^18+48*q^17+66*q^16+84*q^15+101*q^14+109*q^13+111*q^12+107*q^ 11+95*q^10+84*q^9+70*q^8+53*q^7+38*q^6+27*q^5+16*q^4+9*q^3+5*q^2+2*q+1, 1+47*q^ 25+173*q^22+83*q^24+224*q^21+126*q^23+2*q+5*q^2+22*q^26+9*q^3+16*q^4+27*q^5+42* q^6+62*q^7+86*q^8+118*q^9+152*q^10+188*q^11+228*q^12+259*q^13+294*q^14+320*q^15 +336*q^16+q^28+342*q^17+334*q^18+308*q^19+270*q^20+7*q^27, 1+962*q^25+1083*q^22 +1037*q^24+1057*q^21+1078*q^23+2*q+335*q^30+462*q^29+29*q^34+70*q^33+5*q^2+858* q^26+9*q^3+16*q^4+27*q^5+42*q^6+136*q^32+66*q^7+95*q^8+134*q^9+182*q^10+243*q^ 11+315*q^12+396*q^13+484*q^14+578*q^15+669*q^16+596*q^28+772*q^17+861*q^18+944* q^19+1013*q^20+730*q^27+226*q^31+8*q^35+q^36, 1+3139*q^25+100*q^42+2427*q^22+ 2929*q^24+2160*q^21+2687*q^23+2*q+9*q^44+3435*q^30+q^45+1237*q^37+3516*q^29+615 *q^39+213*q^41+2370*q^34+903*q^38+2728*q^33+5*q^2+3323*q^26+9*q^3+16*q^4+27*q^5 +42*q^6+3040*q^32+66*q^7+99*q^8+143*q^9+198*q^10+273*q^11+366*q^12+480*q^13+619 *q^14+778*q^15+963*q^16+37*q^43+3514*q^28+1173*q^17+1400*q^18+1641*q^19+1900*q^ 20+3444*q^27+385*q^40+3279*q^31+1987*q^35+1601*q^36, 1+5803*q^25+7674*q^42+3793 *q^22+5095*q^24+3224*q^21+4427*q^23+2*q+321*q^51+5421*q^44+46*q^53+9494*q^30+ 138*q^52+q^55+10*q^54+4337*q^45+11465*q^37+8793*q^29+3333*q^46+10476*q^39+8739* q^41+11480*q^34+628*q^50+11077*q^38+11165*q^33+5*q^2+6537*q^26+9*q^3+16*q^4+27* q^5+42*q^6+10705*q^32+66*q^7+99*q^8+147*q^9+207*q^10+289*q^11+396*q^12+531*q^13 +699*q^14+911*q^15+1162*q^16+6547*q^43+1078*q^49+8049*q^28+1465*q^17+1819*q^18+ 2232*q^19+2693*q^20+7301*q^27+9686*q^40+10139*q^31+11654*q^35+1684*q^48+11659*q ^36+2443*q^47, 1+8356*q^25+38955*q^42+4921*q^22+7072*q^24+4048*q^21+5932*q^23+2 *q+24877*q^51+39244*q^44+18111*q^53+6635*q^57+16823*q^30+21495*q^52+4620*q^58+ 11801*q^55+1815*q^60+14845*q^54+38649*q^45+31879*q^37+14875*q^29+37551*q^46+11* q^65+35539*q^39+987*q^61+38166*q^41+25441*q^34+28134*q^50+56*q^64+33823*q^38+ 23210*q^33+9043*q^56+5*q^2+9784*q^26+9*q^3+16*q^4+27*q^5+42*q^6+21023*q^32+66*q ^7+99*q^8+147*q^9+468*q^62+211*q^10+298*q^11+412*q^12+561*q^13+750*q^14+991*q^ 15+1291*q^16+q^66+39327*q^43+31132*q^49+13042*q^28+1662*q^17+2113*q^18+2653*q^ 19+3294*q^20+11350*q^27+37010*q^40+185*q^63+18869*q^31+27646*q^35+33749*q^48+ 3013*q^59+29810*q^36+35913*q^47, 1+10445*q^25+90600*q^42+5763*q^22+8633*q^24+ 4655*q^21+7090*q^23+2*q+133843*q^51+103713*q^44+133728*q^53+116969*q^57+24060*q ^30+134420*q^52+109381*q^58+128193*q^55+91051*q^60+131665*q^54+109895*q^45+5196 *q^71+663*q^74+58081*q^37+24647*q^67+20668*q^29+115671*q^46+40797*q^65+12693*q^ 69+70664*q^39+80885*q^61+83920*q^41+41347*q^34+132098*q^50+50159*q^64+64246*q^ 38+36482*q^33+123289*q^56+5*q^2+18107*q^68+12521*q^26+9*q^3+16*q^4+27*q^5+42*q^ 6+31966*q^32+66*q^7+99*q^8+147*q^9+70465*q^62+211*q^10+302*q^11+421*q^12+67*q^ 76+577*q^13+780*q^14+1042*q^15+1371*q^16+32242*q^66+8408*q^70+12*q^77+q^78+ 97251*q^43+129272*q^49+17603*q^28+1791*q^17+2306*q^18+2945*q^19+3717*q^20+14914 *q^27+2950*q^72+77215*q^40+1502*q^73+60126*q^63+27833*q^31+242*q^75+46604*q^35+ 125497*q^48+100652*q^59+52164*q^36+120929*q^47, 1+12045*q^25+154144*q^42+6372*q ^22+13*q^90+9813*q^24+5076*q^21+7938*q^23+2*q+327601*q^51+188361*q^44+367365*q^ 53+433557*q^57+30304*q^30+347801*q^52+445399*q^58+403522*q^55+461061*q^60+q^91+ 386077*q^54+206759*q^45+295956*q^71+199054*q^74+85826*q^37+405602*q^67+25519*q^ 29+225886*q^46+440762*q^65+356058*q^69+110106*q^39+464254*q^61+138440*q^41+ 56796*q^34+67429*q^79+307070*q^50+452425*q^64+97454*q^38+48966*q^33+419501*q^56 +5*q^2+382455*q^68+14670*q^26+9*q^3+16*q^4+27*q^5+42*q^6+41962*q^32+66*q^7+99*q ^8+147*q^9+464042*q^62+211*q^10+302*q^11+425*q^12+139124*q^76+586*q^13+796*q^14 +1072*q^15+1422*q^16+425129*q^66+326986*q^70+112419*q^77+88424*q^78+170789*q^43 +286394*q^49+21355*q^28+1871*q^17+2435*q^18+3138*q^19+4005*q^20+17768*q^27+ 263721*q^72+123761*q^40+231149*q^73+460150*q^63+35765*q^31+168168*q^75+65528*q^ 35+49596*q^80+265876*q^48+14786*q^83+34957*q^81+4648*q^85+2222*q^86+916*q^87+ 310*q^88+79*q^89+8671*q^84+23432*q^82+454667*q^59+75191*q^36+245630*q^47, 1+ 13231*q^25+220909*q^42+6789*q^22+310507*q^90+10663*q^24+5364*q^21+8545*q^23+2*q +25212*q^97+582831*q^51+281698*q^44+693485*q^53+938955*q^57+35351*q^30+96484*q^ 94+636882*q^52+1003362*q^58+812952*q^55+1132689*q^60+243242*q^91+752180*q^54+ 316199*q^45+1619234*q^71+1595529*q^74+41889*q^96+14050*q^98+111716*q^37+1520247 *q^67+29391*q^29+353434*q^46+136253*q^93+7124*q^99+1428755*q^65+1586013*q^69+ 148690*q^39+1196428*q^61+194445*q^41+70271*q^34+1336654*q^79+531292*q^50+ 1375537*q^64+3206*q^100+129151*q^38+59659*q^33+875291*q^56+5*q^2+1556794*q^68+ 16292*q^26+9*q^3+16*q^4+27*q^5+42*q^6+50361*q^32+66*q^7+99*q^8+147*q^9+65293*q^ 95+1258579*q^62+1238*q^101+211*q^10+302*q^11+425*q^12+1525161*q^76+590*q^13+805 *q^14+1088*q^15+1452*q^16+1477244*q^66+1607092*q^70+1473071*q^77+1410018*q^78+ 249998*q^43+390*q^102+482585*q^49+24277*q^28+1922*q^17+2515*q^18+3267*q^19+4198 *q^20+19960*q^27+1621721*q^72+170366*q^40+1613991*q^73+92*q^103+1318572*q^63+ 42319*q^31+1565967*q^75+82423*q^35+14*q^104+1253978*q^80+436629*q^48+q^105+ 965395*q^83+1163334*q^81+759154*q^85+658273*q^86+561515*q^87+470490*q^88+386488 *q^89+862217*q^84+1066499*q^82+1068206*q^59+96162*q^36+185107*q^92+393646*q^47] The number of permutations avoiding, {[1, 4, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}, is given by [1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59280, 227240, 873886, 3370030, 13027730, 50469890] The number of EVEN permutations avoiding, {[1, 4, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}, is given by [1, 1, 3, 11, 39, 143, 540, 2043, 7758, 29635, 113638, 436959, 1684934, 6513815, 25235293] The number of ODD permutations avoiding, {[1, 4, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}, is given by [0, 1, 3, 10, 38, 144, 539, 2039, 7764, 29645, 113602, 436927, 1685096, 6513915, 25234597] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 39, 144, 539, 2043, 7758, 29645, 113602, 436959, 1684934, 6513915, 25234597] ODD:, [0, 1, 3, 10, 38, 143, 540, 2039, 7764, 29635, 113638, 436927, 1685096, 6513815, 25235293] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.779220779, 6.926829268, 9.359592215, 12.05218030, 14.98556887, 18.14482119, 21.51775216, 25.09410724, 28.86505046, 32.82283460, 36.96057865] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.495268426, 2.135801055, 2.873320394, 3.695108384, 4.590104111, 5.550129790, 6.569268758, 7.643132357, 8.768335770, 9.942164437, 11.16236724, 12.42702691] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.23583, 4.56165, 8.25597, 13.6538, 21.0691, 30.8039, 43.1553, 58.4175, 76.8837, 98.8466, 124.598, 154.431] Skewness: , [Float(undefined), 0., 0., -0.004130812756, 0.05662654229, 0.1306126752, 0.1971171820, 0.2526898503, 0.2984443704, 0.3362127221, 0.3676341554, 0.3940297190, 0.4164237327, 0.4356099477, 0.4521977837] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.384387498, 2.509960915, 2.589851891, 2.660301263, 2.726246869, 2.787295853, 2.842776573, 2.892632463, 2.937167402, 2.976843814, 3.012220247, 3.043750330] end of this data For the equivalence class of patterns, { {[1, 3, 2, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, {[1, 3, 2, 4], [3, 4, 1, 2], [4, 2, 1, 3]}, {[1, 3, 2, 4], [3, 4, 1, 2], [3, 2, 4, 1]}, {[2, 3, 1, 4], [2, 1, 4, 3], [4, 2, 3, 1]}, {[1, 4, 2, 3], [2, 1, 4, 3], [4, 2, 3, 1]}, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 2, 3, 1]}, {[2, 1, 4, 3], [3, 1, 2, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [2, 4, 3, 1], [3, 4, 1, 2]}} the member , {[1, 3, 2, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {4}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 2, 0, 0]}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 3, 4], {[0, 2, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 3, 4], {[0, 2, 0, 0, 0]}, {1}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 3, 4, 1], %1, {1}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[1, 3, 2, 4], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[3, 2, 4, 1], %1, {1}], [[3, 4, 2, 1], %1, {3}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {2}], [[2, 5, 3, 4, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[1, 4, 2, 3, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [ [1, 5, 2, 3, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[1, 5, 2, 4, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 5, 3, 4, 2], { [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[1, 2, 4, 3, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [ [1, 3, 5, 4, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 3, 5, 4, 1], { [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[1, 2, 5, 4, 3], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [ [3, 1, 5, 4, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 5, 4, 1], { [0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 1, 5, 4, 3], {[0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 2, 0, 0]}, {3}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {}], [[1, 2, 5, 3, 4], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[1, 2, 4, 3], {[0, 0, 0, 0, 1]}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0]}, {3}], [[2, 1, 4, 3], {[0, 0, 0, 0, 1], [0, 2, 0, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 2, 1], [0, 1, 0, 0]}, {}], [[3, 2, 1], {}, {2}], [[1, 2, 3], {}, {}], [[1, 2, 3, 4], {}, {2}]} %1 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 1]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+6*q^3+3*q^4+3*q^5+q^6, 1+2*q+5*q^2+10*q^3 +13*q^4+13*q^5+12*q^6+8*q^7+4*q^8+4*q^9+q^10, 1+2*q+5*q^2+10*q^3+17*q^4+28*q^5+ 29*q^6+33*q^7+27*q^8+28*q^9+23*q^10+15*q^11+10*q^12+5*q^13+5*q^14+q^15, 1+2*q+5 *q^2+10*q^3+17*q^4+32*q^5+47*q^6+59*q^7+70*q^8+73*q^9+72*q^10+73*q^11+64*q^12+ 54*q^13+48*q^14+42*q^15+28*q^16+18*q^17+12*q^18+6*q^19+6*q^20+q^21, 1+2*q+5*q^2 +10*q^3+17*q^4+32*q^5+51*q^6+80*q^7+101*q^8+134*q^9+152*q^10+164*q^11+177*q^12+ 164*q^13+169*q^14+163*q^15+151*q^16+126*q^17+119*q^18+93*q^19+83*q^20+67*q^21+ 50*q^22+33*q^23+21*q^24+14*q^25+7*q^26+7*q^27+q^28, 1+2*q+5*q^2+10*q^3+17*q^4+ 32*q^5+51*q^6+84*q^7+125*q^8+170*q^9+225*q^10+275*q^11+325*q^12+360*q^13+381*q^ 14+396*q^15+399*q^16+409*q^17+391*q^18+368*q^19+349*q^20+317*q^21+301*q^22+261* q^23+223*q^24+188*q^25+160*q^26+132*q^27+109*q^28+79*q^29+58*q^30+38*q^31+24*q^ 32+16*q^33+8*q^34+8*q^35+q^36, 1+2*q+5*q^2+10*q^3+17*q^4+32*q^5+51*q^6+84*q^7+ 129*q^8+197*q^9+266*q^10+360*q^11+459*q^12+556*q^13+674*q^14+741*q^15+820*q^16+ 866*q^17+921*q^18+929*q^19+942*q^20+934*q^21+911*q^22+903*q^23+844*q^24+787*q^ 25+743*q^26+687*q^27+616*q^28+565*q^29+487*q^30+432*q^31+359*q^32+308*q^33+244* q^34+213*q^35+167*q^36+127*q^37+91*q^38+66*q^39+43*q^40+27*q^41+18*q^42+9*q^43+ 9*q^44+q^45, 1+2*q+5*q^2+10*q^3+17*q^4+32*q^5+51*q^6+84*q^7+129*q^8+201*q^9+296 *q^10+406*q^11+556*q^12+713*q^13+908*q^14+1117*q^15+1309*q^16+1500*q^17+1677*q^ 18+1840*q^19+1953*q^20+2064*q^21+2118*q^22+2169*q^23+2224*q^24+2190*q^25+2157*q ^26+2114*q^27+2053*q^28+1974*q^29+1904*q^30+1776*q^31+1661*q^32+1546*q^33+1397* q^34+1278*q^35+1164*q^36+1035*q^37+897*q^38+794*q^39+664*q^40+567*q^41+473*q^42 +386*q^43+314*q^44+261*q^45+193*q^46+145*q^47+103*q^48+74*q^49+48*q^50+30*q^51+ 20*q^52+10*q^53+10*q^54+q^55, 1+2*q+5*q^2+10*q^3+17*q^4+32*q^5+51*q^6+84*q^7+ 129*q^8+201*q^9+300*q^10+439*q^11+607*q^12+822*q^13+1088*q^14+1389*q^15+1756*q^ 16+2114*q^17+2532*q^18+2913*q^19+3332*q^20+3680*q^21+4016*q^22+4327*q^23+4555*q ^24+4800*q^25+4927*q^26+5052*q^27+5088*q^28+5127*q^29+5113*q^30+5070*q^31+4991* q^32+4855*q^33+4689*q^34+4548*q^35+4334*q^36+4124*q^37+3893*q^38+3644*q^39+3393 *q^40+3132*q^41+2848*q^42+2582*q^43+2347*q^44+2088*q^45+1859*q^46+1625*q^47+ 1414*q^48+1205*q^49+1035*q^50+847*q^51+721*q^52+579*q^53+480*q^54+378*q^55+299* q^56+219*q^57+163*q^58+115*q^59+82*q^60+53*q^61+33*q^62+22*q^63+11*q^64+11*q^65 +q^66, 1+2*q+5*q^2+10*q^3+17*q^4+32*q^5+51*q^6+84*q^7+129*q^8+201*q^9+300*q^10+ 443*q^11+643*q^12+878*q^13+1209*q^14+1592*q^15+2066*q^16+2632*q^17+3257*q^18+ 3965*q^19+4727*q^20+5538*q^21+6322*q^22+7146*q^23+7920*q^24+8668*q^25+9374*q^26 +9974*q^27+10490*q^28+10953*q^29+11336*q^30+11579*q^31+11843*q^32+11932*q^33+ 12009*q^34+12013*q^35+11888*q^36+11767*q^37+11575*q^38+11358*q^39+11027*q^40+ 10691*q^41+10290*q^42+9811*q^43+9386*q^44+8887*q^45+8358*q^46+7888*q^47+7302*q^ 48+6755*q^49+6218*q^50+5685*q^51+5148*q^52+4651*q^53+4159*q^54+3698*q^55+3296*q ^56+2860*q^57+2482*q^58+2135*q^59+1830*q^60+1528*q^61+1284*q^62+1052*q^63+866*q ^64+703*q^65+567*q^66+431*q^67+337*q^68+245*q^69+181*q^70+127*q^71+90*q^72+58*q ^73+36*q^74+24*q^75+12*q^76+12*q^77+q^78, 1+2*q+5*q^2+10*q^3+17*q^4+32*q^5+51*q ^6+84*q^7+129*q^8+201*q^9+300*q^10+443*q^11+647*q^12+917*q^13+1270*q^14+1725*q^ 15+2292*q^16+2980*q^17+3846*q^18+4801*q^19+5960*q^20+7226*q^21+8661*q^22+10146* q^23+11743*q^24+13394*q^25+15032*q^26+16786*q^27+18317*q^28+19885*q^29+21311*q^ 30+22652*q^31+23811*q^32+24888*q^33+25753*q^34+26503*q^35+27161*q^36+27557*q^37 +27886*q^38+28103*q^39+28195*q^40+28121*q^41+28028*q^42+27704*q^43+27276*q^44+ 26841*q^45+26190*q^46+25583*q^47+24878*q^48+23960*q^49+23085*q^50+22089*q^51+ 21098*q^52+19988*q^53+18919*q^54+17772*q^55+16640*q^56+15571*q^57+14394*q^58+ 13248*q^59+12213*q^60+11095*q^61+10074*q^62+9098*q^63+8149*q^64+7277*q^65+6462* q^66+5683*q^67+4963*q^68+4334*q^69+3698*q^70+3167*q^71+2690*q^72+2264*q^73+1861 *q^74+1558*q^75+1247*q^76+1031*q^77+819*q^78+642*q^79+484*q^80+375*q^81+271*q^ 82+199*q^83+139*q^84+98*q^85+63*q^86+39*q^87+26*q^88+13*q^89+13*q^90+q^91, 1+2* q+5*q^2+10*q^3+17*q^4+32*q^5+51*q^6+84*q^7+129*q^8+201*q^9+300*q^10+443*q^11+ 647*q^12+921*q^13+1312*q^14+1791*q^15+2437*q^16+3229*q^17+4232*q^18+5461*q^19+ 6907*q^20+8640*q^21+10624*q^22+12932*q^23+15436*q^24+18218*q^25+21200*q^26+ 24389*q^27+27749*q^28+31168*q^29+34607*q^30+38055*q^31+41471*q^32+44704*q^33+ 47836*q^34+50708*q^35+53437*q^36+55851*q^37+58051*q^38+59948*q^39+61580*q^40+ 63071*q^41+64157*q^42+65092*q^43+65764*q^44+66161*q^45+66323*q^46+66311*q^47+ 66074*q^48+65676*q^49+65060*q^50+64167*q^51+63111*q^52+61919*q^53+60519*q^54+ 58909*q^55+57246*q^56+55382*q^57+53420*q^58+51371*q^59+49098*q^60+46882*q^61+ 44558*q^62+42150*q^63+39738*q^64+37325*q^65+34958*q^66+32537*q^67+30238*q^68+ 27904*q^69+25669*q^70+23526*q^71+21413*q^72+19406*q^73+17566*q^74+15729*q^75+ 14068*q^76+12515*q^77+11063*q^78+9733*q^79+8496*q^80+7377*q^81+6349*q^82+5456*q ^83+4609*q^84+3905*q^85+3266*q^86+2710*q^87+2222*q^88+1821*q^89+1464*q^90+1187* q^91+922*q^92+717*q^93+537*q^94+413*q^95+297*q^96+217*q^97+151*q^98+106*q^99+68 *q^100+42*q^101+28*q^102+14*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, q^6+2*q^5+5*q^4+6*q^3+3*q^2+3*q+1, q^10+2*q^9+5*q^8+ 10*q^7+13*q^6+13*q^5+12*q^4+8*q^3+4*q^2+4*q+1, q^15+2*q^14+5*q^13+10*q^12+17*q^ 11+28*q^10+29*q^9+33*q^8+27*q^7+28*q^6+23*q^5+15*q^4+10*q^3+5*q^2+5*q+1, q^21+2 *q^20+5*q^19+10*q^18+17*q^17+32*q^16+47*q^15+59*q^14+70*q^13+73*q^12+72*q^11+73 *q^10+64*q^9+54*q^8+48*q^7+42*q^6+28*q^5+18*q^4+12*q^3+6*q^2+6*q+1, 1+10*q^25+ 51*q^22+17*q^24+80*q^21+32*q^23+7*q+7*q^2+5*q^26+14*q^3+21*q^4+33*q^5+50*q^6+67 *q^7+83*q^8+93*q^9+119*q^10+126*q^11+151*q^12+163*q^13+169*q^14+164*q^15+177*q^ 16+q^28+164*q^17+152*q^18+134*q^19+101*q^20+2*q^27, 1+275*q^25+381*q^22+325*q^ 24+396*q^21+360*q^23+8*q+51*q^30+84*q^29+5*q^34+10*q^33+8*q^2+225*q^26+16*q^3+ 24*q^4+38*q^5+58*q^6+17*q^32+79*q^7+109*q^8+132*q^9+160*q^10+188*q^11+223*q^12+ 261*q^13+301*q^14+317*q^15+349*q^16+125*q^28+368*q^17+391*q^18+409*q^19+399*q^ 20+170*q^27+32*q^31+2*q^35+q^36, 1+942*q^25+10*q^42+903*q^22+934*q^24+844*q^21+ 911*q^23+9*q+2*q^44+741*q^30+q^45+129*q^37+820*q^29+51*q^39+17*q^41+360*q^34+84 *q^38+459*q^33+9*q^2+929*q^26+18*q^3+27*q^4+43*q^5+66*q^6+556*q^32+91*q^7+127*q ^8+167*q^9+213*q^10+244*q^11+308*q^12+359*q^13+432*q^14+487*q^15+565*q^16+5*q^ 43+866*q^28+616*q^17+687*q^18+743*q^19+787*q^20+921*q^27+32*q^40+674*q^31+266*q ^35+197*q^36, 1+1904*q^25+713*q^42+1546*q^22+1776*q^24+1397*q^21+1661*q^23+10*q +17*q^51+406*q^44+5*q^53+2190*q^30+10*q^52+q^55+2*q^54+296*q^45+1677*q^37+2157* q^29+201*q^46+1309*q^39+908*q^41+2064*q^34+32*q^50+1500*q^38+2118*q^33+10*q^2+ 1974*q^26+20*q^3+30*q^4+48*q^5+74*q^6+2169*q^32+103*q^7+145*q^8+193*q^9+261*q^ 10+314*q^11+386*q^12+473*q^13+567*q^14+664*q^15+794*q^16+556*q^43+51*q^49+2114* q^28+897*q^17+1035*q^18+1164*q^19+1278*q^20+2053*q^27+1117*q^40+2224*q^31+1953* q^35+84*q^48+1840*q^36+129*q^47, 1+3132*q^25+4555*q^42+2347*q^22+2848*q^24+2088 *q^21+2582*q^23+11*q+1389*q^51+4016*q^44+822*q^53+201*q^57+4334*q^30+1088*q^52+ 129*q^58+439*q^55+51*q^60+607*q^54+3680*q^45+5127*q^37+4124*q^29+3332*q^46+2*q^ 65+5052*q^39+32*q^61+4800*q^41+4991*q^34+1756*q^50+5*q^64+5088*q^38+4855*q^33+ 300*q^56+11*q^2+3393*q^26+22*q^3+33*q^4+53*q^5+82*q^6+4689*q^32+115*q^7+163*q^8 +219*q^9+17*q^62+299*q^10+378*q^11+480*q^12+579*q^13+721*q^14+847*q^15+1035*q^ 16+q^66+4327*q^43+2114*q^49+3893*q^28+1205*q^17+1414*q^18+1625*q^19+1859*q^20+ 3644*q^27+4927*q^40+10*q^63+4548*q^31+5070*q^35+2532*q^48+84*q^59+5113*q^36+ 2913*q^47, 1+4651*q^25+11888*q^42+3296*q^22+4159*q^24+2860*q^21+3698*q^23+12*q+ 9974*q^51+12009*q^44+8668*q^53+5538*q^57+7302*q^30+9374*q^52+4727*q^58+7146*q^ 55+3257*q^60+7920*q^54+11932*q^45+84*q^71+17*q^74+10691*q^37+443*q^67+6755*q^29 +11843*q^46+878*q^65+201*q^69+11358*q^39+2632*q^61+11767*q^41+9386*q^34+10490*q ^50+1209*q^64+11027*q^38+8887*q^33+6322*q^56+12*q^2+300*q^68+5148*q^26+24*q^3+ 36*q^4+58*q^5+90*q^6+8358*q^32+127*q^7+181*q^8+245*q^9+2066*q^62+337*q^10+431*q ^11+567*q^12+5*q^76+703*q^13+866*q^14+1052*q^15+1284*q^16+643*q^66+129*q^70+2*q ^77+q^78+12013*q^43+10953*q^49+6218*q^28+1528*q^17+1830*q^18+2135*q^19+2482*q^ 20+5685*q^27+51*q^72+11575*q^40+32*q^73+1592*q^63+7888*q^31+10*q^75+9811*q^35+ 11336*q^48+3965*q^59+10290*q^36+11579*q^47, 1+6462*q^25+23960*q^42+4334*q^22+2* q^90+5683*q^24+3698*q^21+4963*q^23+13*q+28195*q^51+25583*q^44+27886*q^53+25753* q^57+11095*q^30+28103*q^52+24888*q^58+27161*q^55+22652*q^60+q^91+27557*q^54+ 26190*q^45+5960*q^71+2980*q^74+18919*q^37+11743*q^67+10074*q^29+26841*q^46+ 15032*q^65+8661*q^69+21098*q^39+21311*q^61+23085*q^41+15571*q^34+647*q^79+28121 *q^50+16786*q^64+19988*q^38+14394*q^33+26503*q^56+13*q^2+10146*q^68+7277*q^26+ 26*q^3+39*q^4+63*q^5+98*q^6+13248*q^32+139*q^7+199*q^8+271*q^9+19885*q^62+375*q ^10+484*q^11+642*q^12+1725*q^76+819*q^13+1031*q^14+1247*q^15+1558*q^16+13394*q^ 66+7226*q^70+1270*q^77+917*q^78+24878*q^43+28028*q^49+9098*q^28+1861*q^17+2264* q^18+2690*q^19+3167*q^20+8149*q^27+4801*q^72+22089*q^40+3846*q^73+18317*q^63+ 12213*q^31+2292*q^75+16640*q^35+443*q^80+27704*q^48+129*q^83+300*q^81+51*q^85+ 32*q^86+17*q^87+10*q^88+5*q^89+84*q^84+201*q^82+23811*q^59+17772*q^36+27276*q^ 47, 1+8496*q^25+42150*q^42+5456*q^22+1791*q^90+7377*q^24+4609*q^21+6349*q^23+14 *q+129*q^97+60519*q^51+46882*q^44+63111*q^53+66074*q^57+15729*q^30+443*q^94+ 61919*q^52+66311*q^58+65060*q^55+66161*q^60+1312*q^91+64167*q^54+49098*q^45+ 47836*q^71+38055*q^74+201*q^96+84*q^98+30238*q^37+58051*q^67+14068*q^29+51371*q ^46+647*q^93+51*q^99+61580*q^65+53437*q^69+34958*q^39+65764*q^61+39738*q^41+ 23526*q^34+21200*q^79+58909*q^50+63071*q^64+32*q^100+32537*q^38+21413*q^33+ 65676*q^56+14*q^2+55851*q^68+9733*q^26+28*q^3+42*q^4+68*q^5+106*q^6+19406*q^32+ 151*q^7+217*q^8+297*q^9+300*q^95+65092*q^62+17*q^101+413*q^10+537*q^11+717*q^12 +31168*q^76+922*q^13+1187*q^14+1464*q^15+1821*q^16+59948*q^66+50708*q^70+27749* q^77+24389*q^78+44558*q^43+10*q^102+57246*q^49+12515*q^28+2222*q^17+2710*q^18+ 3266*q^19+3905*q^20+11063*q^27+44704*q^72+37325*q^40+41471*q^73+5*q^103+64157*q ^63+17566*q^31+34607*q^75+25669*q^35+2*q^104+18218*q^80+55382*q^48+q^105+10624* q^83+15436*q^81+6907*q^85+5461*q^86+4232*q^87+3229*q^88+2437*q^89+8640*q^84+ 12932*q^82+66323*q^59+27904*q^36+921*q^92+53420*q^47] The number of permutations avoiding, {[1, 3, 2, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 2, 6, 21, 73, 239, 740, 2194, 6298, 17653, 48621, 132199, 356040, 952154, 2533014] The number of EVEN permutations avoiding, {[1, 3, 2, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [1, 1, 3, 10, 36, 117, 370, 1105, 3153, 8817, 24302, 66110, 178032, 476068, 1266495] The number of ODD permutations avoiding, {[1, 3, 2, 4], [3, 4, 1, 2], [4, 1, 3, 2]}, is given by [0, 1, 3, 11, 37, 122, 370, 1089, 3145, 8836, 24319, 66089, 178008, 476086, 1266519] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 10, 36, 122, 370, 1105, 3153, 8836, 24319, 66110, 178032, 476086, 1266519] ODD:, [0, 1, 3, 11, 37, 117, 370, 1089, 3145, 8817, 24302, 66089, 178008, 476068, 1266495] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.481171548, 10.40810811, 13.75888788, 17.52508733, 21.70747182, 26.31264269, 31.35068344, 36.83347377, 42.77349462, 49.18302189] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.480025740, 2.106740650, 2.870762493, 3.765912430, 4.775423812, 5.885006546, 7.084975099, 8.369540461, 9.735657087, 11.18194587, 12.70781887, 14.31284393] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.19048, 4.43836, 8.24128, 14.1821, 22.8047, 34.6333, 50.1969, 70.0492, 94.7830, 125.036, 161.489, 204.858] Skewness: , [Float(undefined), 0., 0., 0.08813469434, 0.1230632343, 0.1356664427, 0.1564356169, 0.1815794807, 0.2043895305, 0.2216768045, 0.2327080870, 0.2379009757, 0.2380873931, 0.2341725107, 0.2269892084] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.481095718, 2.661972429, 2.668425788, 2.645303108, 2.638378125, 2.646303776, 2.660155363, 2.673465041, 2.682957563, 2.687667680, 2.687910354, 2.684598291] end of this data For the equivalence class of patterns, { {[2, 3, 4, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, {[1, 4, 3, 2], [3, 4, 2, 1], [3, 2, 4, 1]}, {[2, 4, 3, 1], [3, 4, 2, 1], [3, 2, 1, 4]}, {[1, 4, 2, 3], [1, 2, 4, 3], [2, 3, 4, 1]}, {[1, 4, 3, 2], [4, 3, 1, 2], [4, 2, 1, 3]}, {[2, 3, 1, 4], [2, 1, 3, 4], [4, 1, 2, 3]}, {[3, 2, 1, 4], [4, 1, 3, 2], [4, 3, 1, 2]}, {[1, 3, 4, 2], [1, 2, 4, 3], [4, 1, 2, 3]}} the member , {[2, 3, 4, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}], [[1, 4, 2, 3], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[3, 4, 1, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 5, 2, 1, 4], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {5}], [ [3, 4, 2, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[4, 5, 2, 1, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {5}], [ [4, 5, 3, 1, 2], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 1, 2], {[0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 1]}, {}], [ [4, 5, 3, 2, 1], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 1]}, {3}], [[3, 4, 2, 1], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {2}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[1, 3, 2], {}, {}], [[1, 4, 3, 2], {}, {2}], [[3, 2, 1], {}, {1}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, 1+2*q+4*q^2+6*q^3+ 10*q^4+12*q^5+14*q^6+13*q^7+9*q^8+4*q^9+q^10, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+ 20*q^6+27*q^7+31*q^8+38*q^9+39*q^10+36*q^11+26*q^12+14*q^13+5*q^14+q^15, 1+2*q+ 4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+30*q^7+39*q^8+53*q^9+64*q^10+79*q^11+89*q^12+ 104*q^13+113*q^14+114*q^15+101*q^16+76*q^17+45*q^18+20*q^19+6*q^20+q^21, 1+2*q+ 4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+31*q^7+41*q^8+57*q^9+72*q^10+94*q^11+117*q^12+ 147*q^13+178*q^14+207*q^15+243*q^16+271*q^17+306*q^18+330*q^19+346*q^20+333*q^ 21+292*q^22+222*q^23+141*q^24+71*q^25+27*q^26+7*q^27+q^28, 1+2*q+4*q^2+6*q^3+10 *q^4+15*q^5+21*q^6+31*q^7+42*q^8+59*q^9+76*q^10+100*q^11+127*q^12+164*q^13+206* q^14+252*q^15+309*q^16+368*q^17+444*q^18+519*q^19+605*q^20+686*q^21+777*q^22+ 856*q^23+945*q^24+1010*q^25+1056*q^26+1055*q^27+991*q^28+853*q^29+656*q^30+434* q^31+239*q^32+105*q^33+35*q^34+8*q^35+q^36, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+21* q^6+31*q^7+42*q^8+60*q^9+78*q^10+104*q^11+133*q^12+174*q^13+221*q^14+274*q^15+ 343*q^16+418*q^17+517*q^18+621*q^19+749*q^20+882*q^21+1042*q^22+1212*q^23+1410* q^24+1618*q^25+1843*q^26+2082*q^27+2316*q^28+2560*q^29+2786*q^30+3016*q^31+3196 *q^32+3324*q^33+3346*q^34+3245*q^35+2970*q^36+2527*q^37+1950*q^38+1330*q^39+778 *q^40+379*q^41+148*q^42+44*q^43+9*q^44+q^45, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+21 *q^6+31*q^7+42*q^8+60*q^9+79*q^10+106*q^11+137*q^12+180*q^13+231*q^14+289*q^15+ 364*q^16+450*q^17+562*q^18+687*q^19+837*q^20+1003*q^21+1204*q^22+1431*q^23+1702 *q^24+2002*q^25+2343*q^26+2723*q^27+3136*q^28+3597*q^29+4097*q^30+4666*q^31+ 5259*q^32+5904*q^33+6564*q^34+7257*q^35+7934*q^36+8625*q^37+9263*q^38+9879*q^39 +10371*q^40+10717*q^41+10810*q^42+10600*q^43+10004*q^44+8982*q^45+7552*q^46+ 5842*q^47+4066*q^48+2488*q^49+1305*q^50+571*q^51+201*q^52+54*q^53+10*q^54+q^55, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+31*q^7+42*q^8+60*q^9+79*q^10+107*q^11+ 139*q^12+184*q^13+237*q^14+299*q^15+379*q^16+471*q^17+593*q^18+730*q^19+900*q^ 20+1088*q^21+1320*q^22+1584*q^23+1907*q^24+2269*q^25+2687*q^26+3168*q^27+3707*q ^28+4331*q^29+5024*q^30+5829*q^31+6702*q^32+7698*q^33+8776*q^34+9975*q^35+11257 *q^36+12680*q^37+14203*q^38+15880*q^39+17649*q^40+19532*q^41+21491*q^42+23497*q ^43+25526*q^44+27527*q^45+29491*q^46+31318*q^47+32994*q^48+34328*q^49+35239*q^ 50+35502*q^51+34985*q^52+33497*q^53+30960*q^54+27325*q^55+22756*q^56+17608*q^57 +12440*q^58+7868*q^59+4365*q^60+2077*q^61+826*q^62+265*q^63+65*q^64+11*q^65+q^ 66, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+31*q^7+42*q^8+60*q^9+79*q^10+107*q^ 11+140*q^12+186*q^13+241*q^14+305*q^15+389*q^16+486*q^17+614*q^18+761*q^19+942* q^20+1149*q^21+1402*q^22+1697*q^23+2056*q^24+2469*q^25+2949*q^26+3503*q^27+4138 *q^28+4876*q^29+5723*q^30+6708*q^31+7807*q^32+9063*q^33+10469*q^34+12059*q^35+ 13821*q^36+15801*q^37+17997*q^38+20467*q^39+23171*q^40+26153*q^41+29398*q^42+ 32933*q^43+36759*q^44+40886*q^45+45345*q^46+50115*q^47+55250*q^48+60650*q^49+ 66346*q^50+72220*q^51+78273*q^52+84351*q^53+90445*q^54+96342*q^55+102025*q^56+ 107222*q^57+111829*q^58+115420*q^59+117737*q^60+118295*q^61+116748*q^62+112662* q^63+105797*q^64+96040*q^65+83582*q^66+69004*q^67+53376*q^68+38117*q^69+24727*q ^70+14320*q^71+7269*q^72+3168*q^73+1156*q^74+341*q^75+77*q^76+12*q^77+q^78, 1+2 *q+4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+31*q^7+42*q^8+60*q^9+79*q^10+107*q^11+140*q ^12+187*q^13+243*q^14+309*q^15+395*q^16+496*q^17+629*q^18+782*q^19+973*q^20+ 1191*q^21+1462*q^22+1777*q^23+2166*q^24+2615*q^25+3145*q^26+3760*q^27+4468*q^28 +5301*q^29+6261*q^30+7395*q^31+8671*q^32+10148*q^33+11813*q^34+13723*q^35+15867 *q^36+18321*q^37+21067*q^38+24192*q^39+27669*q^40+31569*q^41+35907*q^42+40722*q ^43+46047*q^44+51905*q^45+58389*q^46+65498*q^47+73346*q^48+81865*q^49+91179*q^ 50+101230*q^51+112136*q^52+123794*q^53+136362*q^54+149729*q^55+164054*q^56+ 179226*q^57+195344*q^58+212194*q^59+229794*q^60+247917*q^61+266432*q^62+285108* q^63+303724*q^64+321997*q^65+339535*q^66+355997*q^67+370779*q^68+383393*q^69+ 392888*q^70+398489*q^71+399068*q^72+393740*q^73+381488*q^74+361679*q^75+333944* q^76+298624*q^77+256748*q^78+210391*q^79+162585*q^80+117047*q^81+77429*q^82+ 46381*q^83+24768*q^84+11594*q^85+4665*q^86+1574*q^87+430*q^88+90*q^89+13*q^90+q ^91, 1+2*q+4*q^2+6*q^3+10*q^4+15*q^5+21*q^6+31*q^7+42*q^8+60*q^9+79*q^10+107*q^ 11+140*q^12+187*q^13+244*q^14+311*q^15+399*q^16+502*q^17+639*q^18+797*q^19+994* q^20+1222*q^21+1504*q^22+1837*q^23+2245*q^24+2723*q^25+3288*q^26+3953*q^27+4721 *q^28+5626*q^29+6681*q^30+7927*q^31+9352*q^32+11003*q^33+12888*q^34+15054*q^35+ 17519*q^36+20354*q^37+23575*q^38+27260*q^39+31401*q^40+36084*q^41+41338*q^42+ 47249*q^43+53864*q^44+61259*q^45+69508*q^46+78688*q^47+88926*q^48+100243*q^49+ 112759*q^50+126523*q^51+141660*q^52+158222*q^53+176376*q^54+196160*q^55+217772* q^56+241255*q^57+266794*q^58+294370*q^59+324186*q^60+356210*q^61+390561*q^62+ 427246*q^63+466386*q^64+507989*q^65+552074*q^66+598666*q^67+647688*q^68+699133* q^69+752656*q^70+808095*q^71+864912*q^72+922779*q^73+980951*q^74+1038975*q^75+ 1095851*q^76+1150916*q^77+1202830*q^78+1250543*q^79+1292259*q^80+1326450*q^81+ 1350706*q^82+1362904*q^83+1360298*q^84+1340554*q^85+1301267*q^86+1240793*q^87+ 1158181*q^88+1053949*q^89+930297*q^90+791572*q^91+644373*q^92+497368*q^93+ 360241*q^94+242014*q^95+148919*q^96+82820*q^97+41039*q^98+17834*q^99+6669*q^100 +2094*q^101+533*q^102+104*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+2*q^5+q^6, q^10+2*q^9+4*q^8+6 *q^7+10*q^6+12*q^5+14*q^4+13*q^3+9*q^2+4*q+1, q^15+2*q^14+4*q^13+6*q^12+10*q^11 +15*q^10+20*q^9+27*q^8+31*q^7+38*q^6+39*q^5+36*q^4+26*q^3+14*q^2+5*q+1, q^21+2* q^20+4*q^19+6*q^18+10*q^17+15*q^16+21*q^15+30*q^14+39*q^13+53*q^12+64*q^11+79*q ^10+89*q^9+104*q^8+113*q^7+114*q^6+101*q^5+76*q^4+45*q^3+20*q^2+6*q+1, 1+6*q^25 +21*q^22+10*q^24+31*q^21+15*q^23+7*q+27*q^2+4*q^26+71*q^3+141*q^4+222*q^5+292*q ^6+333*q^7+346*q^8+330*q^9+306*q^10+271*q^11+243*q^12+207*q^13+178*q^14+147*q^ 15+117*q^16+q^28+94*q^17+72*q^18+57*q^19+41*q^20+2*q^27, 1+100*q^25+206*q^22+ 127*q^24+252*q^21+164*q^23+8*q+21*q^30+31*q^29+4*q^34+6*q^33+35*q^2+76*q^26+105 *q^3+239*q^4+434*q^5+656*q^6+10*q^32+853*q^7+991*q^8+1055*q^9+1056*q^10+1010*q^ 11+945*q^12+856*q^13+777*q^14+686*q^15+605*q^16+42*q^28+519*q^17+444*q^18+368*q ^19+309*q^20+59*q^27+15*q^31+2*q^35+q^36, 1+749*q^25+6*q^42+1212*q^22+882*q^24+ 1410*q^21+1042*q^23+9*q+2*q^44+274*q^30+q^45+42*q^37+343*q^29+21*q^39+10*q^41+ 104*q^34+31*q^38+133*q^33+44*q^2+621*q^26+148*q^3+379*q^4+778*q^5+1330*q^6+174* q^32+1950*q^7+2527*q^8+2970*q^9+3245*q^10+3346*q^11+3324*q^12+3196*q^13+3016*q^ 14+2786*q^15+2560*q^16+4*q^43+418*q^28+2316*q^17+2082*q^18+1843*q^19+1618*q^20+ 517*q^27+15*q^40+221*q^31+78*q^35+60*q^36, 1+4097*q^25+180*q^42+5904*q^22+4666* q^24+6564*q^21+5259*q^23+10*q+10*q^51+106*q^44+4*q^53+2002*q^30+6*q^52+q^55+2*q ^54+79*q^45+562*q^37+2343*q^29+60*q^46+364*q^39+231*q^41+1003*q^34+15*q^50+450* q^38+1204*q^33+54*q^2+3597*q^26+201*q^3+571*q^4+1305*q^5+2488*q^6+1431*q^32+ 4066*q^7+5842*q^8+7552*q^9+8982*q^10+10004*q^11+10600*q^12+10810*q^13+10717*q^ 14+10371*q^15+9879*q^16+137*q^43+21*q^49+2723*q^28+9263*q^17+8625*q^18+7934*q^ 19+7257*q^20+3136*q^27+289*q^40+1702*q^31+837*q^35+31*q^48+687*q^36+42*q^47, 1+ 19532*q^25+1907*q^42+25526*q^22+21491*q^24+27527*q^21+23497*q^23+11*q+299*q^51+ 1320*q^44+184*q^53+60*q^57+11257*q^30+237*q^52+42*q^58+107*q^55+21*q^60+139*q^ 54+1088*q^45+4331*q^37+12680*q^29+900*q^46+2*q^65+3168*q^39+15*q^61+2269*q^41+ 6702*q^34+379*q^50+4*q^64+3707*q^38+7698*q^33+79*q^56+65*q^2+17649*q^26+265*q^3 +826*q^4+2077*q^5+4365*q^6+8776*q^32+7868*q^7+12440*q^8+17608*q^9+10*q^62+22756 *q^10+27325*q^11+30960*q^12+33497*q^13+34985*q^14+35502*q^15+35239*q^16+q^66+ 1584*q^43+471*q^49+14203*q^28+34328*q^17+32994*q^18+31318*q^19+29491*q^20+15880 *q^27+2687*q^40+6*q^63+9975*q^31+5829*q^35+593*q^48+31*q^59+5024*q^36+730*q^47, 1+84351*q^25+13821*q^42+102025*q^22+90445*q^24+107222*q^21+96342*q^23+12*q+3503 *q^51+10469*q^44+2469*q^53+1149*q^57+55250*q^30+2949*q^52+942*q^58+1697*q^55+ 614*q^60+2056*q^54+9063*q^45+31*q^71+10*q^74+26153*q^37+107*q^67+60650*q^29+ 7807*q^46+186*q^65+60*q^69+20467*q^39+486*q^61+15801*q^41+36759*q^34+4138*q^50+ 241*q^64+23171*q^38+40886*q^33+1402*q^56+77*q^2+79*q^68+78273*q^26+341*q^3+1156 *q^4+3168*q^5+7269*q^6+45345*q^32+14320*q^7+24727*q^8+38117*q^9+389*q^62+53376* q^10+69004*q^11+83582*q^12+4*q^76+96040*q^13+105797*q^14+112662*q^15+116748*q^ 16+140*q^66+42*q^70+2*q^77+q^78+12059*q^43+4876*q^49+66346*q^28+118295*q^17+ 117737*q^18+115420*q^19+111829*q^20+72220*q^27+21*q^72+17997*q^40+15*q^73+305*q ^63+50115*q^31+6*q^75+32933*q^35+5723*q^48+761*q^59+29398*q^36+6708*q^47, 1+ 339535*q^25+81865*q^42+383393*q^22+2*q^90+355997*q^24+392888*q^21+370779*q^23+ 13*q+27669*q^51+65498*q^44+21067*q^53+11813*q^57+247917*q^30+24192*q^52+10148*q ^58+15867*q^55+7395*q^60+q^91+18321*q^54+58389*q^45+973*q^71+496*q^74+136362*q^ 37+2166*q^67+266432*q^29+51905*q^46+3145*q^65+1462*q^69+112136*q^39+6261*q^61+ 91179*q^41+179226*q^34+140*q^79+31569*q^50+3760*q^64+123794*q^38+195344*q^33+ 13723*q^56+90*q^2+1777*q^68+321997*q^26+430*q^3+1574*q^4+4665*q^5+11594*q^6+ 212194*q^32+24768*q^7+46381*q^8+77429*q^9+5301*q^62+117047*q^10+162585*q^11+ 210391*q^12+309*q^76+256748*q^13+298624*q^14+333944*q^15+361679*q^16+2615*q^66+ 1191*q^70+243*q^77+187*q^78+73346*q^43+35907*q^49+285108*q^28+381488*q^17+ 393740*q^18+399068*q^19+398489*q^20+303724*q^27+782*q^72+101230*q^40+629*q^73+ 4468*q^63+229794*q^31+395*q^75+164054*q^35+107*q^80+40722*q^48+42*q^83+79*q^81+ 21*q^85+15*q^86+10*q^87+6*q^88+4*q^89+31*q^84+60*q^82+8671*q^59+149729*q^36+ 46047*q^47, 1+1292259*q^25+427246*q^42+1362904*q^22+311*q^90+1326450*q^24+ 1360298*q^21+1350706*q^23+14*q+42*q^97+176376*q^51+356210*q^44+141660*q^53+ 88926*q^57+1038975*q^30+107*q^94+158222*q^52+78688*q^58+112759*q^55+61259*q^60+ 244*q^91+126523*q^54+324186*q^45+12888*q^71+7927*q^74+60*q^96+31*q^98+647688*q^ 37+23575*q^67+1095851*q^29+294370*q^46+140*q^93+21*q^99+31401*q^65+17519*q^69+ 552074*q^39+53864*q^61+466386*q^41+808095*q^34+3288*q^79+196160*q^50+36084*q^64 +15*q^100+598666*q^38+864912*q^33+100243*q^56+104*q^2+20354*q^68+1250543*q^26+ 533*q^3+2094*q^4+6669*q^5+17834*q^6+922779*q^32+41039*q^7+82820*q^8+148919*q^9+ 79*q^95+47249*q^62+10*q^101+242014*q^10+360241*q^11+497368*q^12+5626*q^76+ 644373*q^13+791572*q^14+930297*q^15+1053949*q^16+27260*q^66+15054*q^70+4721*q^ 77+3953*q^78+390561*q^43+6*q^102+217772*q^49+1150916*q^28+1158181*q^17+1240793* q^18+1301267*q^19+1340554*q^20+1202830*q^27+11003*q^72+507989*q^40+9352*q^73+4* q^103+41338*q^63+980951*q^31+6681*q^75+752656*q^35+2*q^104+2723*q^80+241255*q^ 48+q^105+1504*q^83+2245*q^81+994*q^85+797*q^86+639*q^87+502*q^88+399*q^89+1222* q^84+1837*q^82+69508*q^59+699133*q^36+187*q^92+266794*q^47] The number of permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, is given by [1, 2, 6, 21, 76, 275, 993, 3593, 13068, 47838, 176277, 653538, 2436158, 9124352, 34315674] The number of EVEN permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, is given by [1, 1, 3, 11, 39, 136, 493, 1800, 6545, 23913, 88109, 326776, 1218156, 4562180, 17157635] The number of ODD permutations avoiding, {[2, 3, 4, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, is given by [0, 1, 3, 10, 37, 139, 500, 1793, 6523, 23925, 88168, 326762, 1218002, 4562172, 17158039] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 39, 139, 500, 1800, 6545, 23925, 88168, 326776, 1218156, 4562172, 17158039] ODD:, [0, 1, 3, 10, 37, 136, 493, 1793, 6523, 23913, 88109, 326762, 1218002, 4562180, 17157635] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.142857143, 5.539473684, 8.752727273, 12.81772407, 17.75313109, 23.56772268, 30.26537481, 37.84836365, 46.31899293, 55.68009464, 65.93498410, 77.08724413] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.489190073, 2.117848894, 2.838171918, 3.630464151, 4.475931467, 5.361931658, 6.281693864, 7.232276943, 8.212632631, 9.222358799, 10.26108832, 11.32827816] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.21769, 4.48528, 8.05522, 13.1803, 20.0340, 28.7503, 39.4597, 52.3058, 67.4473, 85.0519, 105.290, 128.330] Skewness: , [Float(undefined), 0., 0., -0.1588977478, -0.3333038179, -0.4778957432, -0.5903742875, -0.6769643603, -0.7432157821, -0.7935004833, -0.8314733764, -0.8601480564, -0.8819446792, -0.8987122271, -0.9118226481] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.435936804, 2.671503304, 2.871950269, 3.063608428, 3.244409631, 3.407523593, 3.548451493, 3.666366478, 3.763108298, 3.841729283, 3.905706486, 3.958165475] end of this data For the equivalence class of patterns, { {[3, 4, 1, 2], [3, 2, 4, 1], [4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 1, 4, 3], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 4, 1, 2], [4, 2, 1, 3]}, {[1, 4, 2, 3], [2, 3, 1, 4], [2, 1, 4, 3]}} the member , {[1, 3, 4, 2], [2, 1, 4, 3], [3, 1, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1]}, {2}], [[1, 3, 2], {[0, 0, 1, 1]}, {}], [[2, 1, 3], {[0, 1, 0, 1], [0, 0, 1, 0]}, {3}], [[2, 3, 1], {[0, 1, 2, 0], [0, 1, 0, 2], [0, 0, 1, 1]}, {2}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 2]}, {}], [[3, 1, 2], {[0, 0, 0, 1]}, {1}], [[1, 4, 2, 3], {[0, 0, 0, 0, 1], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], { [0, 0, 1, 1, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [0, 1, 2, 0, 0], [0, 1, 0, 2, 0], [0, 1, 0, 0, 2]}, {2}], [[3, 2, 1], {[0, 1, 2, 0], [0, 1, 0, 2], [0, 0, 1, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+2*q^2+6*q^3+5*q^4+3*q^5+q^6, 1+4*q+2*q^2+5*q^3+ 8*q^4+11*q^5+14*q^6+15*q^7+9*q^8+4*q^9+q^10, 1+5*q+2*q^2+6*q^3+6*q^4+13*q^5+15* q^6+15*q^7+28*q^8+34*q^9+43*q^10+40*q^11+29*q^12+14*q^13+5*q^14+q^15, 1+6*q+2*q ^2+7*q^3+6*q^4+10*q^5+20*q^6+21*q^7+20*q^8+29*q^9+43*q^10+51*q^11+65*q^12+88*q^ 13+109*q^14+125*q^15+118*q^16+86*q^17+49*q^18+20*q^19+6*q^20+q^21, 1+7*q+2*q^2+ 8*q^3+6*q^4+10*q^5+17*q^6+25*q^7+30*q^8+27*q^9+41*q^10+43*q^11+71*q^12+77*q^13+ 86*q^14+104*q^15+156*q^16+192*q^17+230*q^18+292*q^19+342*q^20+372*q^21+347*q^22 +263*q^23+159*q^24+76*q^25+27*q^26+7*q^27+q^28, 1+8*q+2*q^2+9*q^3+6*q^4+10*q^5+ 18*q^6+20*q^7+34*q^8+39*q^9+41*q^10+49*q^11+60*q^12+75*q^13+107*q^14+115*q^15+ 129*q^16+140*q^17+180*q^18+240*q^19+295*q^20+349*q^21+417*q^22+555*q^23+687*q^ 24+798*q^25+956*q^26+1073*q^27+1124*q^28+1032*q^29+802*q^30+513*q^31+267*q^32+ 111*q^33+35*q^34+8*q^35+q^36, 1+9*q+2*q^2+10*q^3+6*q^4+10*q^5+19*q^6+20*q^7+28* q^8+43*q^9+56*q^10+47*q^11+69*q^12+73*q^13+99*q^14+118*q^15+155*q^16+167*q^17+ 194*q^18+209*q^19+237*q^20+303*q^21+378*q^22+436*q^23+485*q^24+582*q^25+707*q^ 26+938*q^27+1133*q^28+1345*q^29+1596*q^30+1972*q^31+2363*q^32+2731*q^33+3101*q^ 34+3377*q^35+3420*q^36+3111*q^37+2448*q^38+1636*q^39+912*q^40+419*q^41+155*q^42 +44*q^43+9*q^44+q^45, 1+10*q+2*q^2+11*q^3+6*q^4+10*q^5+20*q^6+20*q^7+28*q^8+36* q^9+61*q^10+63*q^11+66*q^12+83*q^13+98*q^14+121*q^15+153*q^16+167*q^17+223*q^18 +249*q^19+277*q^20+299*q^21+348*q^22+406*q^23+485*q^24+591*q^25+654*q^26+729*q^ 27+808*q^28+948*q^29+1153*q^30+1471*q^31+1712*q^32+1980*q^33+2373*q^34+2870*q^ 35+3561*q^36+4251*q^37+4994*q^38+5813*q^39+6924*q^40+8089*q^41+9117*q^42+10069* q^43+10612*q^44+10504*q^45+9445*q^46+7514*q^47+5176*q^48+3049*q^49+1514*q^50+ 625*q^51+209*q^52+54*q^53+10*q^54+q^55, 1+11*q+2*q^2+12*q^3+6*q^4+10*q^5+21*q^6 +20*q^7+28*q^8+36*q^9+54*q^10+67*q^11+84*q^12+79*q^13+109*q^14+120*q^15+155*q^ 16+179*q^17+217*q^18+249*q^19+315*q^20+354*q^21+388*q^22+429*q^23+491*q^24+577* q^25+652*q^26+764*q^27+881*q^28+1019*q^29+1095*q^30+1198*q^31+1341*q^32+1601*q^ 33+1844*q^34+2217*q^35+2553*q^36+2875*q^37+3315*q^38+3906*q^39+4728*q^40+5830*q ^41+6871*q^42+7925*q^43+9300*q^44+11054*q^45+13212*q^46+15549*q^47+18051*q^48+ 20766*q^49+24017*q^50+27353*q^51+30233*q^52+32486*q^53+33455*q^54+32424*q^55+ 28905*q^56+23137*q^57+16336*q^58+10034*q^59+5306*q^60+2384*q^61+896*q^62+274*q^ 63+65*q^64+11*q^65+q^66, 1+12*q+2*q^2+13*q^3+6*q^4+10*q^5+22*q^6+20*q^7+28*q^8+ 36*q^9+55*q^10+58*q^11+88*q^12+99*q^13+104*q^14+133*q^15+152*q^16+179*q^17+231* q^18+257*q^19+309*q^20+360*q^21+441*q^22+479*q^23+547*q^24+604*q^25+684*q^26+ 803*q^27+903*q^28+1013*q^29+1160*q^30+1365*q^31+1502*q^32+1658*q^33+1779*q^34+ 1987*q^35+2277*q^36+2612*q^37+2933*q^38+3407*q^39+3833*q^40+4236*q^41+4730*q^42 +5410*q^43+6344*q^44+7480*q^45+8881*q^46+10302*q^47+11921*q^48+13664*q^49+16028 *q^50+19022*q^51+22752*q^52+26519*q^53+30415*q^54+35255*q^55+41172*q^56+48393*q ^57+56025*q^58+64284*q^59+72942*q^60+82429*q^61+91802*q^62+99514*q^63+104634*q^ 64+105542*q^65+100628*q^66+88965*q^67+71536*q^68+51460*q^69+32708*q^70+18179*q^ 71+8749*q^72+3599*q^73+1244*q^74+351*q^75+77*q^76+12*q^77+q^78, 1+13*q+2*q^2+14 *q^3+6*q^4+10*q^5+23*q^6+20*q^7+28*q^8+36*q^9+56*q^10+58*q^11+78*q^12+103*q^13+ 126*q^14+128*q^15+165*q^16+175*q^17+229*q^18+271*q^19+317*q^20+371*q^21+439*q^ 22+498*q^23+593*q^24+670*q^25+746*q^26+837*q^27+950*q^28+1085*q^29+1224*q^30+ 1404*q^31+1534*q^32+1777*q^33+2012*q^34+2240*q^35+2432*q^36+2672*q^37+2936*q^38 +3318*q^39+3700*q^40+4149*q^41+4636*q^42+5301*q^43+5844*q^44+6348*q^45+7030*q^ 46+7763*q^47+8875*q^48+10208*q^49+11642*q^50+13367*q^51+15231*q^52+17200*q^53+ 19510*q^54+22493*q^55+26264*q^56+30710*q^57+35970*q^58+41681*q^59+47738*q^60+ 54709*q^61+63178*q^62+73608*q^63+86362*q^64+99746*q^65+114123*q^66+131061*q^67+ 151339*q^68+174740*q^69+199796*q^70+226011*q^71+252873*q^72+280451*q^73+306213* q^74+325948*q^75+336430*q^76+333455*q^77+313517*q^78+275231*q^79+221951*q^80+ 162049*q^81+105875*q^82+61315*q^83+31197*q^84+13810*q^85+5249*q^86+1682*q^87+ 441*q^88+90*q^89+13*q^90+q^91, 1+14*q+2*q^2+15*q^3+6*q^4+10*q^5+24*q^6+20*q^7+ 28*q^8+36*q^9+57*q^10+58*q^11+78*q^12+92*q^13+130*q^14+153*q^15+158*q^16+189*q^ 17+224*q^18+267*q^19+331*q^20+378*q^21+449*q^22+513*q^23+607*q^24+677*q^25+807* q^26+909*q^27+1023*q^28+1133*q^29+1278*q^30+1471*q^31+1659*q^32+1877*q^33+2074* q^34+2345*q^35+2621*q^36+2958*q^37+3263*q^38+3601*q^39+3913*q^40+4336*q^41+4796 *q^42+5316*q^43+5885*q^44+6564*q^45+7268*q^46+8141*q^47+8912*q^48+9631*q^49+ 10556*q^50+11619*q^51+12934*q^52+14447*q^53+16195*q^54+18099*q^55+20440*q^56+ 22782*q^57+25206*q^58+28061*q^59+31657*q^60+35990*q^61+41567*q^62+47715*q^63+ 54674*q^64+62415*q^65+70823*q^66+80441*q^67+92275*q^68+106609*q^69+122870*q^70+ 141966*q^71+162681*q^72+185530*q^73+211214*q^74+242166*q^75+279120*q^76+323109* q^77+371039*q^78+422381*q^79+481103*q^80+548589*q^81+624460*q^82+704421*q^83+ 786582*q^84+868127*q^85+947080*q^86+1016165*q^87+1063994*q^88+1080356*q^89+ 1054970*q^90+980094*q^91+855088*q^92+690892*q^93+510425*q^94+341115*q^95+204389 *q^96+108926*q^97+51207*q^98+21025*q^99+7438*q^100+2224*q^101+545*q^102+104*q^ 103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+2*q^4+3*q^5+q^6, q^10+4*q^9+2*q^8+5 *q^7+8*q^6+11*q^5+14*q^4+15*q^3+9*q^2+4*q+1, q^15+5*q^14+2*q^13+6*q^12+6*q^11+ 13*q^10+15*q^9+15*q^8+28*q^7+34*q^6+43*q^5+40*q^4+29*q^3+14*q^2+5*q+1, q^21+6*q ^20+2*q^19+7*q^18+6*q^17+10*q^16+20*q^15+21*q^14+20*q^13+29*q^12+43*q^11+51*q^ 10+65*q^9+88*q^8+109*q^7+125*q^6+118*q^5+86*q^4+49*q^3+20*q^2+6*q+1, 1+8*q^25+ 17*q^22+6*q^24+25*q^21+10*q^23+7*q+27*q^2+2*q^26+76*q^3+159*q^4+263*q^5+347*q^6 +372*q^7+342*q^8+292*q^9+230*q^10+192*q^11+156*q^12+104*q^13+86*q^14+77*q^15+71 *q^16+q^28+43*q^17+41*q^18+27*q^19+30*q^20+7*q^27, 1+49*q^25+107*q^22+60*q^24+ 115*q^21+75*q^23+8*q+18*q^30+20*q^29+2*q^34+9*q^33+35*q^2+41*q^26+111*q^3+267*q ^4+513*q^5+802*q^6+6*q^32+1032*q^7+1124*q^8+1073*q^9+956*q^10+798*q^11+687*q^12 +555*q^13+417*q^14+349*q^15+295*q^16+34*q^28+240*q^17+180*q^18+140*q^19+129*q^ 20+39*q^27+10*q^31+8*q^35+q^36, 1+237*q^25+10*q^42+436*q^22+303*q^24+485*q^21+ 378*q^23+9*q+9*q^44+118*q^30+q^45+28*q^37+155*q^29+19*q^39+6*q^41+47*q^34+20*q^ 38+69*q^33+44*q^2+209*q^26+155*q^3+419*q^4+912*q^5+1636*q^6+73*q^32+2448*q^7+ 3111*q^8+3420*q^9+3377*q^10+3101*q^11+2731*q^12+2363*q^13+1972*q^14+1596*q^15+ 1345*q^16+2*q^43+167*q^28+1133*q^17+938*q^18+707*q^19+582*q^20+194*q^27+10*q^40 +99*q^31+56*q^35+43*q^36, 1+1153*q^25+83*q^42+1980*q^22+1471*q^24+2373*q^21+ 1712*q^23+10*q+6*q^51+63*q^44+2*q^53+591*q^30+11*q^52+q^55+10*q^54+61*q^45+223* q^37+654*q^29+36*q^46+153*q^39+98*q^41+299*q^34+10*q^50+167*q^38+348*q^33+54*q^ 2+948*q^26+209*q^3+625*q^4+1514*q^5+3049*q^6+406*q^32+5176*q^7+7514*q^8+9445*q^ 9+10504*q^10+10612*q^11+10069*q^12+9117*q^13+8089*q^14+6924*q^15+5813*q^16+66*q ^43+20*q^49+729*q^28+4994*q^17+4251*q^18+3561*q^19+2870*q^20+808*q^27+121*q^40+ 485*q^31+277*q^35+20*q^48+249*q^36+28*q^47, 1+5830*q^25+491*q^42+9300*q^22+6871 *q^24+11054*q^21+7925*q^23+11*q+120*q^51+388*q^44+79*q^53+36*q^57+2553*q^30+109 *q^52+28*q^58+67*q^55+21*q^60+84*q^54+354*q^45+1019*q^37+2875*q^29+315*q^46+11* q^65+764*q^39+10*q^61+577*q^41+1341*q^34+155*q^50+2*q^64+881*q^38+1601*q^33+54* q^56+65*q^2+4728*q^26+274*q^3+896*q^4+2384*q^5+5306*q^6+1844*q^32+10034*q^7+ 16336*q^8+23137*q^9+6*q^62+28905*q^10+32424*q^11+33455*q^12+32486*q^13+30233*q^ 14+27353*q^15+24017*q^16+q^66+429*q^43+179*q^49+3315*q^28+20766*q^17+18051*q^18 +15549*q^19+13212*q^20+3906*q^27+652*q^40+12*q^63+2217*q^31+1198*q^35+217*q^48+ 20*q^59+1095*q^36+249*q^47, 1+26519*q^25+2277*q^42+41172*q^22+30415*q^24+48393* q^21+35255*q^23+12*q+803*q^51+1779*q^44+604*q^53+360*q^57+11921*q^30+684*q^52+ 309*q^58+479*q^55+231*q^60+547*q^54+1658*q^45+20*q^71+6*q^74+4236*q^37+58*q^67+ 13664*q^29+1502*q^46+99*q^65+36*q^69+3407*q^39+179*q^61+2612*q^41+6344*q^34+903 *q^50+104*q^64+3833*q^38+7480*q^33+441*q^56+77*q^2+55*q^68+22752*q^26+351*q^3+ 1244*q^4+3599*q^5+8749*q^6+8881*q^32+18179*q^7+32708*q^8+51460*q^9+152*q^62+ 71536*q^10+88965*q^11+100628*q^12+2*q^76+105542*q^13+104634*q^14+99514*q^15+ 91802*q^16+88*q^66+28*q^70+12*q^77+q^78+1987*q^43+1013*q^49+16028*q^28+82429*q^ 17+72942*q^18+64284*q^19+56025*q^20+19022*q^27+22*q^72+2933*q^40+10*q^73+133*q^ 63+10302*q^31+13*q^75+5410*q^35+1160*q^48+257*q^59+4730*q^36+1365*q^47, 1+ 114123*q^25+10208*q^42+174740*q^22+13*q^90+131061*q^24+199796*q^21+151339*q^23+ 13*q+3700*q^51+7763*q^44+2936*q^53+2012*q^57+54709*q^30+3318*q^52+1777*q^58+ 2432*q^55+1404*q^60+q^91+2672*q^54+7030*q^45+317*q^71+175*q^74+19510*q^37+593*q ^67+63178*q^29+6348*q^46+746*q^65+439*q^69+15231*q^39+1224*q^61+11642*q^41+ 30710*q^34+78*q^79+4149*q^50+837*q^64+17200*q^38+35970*q^33+2240*q^56+90*q^2+ 498*q^68+99746*q^26+441*q^3+1682*q^4+5249*q^5+13810*q^6+41681*q^32+31197*q^7+ 61315*q^8+105875*q^9+1085*q^62+162049*q^10+221951*q^11+275231*q^12+128*q^76+ 313517*q^13+333455*q^14+336430*q^15+325948*q^16+670*q^66+371*q^70+126*q^77+103* q^78+8875*q^43+4636*q^49+73608*q^28+306213*q^17+280451*q^18+252873*q^19+226011* q^20+86362*q^27+271*q^72+13367*q^40+229*q^73+950*q^63+47738*q^31+165*q^75+26264 *q^35+58*q^80+5301*q^48+28*q^83+56*q^81+23*q^85+10*q^86+6*q^87+14*q^88+2*q^89+ 20*q^84+36*q^82+1534*q^59+22493*q^36+5844*q^47, 1+481103*q^25+47715*q^42+704421 *q^22+153*q^90+548589*q^24+786582*q^21+624460*q^23+14*q+28*q^97+16195*q^51+ 35990*q^44+12934*q^53+8912*q^57+242166*q^30+58*q^94+14447*q^52+8141*q^58+10556* q^55+6564*q^60+130*q^91+11619*q^54+31657*q^45+2074*q^71+1471*q^74+36*q^96+20*q^ 98+92275*q^37+3263*q^67+279120*q^29+28061*q^46+78*q^93+24*q^99+3913*q^65+2621*q ^69+70823*q^39+5885*q^61+54674*q^41+141966*q^34+807*q^79+18099*q^50+4336*q^64+ 10*q^100+80441*q^38+162681*q^33+9631*q^56+104*q^2+2958*q^68+422381*q^26+545*q^3 +2224*q^4+7438*q^5+21025*q^6+185530*q^32+51207*q^7+108926*q^8+204389*q^9+57*q^ 95+5316*q^62+6*q^101+341115*q^10+510425*q^11+690892*q^12+1133*q^76+855088*q^13+ 980094*q^14+1054970*q^15+1080356*q^16+3601*q^66+2345*q^70+1023*q^77+909*q^78+ 41567*q^43+15*q^102+20440*q^49+323109*q^28+1063994*q^17+1016165*q^18+947080*q^ 19+868127*q^20+371039*q^27+1877*q^72+62415*q^40+1659*q^73+2*q^103+4796*q^63+ 211214*q^31+1278*q^75+122870*q^35+14*q^104+677*q^80+22782*q^48+q^105+449*q^83+ 607*q^81+331*q^85+267*q^86+224*q^87+189*q^88+158*q^89+378*q^84+513*q^82+7268*q^ 59+106609*q^36+92*q^92+25206*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [3, 1, 2, 4]}, is given by [1, 2, 6, 21, 74, 257, 883, 3019, 10306, 35174, 120063, 409878, 1399367, 4777721, 16312213] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [3, 1, 2, 4]}, is given by [1, 1, 3, 9, 35, 129, 439, 1516, 5162, 17573, 60030, 204928, 699646, 2388894, 8156140] The number of ODD permutations avoiding, {[1, 3, 4, 2], [2, 1, 4, 3], [3, 1, 2, 4]}, is given by [0, 1, 3, 12, 39, 128, 444, 1503, 5144, 17601, 60033, 204950, 699721, 2388827, 8156073] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 9, 35, 128, 444, 1516, 5162, 17601, 60033, 204928, 699646, 2388827, 8156073] ODD:, [0, 1, 3, 12, 39, 129, 439, 1503, 5144, 17573, 60030, 204950, 699721, 2388894, 8156140] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.142857143, 5.635135135, 9.050583658, 13.43374858, 18.80987082, 25.18959829, 32.57477114, 40.96380234, 50.35475434, 60.74637390, 72.13809032, 84.52973272] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.520830421, 2.172147549, 2.888554101, 3.630884617, 4.365044755, 5.068438325, 5.730100291, 6.347696571, 6.923780708, 7.462878641, 7.969790997, 8.448847236] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.31293, 4.71822, 8.34374, 13.1833, 19.0536, 25.6891, 32.8340, 40.2933, 47.9387, 55.6946, 63.5176, 71.3830] Skewness: , [Float(undefined), 0., 0., -0.2420131140, -0.5395922505, -0.7827615387, -0.9821651815, -1.152993277, -1.301022911, -1.427012791, -1.530681133, -1.612545139, -1.674194781, -1.718113378, -1.747128250] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.396775778, 2.861272329, 3.362708056, 3.907689004, 4.503805291, 5.141773712, 5.797221868, 6.438976218, 7.037451192, 7.570162575, 8.023724443, 8.393430601] end of this data For the equivalence class of patterns, { {[2, 4, 1, 3], [2, 3, 1, 4], [4, 3, 1, 2]}, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 3, 1, 2]}, {[1, 2, 4, 3], [3, 2, 4, 1], [3, 1, 4, 2]}, {[2, 4, 3, 1], [2, 4, 1, 3], [2, 1, 3, 4]}, {[1, 4, 2, 3], [2, 4, 1, 3], [3, 4, 2, 1]}, {[1, 2, 4, 3], [2, 4, 1, 3], [4, 2, 1, 3]}, {[2, 1, 3, 4], [3, 1, 4, 2], [4, 1, 3, 2]}, {[3, 4, 2, 1], [3, 1, 4, 2], [3, 1, 2, 4]}} the member , {[1, 3, 4, 2], [3, 1, 4, 2], [4, 3, 1, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], %1, {1}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}], [[2, 1, 3], {[0, 1, 0, 0]}, {}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}], [[4, 2, 3, 1], %1, {2}], [[4, 5, 2, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[3, 4, 1, 2], {}, {}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0]}, {3}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {3}], [[4, 5, 1, 3, 2], {[0, 0, 1, 0, 0, 0]}, {4}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}], [[2, 4, 3, 1], %1, {1}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0]}, {1}], [[3, 2, 4, 1], %1, {1}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], %1, {1}], [[2, 4, 1, 3], {[0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], %1, {2}], [ [3, 4, 1, 2, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[3, 5, 1, 2, 4], {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {}, {}], [[3, 1, 2], {}, {}], [[4, 5, 1, 2, 3], {[0, 1, 0, 0, 0, 0]}, {3}], [[2, 1, 3, 4], %1, {3}]} %1 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+7*q^2+10*q^3 +13*q^4+12*q^5+11*q^6+9*q^7+5*q^8+2*q^9+q^10, 1+5*q+11*q^2+18*q^3+26*q^4+30*q^5 +33*q^6+34*q^7+30*q^8+26*q^9+20*q^10+15*q^11+9*q^12+5*q^13+2*q^14+q^15, 1+6*q+ 16*q^2+30*q^3+48*q^4+63*q^5+75*q^6+88*q^7+91*q^8+90*q^9+89*q^10+78*q^11+68*q^12 +60*q^13+43*q^14+33*q^15+24*q^16+15*q^17+9*q^18+5*q^19+2*q^20+q^21, 1+7*q+22*q^ 2+47*q^3+83*q^4+122*q^5+157*q^6+194*q^7+222*q^8+240*q^9+257*q^10+260*q^11+252*q ^12+242*q^13+223*q^14+191*q^15+173*q^16+146*q^17+118*q^18+93*q^19+70*q^20+50*q^ 21+37*q^22+24*q^23+15*q^24+9*q^25+5*q^26+2*q^27+q^28, 1+8*q+29*q^2+70*q^3+136*q ^4+221*q^5+310*q^6+404*q^7+491*q^8+562*q^9+634*q^10+689*q^11+725*q^12+749*q^13+ 745*q^14+721*q^15+691*q^16+643*q^17+593*q^18+529*q^19+463*q^20+406*q^21+345*q^ 22+280*q^23+233*q^24+182*q^25+139*q^26+103*q^27+77*q^28+54*q^29+37*q^30+24*q^31 +15*q^32+9*q^33+5*q^34+2*q^35+q^36, 1+9*q+37*q^2+100*q^3+213*q^4+379*q^5+579*q^ 6+803*q^7+1033*q^8+1241*q^9+1447*q^10+1642*q^11+1810*q^12+1973*q^13+2092*q^14+ 2149*q^15+2186*q^16+2174*q^17+2113*q^18+2049*q^19+1932*q^20+1803*q^21+1680*q^22 +1521*q^23+1351*q^24+1214*q^25+1050*q^26+910*q^27+772*q^28+640*q^29+529*q^30+ 433*q^31+335*q^32+265*q^33+203*q^34+149*q^35+110*q^36+81*q^37+54*q^38+37*q^39+ 24*q^40+15*q^41+9*q^42+5*q^43+2*q^44+q^45, 1+10*q+46*q^2+138*q^3+321*q^4+621*q^ 5+1029*q^6+1525*q^7+2079*q^8+2631*q^9+3177*q^10+3717*q^11+4224*q^12+4741*q^13+ 5237*q^14+5649*q^15+6000*q^16+6267*q^17+6404*q^18+6460*q^19+6435*q^20+6306*q^21 +6149*q^22+5934*q^23+5627*q^24+5303*q^25+4941*q^26+4551*q^27+4146*q^28+3752*q^ 29+3339*q^30+2985*q^31+2626*q^32+2265*q^33+1948*q^34+1661*q^35+1384*q^36+1158*q ^37+945*q^38+762*q^39+613*q^40+484*q^41+367*q^42+286*q^43+213*q^44+156*q^45+114 *q^46+81*q^47+54*q^48+37*q^49+24*q^50+15*q^51+9*q^52+5*q^53+2*q^54+q^55, 1+11*q +56*q^2+185*q^3+468*q^4+979*q^5+1751*q^6+2775*q^7+4012*q^8+5359*q^9+6748*q^10+ 8156*q^11+9524*q^12+10909*q^13+12332*q^14+13702*q^15+15040*q^16+16287*q^17+ 17322*q^18+18166*q^19+18767*q^20+19097*q^21+19286*q^22+19276*q^23+19104*q^24+ 18777*q^25+18259*q^26+17631*q^27+16885*q^28+15977*q^29+15031*q^30+14035*q^31+ 13000*q^32+11953*q^33+10934*q^34+9888*q^35+8915*q^36+7958*q^37+7062*q^38+6200*q ^39+5413*q^40+4671*q^41+4028*q^42+3430*q^43+2880*q^44+2404*q^45+1997*q^46+1629* q^47+1333*q^48+1063*q^49+842*q^50+664*q^51+516*q^52+388*q^53+296*q^54+220*q^55+ 160*q^56+114*q^57+81*q^58+54*q^59+37*q^60+24*q^61+15*q^62+9*q^63+5*q^64+2*q^65+ q^66, 1+12*q+67*q^2+242*q^3+663*q^4+1493*q^5+2869*q^6+4856*q^7+7445*q^8+10500*q ^9+13846*q^10+17370*q^11+20920*q^12+24494*q^13+28183*q^14+31906*q^15+35715*q^16 +39611*q^17+43357*q^18+46881*q^19+50060*q^20+52623*q^21+54704*q^22+56324*q^23+ 57375*q^24+58087*q^25+58363*q^26+58151*q^27+57640*q^28+56686*q^29+55244*q^30+ 53565*q^31+51550*q^32+49189*q^33+46791*q^34+44160*q^35+41421*q^36+38717*q^37+ 35944*q^38+33163*q^39+30467*q^40+27734*q^41+25152*q^42+22721*q^43+20357*q^44+ 18111*q^45+16099*q^46+14153*q^47+12401*q^48+10794*q^49+9319*q^50+7987*q^51+6825 *q^52+5754*q^53+4846*q^54+4053*q^55+3342*q^56+2741*q^57+2244*q^58+1800*q^59+ 1447*q^60+1143*q^61+893*q^62+696*q^63+537*q^64+398*q^65+303*q^66+224*q^67+160*q ^68+114*q^69+81*q^70+54*q^71+37*q^72+24*q^73+15*q^74+9*q^75+5*q^76+2*q^77+q^78, 1+13*q+79*q^2+310*q^3+916*q^4+2212*q^5+4548*q^6+8203*q^7+13326*q^8+19834*q^9+ 27443*q^10+35858*q^11+44715*q^12+53788*q^13+63137*q^14+72691*q^15+82536*q^16+ 92870*q^17+103490*q^18+114265*q^19+124989*q^20+135052*q^21+144214*q^22+152394*q ^23+159273*q^24+165005*q^25+169698*q^26+173266*q^27+175853*q^28+177512*q^29+ 178014*q^30+177518*q^31+176003*q^32+173306*q^33+169577*q^34+165071*q^35+159727* q^36+153840*q^37+147519*q^38+140709*q^39+133637*q^40+126392*q^41+118822*q^42+ 111291*q^43+103773*q^44+96239*q^45+88803*q^46+81700*q^47+74689*q^48+68095*q^49+ 61801*q^50+55801*q^51+50178*q^52+44910*q^53+39904*q^54+35410*q^55+31240*q^56+ 27370*q^57+23885*q^58+20777*q^59+17911*q^60+15422*q^61+13166*q^62+11174*q^63+ 9439*q^64+7925*q^65+6585*q^66+5479*q^67+4516*q^68+3681*q^69+2984*q^70+2411*q^71 +1914*q^72+1527*q^73+1194*q^74+925*q^75+717*q^76+547*q^77+405*q^78+307*q^79+224 *q^80+160*q^81+114*q^82+81*q^83+54*q^84+37*q^85+24*q^86+15*q^87+9*q^88+5*q^89+2 *q^90+q^91, 1+14*q+92*q^2+390*q^3+1238*q^4+3195*q^5+7003*q^6+13425*q^7+23078*q^ 8+36214*q^9+52602*q^10+71735*q^11+92870*q^12+115262*q^13+138629*q^14+162742*q^ 15+187586*q^16+213617*q^17+240885*q^18+269435*q^19+299301*q^20+329600*q^21+ 359548*q^22+388592*q^23+415603*q^24+440238*q^25+462374*q^26+481623*q^27+498467* q^28+512886*q^29+524740*q^30+534392*q^31+541567*q^32+545986*q^33+547706*q^34+ 546284*q^35+541964*q^36+535131*q^37+525730*q^38+514172*q^39+500842*q^40+485851* q^41+469455*q^42+451914*q^43+433296*q^44+413904*q^45+393833*q^46+373197*q^47+ 352219*q^48+331260*q^49+310257*q^50+289638*q^51+269476*q^52+249780*q^53+230636* q^54+212325*q^55+194664*q^56+177910*q^57+161965*q^58+146847*q^59+132637*q^60+ 119450*q^61+107048*q^62+95660*q^63+85133*q^64+75471*q^65+66626*q^66+58604*q^67+ 51249*q^68+44722*q^69+38821*q^70+33518*q^71+28830*q^72+24722*q^73+21040*q^74+ 17867*q^75+15064*q^76+12647*q^77+10556*q^78+8766*q^79+7219*q^80+5944*q^81+4851* q^82+3920*q^83+3151*q^84+2525*q^85+1994*q^86+1578*q^87+1226*q^88+946*q^89+727*q ^90+554*q^91+409*q^92+307*q^93+224*q^94+160*q^95+114*q^96+81*q^97+54*q^98+37*q^ 99+24*q^100+15*q^101+9*q^102+5*q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+ 10*q^7+13*q^6+12*q^5+11*q^4+9*q^3+5*q^2+2*q+1, q^15+5*q^14+11*q^13+18*q^12+26*q ^11+30*q^10+33*q^9+34*q^8+30*q^7+26*q^6+20*q^5+15*q^4+9*q^3+5*q^2+2*q+1, q^21+6 *q^20+16*q^19+30*q^18+48*q^17+63*q^16+75*q^15+88*q^14+91*q^13+90*q^12+89*q^11+ 78*q^10+68*q^9+60*q^8+43*q^7+33*q^6+24*q^5+15*q^4+9*q^3+5*q^2+2*q+1, 1+47*q^25+ 157*q^22+83*q^24+194*q^21+122*q^23+2*q+5*q^2+22*q^26+9*q^3+15*q^4+24*q^5+37*q^6 +50*q^7+70*q^8+93*q^9+118*q^10+146*q^11+173*q^12+191*q^13+223*q^14+242*q^15+252 *q^16+q^28+260*q^17+257*q^18+240*q^19+222*q^20+7*q^27, 1+689*q^25+745*q^22+725* q^24+721*q^21+749*q^23+2*q+310*q^30+404*q^29+29*q^34+70*q^33+5*q^2+634*q^26+9*q ^3+15*q^4+24*q^5+37*q^6+136*q^32+54*q^7+77*q^8+103*q^9+139*q^10+182*q^11+233*q^ 12+280*q^13+345*q^14+406*q^15+463*q^16+491*q^28+529*q^17+593*q^18+643*q^19+691* q^20+562*q^27+221*q^31+8*q^35+q^36, 1+1932*q^25+100*q^42+1521*q^22+1803*q^24+ 1351*q^21+1680*q^23+2*q+9*q^44+2149*q^30+q^45+1033*q^37+2186*q^29+579*q^39+213* q^41+1642*q^34+803*q^38+1810*q^33+5*q^2+2049*q^26+9*q^3+15*q^4+24*q^5+37*q^6+ 1973*q^32+54*q^7+81*q^8+110*q^9+149*q^10+203*q^11+265*q^12+335*q^13+433*q^14+ 529*q^15+640*q^16+37*q^43+2174*q^28+772*q^17+910*q^18+1050*q^19+1214*q^20+2113* q^27+379*q^40+2092*q^31+1447*q^35+1241*q^36, 1+3339*q^25+4741*q^42+2265*q^22+ 2985*q^24+1948*q^21+2626*q^23+2*q+321*q^51+3717*q^44+46*q^53+5303*q^30+138*q^52 +q^55+10*q^54+3177*q^45+6404*q^37+4941*q^29+2631*q^46+6000*q^39+5237*q^41+6306* q^34+621*q^50+6267*q^38+6149*q^33+5*q^2+3752*q^26+9*q^3+15*q^4+24*q^5+37*q^6+ 5934*q^32+54*q^7+81*q^8+114*q^9+156*q^10+213*q^11+286*q^12+367*q^13+484*q^14+ 613*q^15+762*q^16+4224*q^43+1029*q^49+4551*q^28+945*q^17+1158*q^18+1384*q^19+ 1661*q^20+4146*q^27+5649*q^40+5627*q^31+6435*q^35+1525*q^48+6460*q^36+2079*q^47 , 1+4671*q^25+19104*q^42+2880*q^22+4028*q^24+2404*q^21+3430*q^23+2*q+13702*q^51 +19286*q^44+10909*q^53+5359*q^57+8915*q^30+12332*q^52+4012*q^58+8156*q^55+1751* q^60+9524*q^54+19097*q^45+15977*q^37+7958*q^29+18767*q^46+11*q^65+17631*q^39+ 979*q^61+18777*q^41+13000*q^34+15040*q^50+56*q^64+16885*q^38+11953*q^33+6748*q^ 56+5*q^2+5413*q^26+9*q^3+15*q^4+24*q^5+37*q^6+10934*q^32+54*q^7+81*q^8+114*q^9+ 468*q^62+160*q^10+220*q^11+296*q^12+388*q^13+516*q^14+664*q^15+842*q^16+q^66+ 19276*q^43+16287*q^49+7062*q^28+1063*q^17+1333*q^18+1629*q^19+1997*q^20+6200*q^ 27+18259*q^40+185*q^63+9888*q^31+14035*q^35+17322*q^48+2775*q^59+15031*q^36+ 18166*q^47, 1+5754*q^25+41421*q^42+3342*q^22+4846*q^24+2741*q^21+4053*q^23+2*q+ 58151*q^51+46791*q^44+58087*q^53+52623*q^57+12401*q^30+58363*q^52+50060*q^58+ 56324*q^55+43357*q^60+57375*q^54+49189*q^45+4856*q^71+663*q^74+27734*q^37+17370 *q^67+10794*q^29+51550*q^46+24494*q^65+10500*q^69+33163*q^39+39611*q^61+38717*q ^41+20357*q^34+57640*q^50+28183*q^64+30467*q^38+18111*q^33+54704*q^56+5*q^2+ 13846*q^68+6825*q^26+9*q^3+15*q^4+24*q^5+37*q^6+16099*q^32+54*q^7+81*q^8+114*q^ 9+35715*q^62+160*q^10+224*q^11+303*q^12+67*q^76+398*q^13+537*q^14+696*q^15+893* q^16+20920*q^66+7445*q^70+12*q^77+q^78+44160*q^43+56686*q^49+9319*q^28+1143*q^ 17+1447*q^18+1800*q^19+2244*q^20+7987*q^27+2869*q^72+35944*q^40+1493*q^73+31906 *q^63+14153*q^31+242*q^75+22721*q^35+55244*q^48+46881*q^59+25152*q^36+53565*q^ 47, 1+6585*q^25+68095*q^42+3681*q^22+13*q^90+5479*q^24+2984*q^21+4516*q^23+2*q+ 133637*q^51+81700*q^44+147519*q^53+169577*q^57+15422*q^30+140709*q^52+173306*q^ 58+159727*q^55+177518*q^60+q^91+153840*q^54+88803*q^45+124989*q^71+92870*q^74+ 39904*q^37+159273*q^67+13166*q^29+96239*q^46+169698*q^65+144214*q^69+50178*q^39 +178014*q^61+61801*q^41+27370*q^34+44715*q^79+126392*q^50+173266*q^64+44910*q^ 38+23885*q^33+165071*q^56+5*q^2+152394*q^68+7925*q^26+9*q^3+15*q^4+24*q^5+37*q^ 6+20777*q^32+54*q^7+81*q^8+114*q^9+177512*q^62+160*q^10+224*q^11+307*q^12+72691 *q^76+405*q^13+547*q^14+717*q^15+925*q^16+165005*q^66+135052*q^70+63137*q^77+ 53788*q^78+74689*q^43+118822*q^49+11174*q^28+1194*q^17+1527*q^18+1914*q^19+2411 *q^20+9439*q^27+114265*q^72+55801*q^40+103490*q^73+175853*q^63+17911*q^31+82536 *q^75+31240*q^35+35858*q^80+111291*q^48+13326*q^83+27443*q^81+4548*q^85+2212*q^ 86+916*q^87+310*q^88+79*q^89+8203*q^84+19834*q^82+176003*q^59+35410*q^36+103773 *q^47, 1+7219*q^25+95660*q^42+3920*q^22+162742*q^90+5944*q^24+3151*q^21+4851*q^ 23+2*q+23078*q^97+230636*q^51+119450*q^44+269476*q^53+352219*q^57+17867*q^30+ 71735*q^94+249780*q^52+373197*q^58+310257*q^55+413904*q^60+138629*q^91+289638*q ^54+132637*q^45+547706*q^71+534392*q^74+36214*q^96+13425*q^98+51249*q^37+525730 *q^67+15064*q^29+146847*q^46+92870*q^93+7003*q^99+500842*q^65+541964*q^69+66626 *q^39+433296*q^61+85133*q^41+33518*q^34+462374*q^79+212325*q^50+485851*q^64+ 3195*q^100+58604*q^38+28830*q^33+331260*q^56+5*q^2+535131*q^68+8766*q^26+9*q^3+ 15*q^4+24*q^5+37*q^6+24722*q^32+54*q^7+81*q^8+114*q^9+52602*q^95+451914*q^62+ 1238*q^101+160*q^10+224*q^11+307*q^12+512886*q^76+409*q^13+554*q^14+727*q^15+ 946*q^16+514172*q^66+546284*q^70+498467*q^77+481623*q^78+107048*q^43+390*q^102+ 194664*q^49+12647*q^28+1226*q^17+1578*q^18+1994*q^19+2525*q^20+10556*q^27+ 545986*q^72+75471*q^40+541567*q^73+92*q^103+469455*q^63+21040*q^31+524740*q^75+ 38821*q^35+14*q^104+440238*q^80+177910*q^48+q^105+359548*q^83+415603*q^81+ 299301*q^85+269435*q^86+240885*q^87+213617*q^88+187586*q^89+329600*q^84+388592* q^82+393833*q^59+44722*q^36+115262*q^92+161965*q^47] The number of permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 3, 1, 2]}, is given by [1, 2, 6, 21, 75, 266, 935, 3263, 11326, 39155, 134955, 464094, 1593231, 5462447, 18709694] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 3, 1, 2]}, is given by [1, 1, 3, 11, 38, 132, 466, 1636, 5670, 19562, 67454, 232101, 796689, 2731048, 9354635] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 3, 1, 2]}, is given by [0, 1, 3, 10, 37, 134, 469, 1627, 5656, 19593, 67501, 231993, 796542, 2731399, 9355059] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 11, 38, 134, 469, 1636, 5670, 19593, 67501, 232101, 796689, 2731399, 9355059] ODD:, [0, 1, 3, 10, 37, 132, 466, 1627, 5656, 19562, 67454, 231993, 796542, 2731048, 9354635] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.952380952, 4.760000000, 6.864661654, 9.237433155, 11.86362243, 14.73547590, 17.84893372, 21.20208218, 24.79431107, 28.62580128, 32.69719468, 37.00937797] The standard deviation for each n is given by [0., 0.5000000000, 0.9574271080, 1.495268426, 2.159259132, 2.937396819, 3.812348336, 4.772042650, 5.808327349, 6.915273303, 8.088299156, 9.323750550, 10.61866041, 11.97058207, 13.37746147] The centralized moments are Second: , [0., 0.250000, 0.916667, 2.23583, 4.66240, 8.62830, 14.5340, 22.7724, 33.7367, 47.8210, 65.4206, 86.9323, 112.756, 143.295, 178.956] Skewness: , [Float(undefined), 0., 0., -0.004130812756, 0.08053774120, 0.1703588141, 0.2401573571, 0.2904135721, 0.3258572646, 0.3505372204, 0.3673365589, 0.3782834951, 0.3848367943, 0.3880826014, 0.3888451054] Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.384387498, 2.467773703, 2.538068376, 2.611041592, 2.678206069, 2.736193961, 2.784937996, 2.825329426, 2.858430260, 2.885287375, 2.906850996, 2.924039630] end of this data For the equivalence class of patterns, { {[1, 2, 4, 3], [2, 3, 4, 1], [2, 3, 1, 4]}, {[3, 4, 2, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, {[3, 2, 4, 1], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 4, 2, 3], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 4, 3, 1], [4, 3, 1, 2]}, {[1, 4, 3, 2], [3, 4, 2, 1], [4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 3, 4, 1], [2, 1, 3, 4]}, {[1, 2, 4, 3], [3, 1, 2, 4], [4, 1, 2, 3]}} the member , {[1, 4, 2, 3], [2, 1, 3, 4], [4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}], [[3, 4, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 3], {}, {2}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[3, 2, 1], {}, {}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {2}], [[3, 4, 2, 1], {}, {1}], [[4, 3, 2, 1], {}, {2}], [[4, 3, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 0, 1, 0]}, {3}], [[4, 2, 1, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 2, 1, 4], {[0, 0, 0, 0, 1]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, 1+2*q+3*q^2+6*q^3+ 10*q^4+13*q^5+14*q^6+13*q^7+9*q^8+4*q^9+q^10, 1+2*q+3*q^2+5*q^3+8*q^4+13*q^5+18 *q^6+26*q^7+36*q^8+41*q^9+40*q^10+36*q^11+26*q^12+14*q^13+5*q^14+q^15, 1+2*q+3* q^2+5*q^3+8*q^4+10*q^5+16*q^6+23*q^7+33*q^8+43*q^9+58*q^10+76*q^11+99*q^12+119* q^13+125*q^14+118*q^15+102*q^16+76*q^17+45*q^18+20*q^19+6*q^20+q^21, 1+2*q+3*q^ 2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+27*q^8+38*q^9+52*q^10+68*q^11+88*q^12+118*q^ 13+152*q^14+187*q^15+233*q^16+289*q^17+345*q^18+380*q^19+381*q^20+350*q^21+297* q^22+223*q^23+141*q^24+71*q^25+27*q^26+7*q^27+q^28, 1+2*q+3*q^2+5*q^3+8*q^4+10* q^5+16*q^6+21*q^7+28*q^8+37*q^9+47*q^10+62*q^11+78*q^12+102*q^13+131*q^14+166*q ^15+206*q^16+267*q^17+333*q^18+419*q^19+517*q^20+626*q^21+742*q^22+883*q^23+ 1031*q^24+1149*q^25+1199*q^26+1162*q^27+1048*q^28+876*q^29+662*q^30+435*q^31+ 239*q^32+105*q^33+35*q^34+8*q^35+q^36, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21 *q^7+28*q^8+38*q^9+48*q^10+62*q^11+78*q^12+99*q^13+121*q^14+152*q^15+187*q^16+ 234*q^17+290*q^18+359*q^19+440*q^20+541*q^21+668*q^22+816*q^23+1001*q^24+1230*q ^25+1493*q^26+1784*q^27+2100*q^28+2439*q^29+2803*q^30+3190*q^31+3534*q^32+3748* q^33+3762*q^34+3558*q^35+3157*q^36+2613*q^37+1980*q^38+1337*q^39+779*q^40+379*q ^41+148*q^42+44*q^43+9*q^44+q^45, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+ 28*q^8+38*q^9+49*q^10+63*q^11+80*q^12+101*q^13+121*q^14+155*q^15+189*q^16+227*q ^17+277*q^18+335*q^19+402*q^20+489*q^21+594*q^22+716*q^23+859*q^24+1029*q^25+ 1247*q^26+1505*q^27+1821*q^28+2177*q^29+2618*q^30+3139*q^31+3774*q^32+4496*q^33 +5327*q^34+6215*q^35+7159*q^36+8138*q^37+9160*q^38+10187*q^39+11138*q^40+11834* q^41+12089*q^42+11793*q^43+10928*q^44+9568*q^45+7855*q^46+5965*q^47+4104*q^48+ 2496*q^49+1306*q^50+571*q^51+201*q^52+54*q^53+10*q^54+q^55, 1+2*q+3*q^2+5*q^3+8 *q^4+10*q^5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64*q^11+81*q^12+103*q^13+125*q^ 14+157*q^15+192*q^16+234*q^17+283*q^18+340*q^19+407*q^20+487*q^21+578*q^22+686* q^23+811*q^24+960*q^25+1143*q^26+1349*q^27+1600*q^28+1887*q^29+2228*q^30+2630*q ^31+3126*q^32+3706*q^33+4390*q^34+5204*q^35+6172*q^36+7318*q^37+8659*q^38+10205 *q^39+12011*q^40+14117*q^41+16485*q^42+19068*q^43+21825*q^44+24682*q^45+27588*q ^46+30501*q^47+33356*q^48+35961*q^49+38006*q^50+39060*q^51+38773*q^52+36966*q^ 53+33678*q^54+29139*q^55+23768*q^56+18072*q^57+12609*q^58+7915*q^59+4374*q^60+ 2078*q^61+826*q^62+265*q^63+65*q^64+11*q^65+q^66, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^ 5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64*q^11+82*q^12+104*q^13+127*q^14+161*q^ 15+196*q^16+239*q^17+290*q^18+349*q^19+418*q^20+501*q^21+593*q^22+700*q^23+827* q^24+970*q^25+1139*q^26+1331*q^27+1553*q^28+1810*q^29+2110*q^30+2454*q^31+2864* q^32+3340*q^33+3891*q^34+4514*q^35+5256*q^36+6136*q^37+7160*q^38+8358*q^39+9743 *q^40+11389*q^41+13337*q^42+15660*q^43+18342*q^44+21472*q^45+25077*q^46+29278*q ^47+34076*q^48+39597*q^49+45794*q^50+52771*q^51+60412*q^52+68620*q^53+77155*q^ 54+85906*q^55+94629*q^56+103155*q^57+111235*q^58+118503*q^59+124279*q^60+127678 *q^61+127781*q^62+123938*q^63+115948*q^64+104088*q^65+89134*q^66+72295*q^67+ 55021*q^68+38797*q^69+24952*q^70+14377*q^71+7279*q^72+3169*q^73+1156*q^74+341*q ^75+77*q^76+12*q^77+q^78, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+ 38*q^9+49*q^10+64*q^11+82*q^12+105*q^13+128*q^14+163*q^15+200*q^16+243*q^17+297 *q^18+358*q^19+427*q^20+515*q^21+611*q^22+720*q^23+855*q^24+1001*q^25+1167*q^26 +1368*q^27+1593*q^28+1844*q^29+2142*q^30+2474*q^31+2855*q^32+3291*q^33+3793*q^ 34+4353*q^35+5008*q^36+5760*q^37+6626*q^38+7627*q^39+8765*q^40+10067*q^41+11564 *q^42+13292*q^43+15305*q^44+17670*q^45+20378*q^46+23495*q^47+27113*q^48+31371*q ^49+36319*q^50+42090*q^51+48779*q^52+56564*q^53+65580*q^54+75971*q^55+87831*q^ 56+101280*q^57+116472*q^58+133594*q^59+152751*q^60+173947*q^61+197022*q^62+ 221734*q^63+247714*q^64+274438*q^65+301300*q^66+327723*q^67+352999*q^68+376370* q^69+396841*q^70+413016*q^71+422907*q^72+424369*q^73+415530*q^74+395371*q^75+ 363948*q^76+322554*q^77+273686*q^78+220888*q^79+168202*q^80+119597*q^81+78391*q ^82+46673*q^83+24836*q^84+11605*q^85+4666*q^86+1574*q^87+430*q^88+90*q^89+13*q^ 90+q^91, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64* q^11+82*q^12+105*q^13+129*q^14+164*q^15+202*q^16+247*q^17+301*q^18+365*q^19+438 *q^20+526*q^21+625*q^22+741*q^23+879*q^24+1032*q^25+1208*q^26+1411*q^27+1644*q^ 28+1908*q^29+2210*q^30+2551*q^31+2939*q^32+3378*q^33+3882*q^34+4444*q^35+5083*q ^36+5806*q^37+6629*q^38+7549*q^39+8597*q^40+9787*q^41+11132*q^42+12660*q^43+ 14403*q^44+16392*q^45+18643*q^46+21197*q^47+24087*q^48+27398*q^49+31197*q^50+ 35546*q^51+40508*q^52+46199*q^53+52739*q^54+60262*q^55+68927*q^56+78906*q^57+ 90456*q^58+103759*q^59+119081*q^60+136726*q^61+157173*q^62+180641*q^63+207572*q ^64+238235*q^65+273121*q^66+312467*q^67+356735*q^68+405987*q^69+460661*q^70+ 520853*q^71+586844*q^72+658202*q^73+734462*q^74+814410*q^75+896988*q^76+980602* q^77+1063609*q^78+1143911*q^79+1219473*q^80+1287816*q^81+1346212*q^82+1391207*q ^83+1418463*q^84+1422897*q^85+1399738*q^86+1345632*q^87+1259542*q^88+1143306*q^ 89+1001861*q^90+843162*q^91+677482*q^92+516042*q^93+369371*q^94+245818*q^95+ 150241*q^96+83191*q^97+41119*q^98+17846*q^99+6670*q^100+2094*q^101+533*q^102+ 104*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+2*q^5+q^6, q^10+2*q^9+3*q^8+6 *q^7+10*q^6+13*q^5+14*q^4+13*q^3+9*q^2+4*q+1, q^15+2*q^14+3*q^13+5*q^12+8*q^11+ 13*q^10+18*q^9+26*q^8+36*q^7+41*q^6+40*q^5+36*q^4+26*q^3+14*q^2+5*q+1, q^21+2*q ^20+3*q^19+5*q^18+8*q^17+10*q^16+16*q^15+23*q^14+33*q^13+43*q^12+58*q^11+76*q^ 10+99*q^9+119*q^8+125*q^7+118*q^6+102*q^5+76*q^4+45*q^3+20*q^2+6*q+1, 1+5*q^25+ 16*q^22+8*q^24+21*q^21+10*q^23+7*q+27*q^2+3*q^26+71*q^3+141*q^4+223*q^5+297*q^6 +350*q^7+381*q^8+380*q^9+345*q^10+289*q^11+233*q^12+187*q^13+152*q^14+118*q^15+ 88*q^16+q^28+68*q^17+52*q^18+38*q^19+27*q^20+2*q^27, 1+62*q^25+131*q^22+78*q^24 +166*q^21+102*q^23+8*q+16*q^30+21*q^29+3*q^34+5*q^33+35*q^2+47*q^26+105*q^3+239 *q^4+435*q^5+662*q^6+8*q^32+876*q^7+1048*q^8+1162*q^9+1199*q^10+1149*q^11+1031* q^12+883*q^13+742*q^14+626*q^15+517*q^16+28*q^28+419*q^17+333*q^18+267*q^19+206 *q^20+37*q^27+10*q^31+2*q^35+q^36, 1+440*q^25+5*q^42+816*q^22+541*q^24+1001*q^ 21+668*q^23+9*q+2*q^44+152*q^30+q^45+28*q^37+187*q^29+16*q^39+8*q^41+62*q^34+21 *q^38+78*q^33+44*q^2+359*q^26+148*q^3+379*q^4+779*q^5+1337*q^6+99*q^32+1980*q^7 +2613*q^8+3157*q^9+3558*q^10+3762*q^11+3748*q^12+3534*q^13+3190*q^14+2803*q^15+ 2439*q^16+3*q^43+234*q^28+2100*q^17+1784*q^18+1493*q^19+1230*q^20+290*q^27+10*q ^40+121*q^31+48*q^35+38*q^36, 1+2618*q^25+101*q^42+4496*q^22+3139*q^24+5327*q^ 21+3774*q^23+10*q+8*q^51+63*q^44+3*q^53+1029*q^30+5*q^52+q^55+2*q^54+49*q^45+ 277*q^37+1247*q^29+38*q^46+189*q^39+121*q^41+489*q^34+10*q^50+227*q^38+594*q^33 +54*q^2+2177*q^26+201*q^3+571*q^4+1306*q^5+2496*q^6+716*q^32+4104*q^7+5965*q^8+ 7855*q^9+9568*q^10+10928*q^11+11793*q^12+12089*q^13+11834*q^14+11138*q^15+10187 *q^16+80*q^43+16*q^49+1505*q^28+9160*q^17+8138*q^18+7159*q^19+6215*q^20+1821*q^ 27+155*q^40+859*q^31+402*q^35+21*q^48+335*q^36+28*q^47, 1+14117*q^25+811*q^42+ 21825*q^22+16485*q^24+24682*q^21+19068*q^23+11*q+157*q^51+578*q^44+103*q^53+38* q^57+6172*q^30+125*q^52+28*q^58+64*q^55+16*q^60+81*q^54+487*q^45+1887*q^37+7318 *q^29+407*q^46+2*q^65+1349*q^39+10*q^61+960*q^41+3126*q^34+192*q^50+3*q^64+1600 *q^38+3706*q^33+49*q^56+65*q^2+12011*q^26+265*q^3+826*q^4+2078*q^5+4374*q^6+ 4390*q^32+7915*q^7+12609*q^8+18072*q^9+8*q^62+23768*q^10+29139*q^11+33678*q^12+ 36966*q^13+38773*q^14+39060*q^15+38006*q^16+q^66+686*q^43+234*q^49+8659*q^28+ 35961*q^17+33356*q^18+30501*q^19+27588*q^20+10205*q^27+1143*q^40+5*q^63+5204*q^ 31+2630*q^35+283*q^48+21*q^59+2228*q^36+340*q^47, 1+68620*q^25+5256*q^42+94629* q^22+77155*q^24+103155*q^21+85906*q^23+12*q+1331*q^51+3891*q^44+970*q^53+501*q^ 57+34076*q^30+1139*q^52+418*q^58+700*q^55+290*q^60+827*q^54+3340*q^45+21*q^71+8 *q^74+11389*q^37+64*q^67+39597*q^29+2864*q^46+104*q^65+38*q^69+8358*q^39+239*q^ 61+6136*q^41+18342*q^34+1553*q^50+127*q^64+9743*q^38+21472*q^33+593*q^56+77*q^2 +49*q^68+60412*q^26+341*q^3+1156*q^4+3169*q^5+7279*q^6+25077*q^32+14377*q^7+ 24952*q^8+38797*q^9+196*q^62+55021*q^10+72295*q^11+89134*q^12+3*q^76+104088*q^ 13+115948*q^14+123938*q^15+127781*q^16+82*q^66+28*q^70+2*q^77+q^78+4514*q^43+ 1810*q^49+45794*q^28+127678*q^17+124279*q^18+118503*q^19+111235*q^20+52771*q^27 +16*q^72+7160*q^40+10*q^73+161*q^63+29278*q^31+5*q^75+15660*q^35+2110*q^48+349* q^59+13337*q^36+2454*q^47, 1+301300*q^25+31371*q^42+376370*q^22+2*q^90+327723*q ^24+396841*q^21+352999*q^23+13*q+8765*q^51+23495*q^44+6626*q^53+3793*q^57+ 173947*q^30+7627*q^52+3291*q^58+5008*q^55+2474*q^60+q^91+5760*q^54+20378*q^45+ 427*q^71+243*q^74+65580*q^37+855*q^67+197022*q^29+17670*q^46+1167*q^65+611*q^69 +48779*q^39+2142*q^61+36319*q^41+101280*q^34+82*q^79+10067*q^50+1368*q^64+56564 *q^38+116472*q^33+4353*q^56+90*q^2+720*q^68+274438*q^26+430*q^3+1574*q^4+4666*q ^5+11605*q^6+133594*q^32+24836*q^7+46673*q^8+78391*q^9+1844*q^62+119597*q^10+ 168202*q^11+220888*q^12+163*q^76+273686*q^13+322554*q^14+363948*q^15+395371*q^ 16+1001*q^66+515*q^70+128*q^77+105*q^78+27113*q^43+11564*q^49+221734*q^28+ 415530*q^17+424369*q^18+422907*q^19+413016*q^20+247714*q^27+358*q^72+42090*q^40 +297*q^73+1593*q^63+152751*q^31+200*q^75+87831*q^35+64*q^80+13292*q^48+28*q^83+ 49*q^81+16*q^85+10*q^86+8*q^87+5*q^88+3*q^89+21*q^84+38*q^82+2855*q^59+75971*q^ 36+15305*q^47, 1+1219473*q^25+180641*q^42+1391207*q^22+164*q^90+1287816*q^24+ 1418463*q^21+1346212*q^23+14*q+28*q^97+52739*q^51+136726*q^44+40508*q^53+24087* q^57+814410*q^30+64*q^94+46199*q^52+21197*q^58+31197*q^55+16392*q^60+129*q^91+ 35546*q^54+119081*q^45+3882*q^71+2551*q^74+38*q^96+21*q^98+356735*q^37+6629*q^ 67+896988*q^29+103759*q^46+82*q^93+16*q^99+8597*q^65+5083*q^69+273121*q^39+ 14403*q^61+207572*q^41+520853*q^34+1208*q^79+60262*q^50+9787*q^64+10*q^100+ 312467*q^38+586844*q^33+27398*q^56+104*q^2+5806*q^68+1143911*q^26+533*q^3+2094* q^4+6670*q^5+17846*q^6+658202*q^32+41119*q^7+83191*q^8+150241*q^9+49*q^95+12660 *q^62+8*q^101+245818*q^10+369371*q^11+516042*q^12+1908*q^76+677482*q^13+843162* q^14+1001861*q^15+1143306*q^16+7549*q^66+4444*q^70+1644*q^77+1411*q^78+157173*q ^43+5*q^102+68927*q^49+980602*q^28+1259542*q^17+1345632*q^18+1399738*q^19+ 1422897*q^20+1063609*q^27+3378*q^72+238235*q^40+2939*q^73+3*q^103+11132*q^63+ 734462*q^31+2210*q^75+460661*q^35+2*q^104+1032*q^80+78906*q^48+q^105+625*q^83+ 879*q^81+438*q^85+365*q^86+301*q^87+247*q^88+202*q^89+5