Single 3-patterns

             There all together, 2, different equivalence classes

       For the equivalence class of patterns, {{[3, 2, 1]}, {[1, 2, 3]}}

              the member , {[3, 2, 1]}, has a scheme of depth , 2

                                  here it is:

  {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {}, {1}]}

                 Using  the scheme, the first, , 15, terms are

[1, 1+q, 1+2*q+2*q^2, 1+3*q+5*q^2+4*q^3+q^4, 1+4*q+9*q^2+12*q^3+10*q^4+4*q^5+2*
q^6, 1+5*q+14*q^2+25*q^3+31*q^4+26*q^5+16*q^6+9*q^7+4*q^8+q^9, 1+6*q+20*q^2+44*
q^3+70*q^4+82*q^5+74*q^6+54*q^7+38*q^8+22*q^9+12*q^10+4*q^11+2*q^12, 1+7*q+27*q
^2+70*q^3+134*q^4+196*q^5+227*q^6+215*q^7+179*q^8+139*q^9+99*q^10+64*q^11+38*q^
12+20*q^13+9*q^14+4*q^15+q^16, 1+8*q+35*q^2+104*q^3+231*q^4+400*q^5+558*q^6+644
*q^7+641*q^8+576*q^9+488*q^10+384*q^11+288*q^12+200*q^13+134*q^14+80*q^15+48*q^
16+24*q^17+12*q^18+4*q^19+2*q^20, 1+9*q+44*q^2+147*q^3+370*q^4+735*q^5+1190*q^6
+1609*q^7+1870*q^8+1931*q^9+1839*q^10+1651*q^11+1406*q^12+1143*q^13+891*q^14+
665*q^15+475*q^16+325*q^17+212*q^18+131*q^19+76*q^20+42*q^21+20*q^22+9*q^23+4*q
^24+q^25, 1+10*q+54*q^2+200*q^3+561*q^4+1252*q^5+2296*q^6+3542*q^7+4704*q^8+
5514*q^9+5872*q^10+5828*q^11+5488*q^12+4932*q^13+4272*q^14+3568*q^15+2894*q^16+
2264*q^17+1728*q^18+1266*q^19+906*q^20+620*q^21+414*q^22+258*q^23+160*q^24+90*q
^25+50*q^26+24*q^27+12*q^28+4*q^29+2*q^30, 1+11*q+65*q^2+264*q^3+815*q^4+2013*q
^5+4110*q^6+7099*q^7+10586*q^8+13907*q^9+16461*q^10+17958*q^11+18420*q^12+17995
*q^13+16882*q^14+15309*q^15+13481*q^16+11557*q^17+9664*q^18+7889*q^19+6286*q^20
+4890*q^21+3715*q^22+2751*q^23+1981*q^24+1391*q^25+947*q^26+624*q^27+397*q^28+
244*q^29+143*q^30+80*q^31+42*q^32+20*q^33+9*q^34+4*q^35+q^36, 1+12*q+77*q^2+340
*q^3+1144*q^4+3092*q^5+6939*q^6+13232*q^7+21849*q^8+31792*q^9+41516*q^10+49588*
q^11+55188*q^12+58104*q^13+58511*q^14+56772*q^15+53422*q^16+48940*q^17+43815*q^
18+38380*q^19+32982*q^20+27776*q^21+22986*q^22+18648*q^23+14872*q^24+11612*q^25
+8918*q^26+6692*q^27+4934*q^28+3544*q^29+2504*q^30+1712*q^31+1152*q^32+744*q^33
+472*q^34+284*q^35+170*q^36+92*q^37+50*q^38+24*q^39+12*q^40+4*q^41+2*q^42, 1+13
*q+90*q^2+429*q^3+1561*q^4+4576*q^5+11176*q^6+23274*q^7+42085*q^8+67137*q^9+
95973*q^10+124920*q^11+150466*q^12+170232*q^13+183141*q^14+189076*q^15+188636*q
^16+182869*q^17+172988*q^18+160178*q^19+145522*q^20+129926*q^21+114125*q^22+
98701*q^23+84089*q^24+70578*q^25+58371*q^26+47576*q^27+38195*q^28+30200*q^29+
23513*q^30+18018*q^31+13578*q^32+10062*q^33+7319*q^34+5225*q^35+3659*q^36+2507*
q^37+1676*q^38+1095*q^39+696*q^40+429*q^41+256*q^42+147*q^43+80*q^44+42*q^45+20
*q^46+9*q^47+4*q^48+q^49, 1+14*q+104*q^2+532*q^3+2080*q^4+6566*q^5+17314*q^6+
39038*q^7+76612*q^8+132836*q^9+206340*q^10+291134*q^11+378374*q^12+459178*q^13+
526758*q^14+577028*q^15+608386*q^16+621218*q^17+617436*q^18+599644*q^19+570856*
q^20+533902*q^21+491534*q^22+445934*q^23+399180*q^24+352682*q^25+307810*q^26+
265338*q^27+226092*q^28+190296*q^29+158338*q^30+130122*q^31+105712*q^32+84772*q
^33+67190*q^34+52520*q^35+40562*q^36+30864*q^37+23190*q^38+17130*q^39+12492*q^
40+8938*q^41+6306*q^42+4350*q^43+2960*q^44+1956*q^45+1276*q^46+802*q^47+498*q^
48+294*q^49+172*q^50+92*q^51+50*q^52+24*q^53+12*q^54+4*q^55+2*q^56]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, q*(q^2+2*q+2), q^2*(q^4+3*q^3+5*q^2+4*q+1), q^4*(q^6+4*q^5+9*q^4+12*q^
3+10*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+25*q^6+31*q^5+26*q^4+16*q^3+9*q^2+4*q+1)
, q^9*(q^12+6*q^11+20*q^10+44*q^9+70*q^8+82*q^7+74*q^6+54*q^5+38*q^4+22*q^3+12*
q^2+4*q+2), q^12*(q^16+7*q^15+27*q^14+70*q^13+134*q^12+196*q^11+227*q^10+215*q^
9+179*q^8+139*q^7+99*q^6+64*q^5+38*q^4+20*q^3+9*q^2+4*q+1), q^16*(q^20+8*q^19+
35*q^18+104*q^17+231*q^16+400*q^15+558*q^14+644*q^13+641*q^12+576*q^11+488*q^10
+384*q^9+288*q^8+200*q^7+134*q^6+80*q^5+48*q^4+24*q^3+12*q^2+4*q+2), q^20*(1+q^
25+147*q^22+9*q^24+370*q^21+44*q^23+4*q+9*q^2+20*q^3+42*q^4+76*q^5+131*q^6+212*
q^7+325*q^8+475*q^9+665*q^10+891*q^11+1143*q^12+1406*q^13+1651*q^14+1839*q^15+
1931*q^16+1870*q^17+1609*q^18+1190*q^19+735*q^20), q^25*(2+1252*q^25+4704*q^22+
2296*q^24+5514*q^21+3542*q^23+4*q+12*q^2+54*q^28+q^30+200*q^27+10*q^29+561*q^26
+24*q^3+50*q^4+90*q^5+160*q^6+258*q^7+414*q^8+620*q^9+906*q^10+1266*q^11+1728*q
^12+2264*q^13+2894*q^14+3568*q^15+4272*q^16+4932*q^17+5488*q^18+5828*q^19+5872*
q^20), q^30*(1+17958*q^25+16882*q^22+18420*q^24+15309*q^21+17995*q^23+4*q+q^36+
11*q^35+2013*q^31+65*q^34+264*q^33+815*q^32+9*q^2+10586*q^28+4110*q^30+13907*q^
27+7099*q^29+16461*q^26+20*q^3+42*q^4+80*q^5+143*q^6+244*q^7+397*q^8+624*q^9+
947*q^10+1391*q^11+1981*q^12+2751*q^13+3715*q^14+4890*q^15+6286*q^16+7889*q^17+
9664*q^18+11557*q^19+13481*q^20), q^36*(2+48940*q^25+32982*q^22+43815*q^24+
27776*q^21+38380*q^23+4*q+6939*q^36+13232*q^35+49588*q^31+21849*q^34+31792*q^33
+41516*q^32+77*q^40+340*q^39+1144*q^38+3092*q^37+q^42+12*q^41+12*q^2+58511*q^28
+55188*q^30+56772*q^27+58104*q^29+53422*q^26+24*q^3+50*q^4+92*q^5+170*q^6+284*q
^7+472*q^8+744*q^9+1152*q^10+1712*q^11+2504*q^12+3544*q^13+4934*q^14+6692*q^15+
8918*q^16+11612*q^17+14872*q^18+18648*q^19+22986*q^20), q^42*(1+84089*q^25+
47576*q^22+70578*q^24+38195*q^21+58371*q^23+4*q+170232*q^36+183141*q^35+172988*
q^31+189076*q^34+188636*q^33+182869*q^32+13*q^48+67137*q^40+95973*q^39+124920*q
^38+150466*q^37+23274*q^42+42085*q^41+9*q^2+129926*q^28+160178*q^30+114125*q^27
+145522*q^29+98701*q^26+20*q^3+42*q^4+80*q^5+147*q^6+256*q^7+429*q^8+696*q^9+
11176*q^43+90*q^47+429*q^46+1561*q^45+4576*q^44+1095*q^10+1676*q^11+2507*q^12+
3659*q^13+5225*q^14+7319*q^15+10062*q^16+q^49+13578*q^17+18018*q^18+23513*q^19+
30200*q^20), q^49*(2+130122*q^25+67190*q^22+105712*q^24+52520*q^21+84772*q^23+4
*q+570856*q^36+533902*q^35+352682*q^31+491534*q^34+445934*q^33+399180*q^32+
76612*q^48+608386*q^40+621218*q^39+617436*q^38+599644*q^37+526758*q^42+577028*q
^41+12*q^2+226092*q^28+307810*q^30+190296*q^27+265338*q^29+158338*q^26+24*q^3+
50*q^4+92*q^5+172*q^6+294*q^7+498*q^8+802*q^9+459178*q^43+132836*q^47+206340*q^
46+291134*q^45+378374*q^44+1276*q^10+1956*q^11+2960*q^12+4350*q^13+6306*q^14+
8938*q^15+12492*q^16+39038*q^49+17130*q^17+23190*q^18+30864*q^19+40562*q^20+
17314*q^50+6566*q^51+532*q^53+2080*q^52+14*q^55+104*q^54+q^56)]


         The number of permutations avoiding, {[3, 2, 1]}, is given by

[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440,

    9694845]

      The number of EVEN permutations avoiding, {[3, 2, 1]},  is given by

[1, 1, 3, 7, 22, 66, 217, 715, 2438, 8398, 29414, 104006, 371516, 1337220,

    4847637]

       The number of ODD permutations avoiding, {[3, 2, 1]},  is given by

[0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220,

    4847208]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 66, 212, 715, 2438, 8398, 29372, 104006, 371516,

    1337220, 4847208]

 ODD:, [0, 1, 3, 7, 20, 66, 217, 715, 2424, 8398, 29414, 104006, 371384,

    1337220, 4847637]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.095238095, 4.257575758,

    5.547785548, 6.957342657, 8.479226656, 10.10752560, 11.83717212,

    13.66375978, 15.58341096, 17.59267884, 19.68847320]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.359438326, 1.726009639,

    2.128872665, 2.565014470, 3.031963342, 3.527693055, 4.050520890,

    4.599027319, 5.171996670, 5.768373707, 6.387231597]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.84807, 2.97911, 4.53210, 6.57930,

    9.19280, 12.4446, 16.4067, 21.1511, 26.7495, 33.2741, 40.7967]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.05483742234,

    0.2001877412, 0.3028109183, 0.3760488113, 0.4295852896, 0.4697419753,

    0.5006136303, 0.5248801009, 0.5443338024, 0.5602036524, 0.5733410603]

Kurtosis: , [Float(undefined), 1.000000000, 1.846938776, 2.460732780,

    2.673976598, 2.793738671, 2.882969845, 2.955701316, 3.016487096,

    3.067720818, 3.111195509, 3.148318059, 3.180302311, 3.208002109,

    3.232170513]

                                end of this data



For the equivalence class of patterns,

    {{[1, 3, 2]}, {[2, 1, 3]}, {[2, 3, 1]}, {[3, 1, 2]}}

              the member , {[1, 3, 2]}, has a scheme of depth , 2

                                  here it is:

  {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {1}]}

                 Using  the scheme, the first, , 15, terms are

[1, 1+q, 1+q+2*q^2+q^3, 1+q+2*q^2+3*q^3+3*q^4+3*q^5+q^6, 1+q+2*q^2+3*q^3+5*q^4+
5*q^5+7*q^6+7*q^7+6*q^8+4*q^9+q^10, 1+q+2*q^2+3*q^3+5*q^4+7*q^5+9*q^6+11*q^7+14
*q^8+16*q^9+16*q^10+17*q^11+14*q^12+10*q^13+5*q^14+q^15, 1+q+2*q^2+3*q^3+5*q^4+
7*q^5+11*q^6+13*q^7+18*q^8+22*q^9+28*q^10+32*q^11+37*q^12+40*q^13+44*q^14+43*q^
15+40*q^16+35*q^17+25*q^18+15*q^19+6*q^20+q^21, 1+q+2*q^2+3*q^3+5*q^4+7*q^5+11*
q^6+15*q^7+20*q^8+26*q^9+34*q^10+42*q^11+53*q^12+63*q^13+73*q^14+85*q^15+96*q^
16+106*q^17+113*q^18+118*q^19+118*q^20+115*q^21+102*q^22+86*q^23+65*q^24+41*q^
25+21*q^26+7*q^27+q^28, 1+q+2*q^2+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+22*q^8+28*q^9
+38*q^10+48*q^11+63*q^12+77*q^13+97*q^14+116*q^15+139*q^16+162*q^17+190*q^18+
215*q^19+245*q^20+268*q^21+293*q^22+314*q^23+331*q^24+338*q^25+338*q^26+326*q^
27+303*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+
q+2*q^2+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+22*q^8+30*q^9+40*q^10+52*q^11+69*q^12+
87*q^13+111*q^14+138*q^15+171*q^16+207*q^17+249*q^18+295*q^19+348*q^20+405*q^21
+466*q^22+531*q^23+598*q^24+665*q^25+734*q^26+801*q^27+862*q^28+918*q^29+958*q^
30+990*q^31+1003*q^32+995*q^33+959*q^34+901*q^35+813*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+q+2*q^2+3*q^3+5*q^4+7*q^
5+11*q^6+15*q^7+22*q^8+30*q^9+42*q^10+54*q^11+73*q^12+93*q^13+121*q^14+152*q^15
+193*q^16+237*q^17+295*q^18+356*q^19+431*q^20+513*q^21+611*q^22+714*q^23+837*q^
24+964*q^25+1109*q^26+1257*q^27+1422*q^28+1588*q^29+1770*q^30+1947*q^31+2131*q^
32+2307*q^33+2481*q^34+2636*q^35+2784*q^36+2900*q^37+2990*q^38+3037*q^39+3039*q
^40+2992*q^41+2887*q^42+2717*q^43+2486*q^44+2203*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+q+2*q^2+3*q^3+
5*q^4+7*q^5+11*q^6+15*q^7+22*q^8+30*q^9+42*q^10+56*q^11+75*q^12+97*q^13+127*q^
14+162*q^15+207*q^16+259*q^17+325*q^18+400*q^19+493*q^20+598*q^21+722*q^22+864*
q^23+1030*q^24+1215*q^25+1428*q^26+1664*q^27+1929*q^28+2220*q^29+2540*q^30+2885
*q^31+3263*q^32+3664*q^33+4092*q^34+4543*q^35+5012*q^36+5494*q^37+5986*q^38+
6483*q^39+6975*q^40+7459*q^41+7914*q^42+8338*q^43+8714*q^44+9031*q^45+9269*q^46
+9427*q^47+9466*q^48+9391*q^49+9173*q^50+8818*q^51+8315*q^52+7668*q^53+6881*q^
54+5991*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+q+2*q^2+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+22*
q^8+30*q^9+42*q^10+56*q^11+77*q^12+99*q^13+131*q^14+168*q^15+217*q^16+273*q^17+
347*q^18+430*q^19+537*q^20+658*q^21+808*q^22+977*q^23+1183*q^24+1413*q^25+1689*
q^26+1997*q^27+2358*q^28+2757*q^29+3221*q^30+3727*q^31+4307*q^32+4934*q^33+5640
*q^34+6402*q^35+7249*q^36+8149*q^37+9138*q^38+10180*q^39+11305*q^40+12477*q^41+
13724*q^42+15006*q^43+16354*q^44+17715*q^45+19118*q^46+20511*q^47+21909*q^48+
23260*q^49+24581*q^50+25813*q^51+26959*q^52+27962*q^53+28808*q^54+29448*q^55+
29868*q^56+30009*q^57+29861*q^58+29382*q^59+28542*q^60+27338*q^61+25761*q^62+
23837*q^63+21591*q^64+19078*q^65+16371*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+q+2*q^2+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+22*q^8+30*q^9+42*q^10+56*q^11+77*q^12
+101*q^13+133*q^14+172*q^15+223*q^16+283*q^17+361*q^18+452*q^19+567*q^20+702*q^
21+868*q^22+1061*q^23+1297*q^24+1568*q^25+1890*q^26+2263*q^27+2701*q^28+3200*q^
29+3782*q^30+4440*q^31+5196*q^32+6048*q^33+7014*q^34+8092*q^35+9305*q^36+10646*
q^37+12139*q^38+13781*q^39+15591*q^40+17560*q^41+19713*q^42+22037*q^43+24547*q^
44+27237*q^45+30110*q^46+33160*q^47+36394*q^48+39788*q^49+43339*q^50+47033*q^51
+50849*q^52+54763*q^53+58754*q^54+62789*q^55+66834*q^56+70857*q^57+74795*q^58+
78616*q^59+82245*q^60+85634*q^61+88701*q^62+91404*q^63+93631*q^64+95331*q^65+
96392*q^66+96757*q^67+96331*q^68+95073*q^69+92889*q^70+89785*q^71+85706*q^72+
80687*q^73+74779*q^74+68086*q^75+60747*q^76+52949*q^77+44917*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+q+2*q^2+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+22
*q^8+30*q^9+42*q^10+56*q^11+77*q^12+101*q^13+135*q^14+174*q^15+227*q^16+289*q^
17+371*q^18+466*q^19+589*q^20+732*q^21+912*q^22+1121*q^23+1381*q^24+1680*q^25+
2046*q^26+2466*q^27+2970*q^28+3548*q^29+4235*q^30+5015*q^31+5933*q^32+6971*q^33
+8177*q^34+9534*q^35+11097*q^36+12842*q^37+14835*q^38+17046*q^39+19546*q^40+
22309*q^41+25407*q^42+28804*q^43+32586*q^44+36708*q^45+41256*q^46+46185*q^47+
51579*q^48+57383*q^49+63688*q^50+70427*q^51+77689*q^52+85401*q^53+93644*q^54+
102324*q^55+111531*q^56+121150*q^57+131254*q^58+141727*q^59+152622*q^60+163811*
q^61+175338*q^62+187037*q^63+198938*q^64+210874*q^65+222835*q^66+234633*q^67+
246251*q^68+257471*q^69+268246*q^70+278345*q^71+287671*q^72+295990*q^73+303177*
q^74+308983*q^75+313274*q^76+315815*q^77+316429*q^78+314915*q^79+311126*q^80+
304914*q^81+296205*q^82+284948*q^83+271150*q^84+254938*q^85+236451*q^86+216001*
q^87+193963*q^88+170831*q^89+147160*q^90+123598*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, q^3+q^2+2*q+1, q^6+q^5+2*q^4+3*q^3+3*q^2+3*q+1, q^10+q^9+2*q^8+3*q^7+5
*q^6+5*q^5+7*q^4+7*q^3+6*q^2+4*q+1, q^15+q^14+2*q^13+3*q^12+5*q^11+7*q^10+9*q^9
+11*q^8+14*q^7+16*q^6+16*q^5+17*q^4+14*q^3+10*q^2+5*q+1, q^21+q^20+2*q^19+3*q^
18+5*q^17+7*q^16+11*q^15+13*q^14+18*q^13+22*q^12+28*q^11+32*q^10+37*q^9+40*q^8+
44*q^7+43*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+15*q^
21+7*q^23+7*q+21*q^2+q^28+q^27+2*q^26+41*q^3+65*q^4+86*q^5+102*q^6+115*q^7+118*
q^8+118*q^9+113*q^10+106*q^11+96*q^12+85*q^13+73*q^14+63*q^15+53*q^16+42*q^17+
34*q^18+26*q^19+20*q^20, 1+48*q^25+97*q^22+63*q^24+116*q^21+77*q^23+8*q+q^36+q^
35+7*q^31+2*q^34+3*q^33+5*q^32+28*q^2+22*q^28+11*q^30+28*q^27+15*q^29+38*q^26+
63*q^3+112*q^4+167*q^5+219*q^6+268*q^7+303*q^8+326*q^9+338*q^10+338*q^11+331*q^
12+314*q^13+293*q^14+268*q^15+245*q^16+215*q^17+190*q^18+162*q^19+139*q^20, 1+
348*q^25+531*q^22+405*q^24+598*q^21+466*q^23+9*q+30*q^36+40*q^35+111*q^31+52*q^
34+69*q^33+87*q^32+7*q^40+11*q^39+15*q^38+22*q^37+3*q^42+5*q^41+36*q^2+207*q^28
+138*q^30+249*q^27+171*q^29+295*q^26+92*q^3+182*q^4+301*q^5+434*q^6+574*q^7+704
*q^8+813*q^9+2*q^43+q^45+q^44+901*q^10+959*q^11+995*q^12+1003*q^13+990*q^14+958
*q^15+918*q^16+862*q^17+801*q^18+734*q^19+665*q^20, 1+1770*q^25+2307*q^22+1947*
q^24+2481*q^21+2131*q^23+10*q+356*q^36+431*q^35+837*q^31+513*q^34+611*q^33+714*
q^32+15*q^48+152*q^40+193*q^39+237*q^38+295*q^37+93*q^42+121*q^41+45*q^2+1257*q
^28+964*q^30+1422*q^27+1109*q^29+1588*q^26+129*q^3+282*q^4+512*q^5+806*q^6+1149
*q^7+1515*q^8+1871*q^9+73*q^43+22*q^47+30*q^46+42*q^45+54*q^44+2203*q^10+2486*q
^11+2717*q^12+2887*q^13+2992*q^14+3039*q^15+3037*q^16+11*q^49+2990*q^17+2900*q^
18+2784*q^19+2636*q^20+7*q^50+5*q^51+2*q^53+3*q^52+q^55+q^54, 1+7459*q^25+8714*
q^22+7914*q^24+9031*q^21+8338*q^23+11*q+15*q^59+22*q^58+30*q^57+11*q^60+7*q^61+
5*q^62+2*q^64+3*q^63+q^66+2540*q^36+2885*q^35+4543*q^31+3263*q^34+3664*q^33+
4092*q^32+q^65+325*q^48+1428*q^40+1664*q^39+1929*q^38+2220*q^37+1030*q^42+1215*
q^41+55*q^2+5986*q^28+5012*q^30+6483*q^27+5494*q^29+6975*q^26+175*q^3+420*q^4+
831*q^5+1419*q^6+2174*q^7+3065*q^8+4032*q^9+864*q^43+400*q^47+493*q^46+598*q^45
+722*q^44+5027*q^10+5991*q^11+6881*q^12+7668*q^13+8315*q^14+8818*q^15+9173*q^16
+259*q^49+9391*q^17+9466*q^18+9427*q^19+9269*q^20+207*q^50+162*q^51+97*q^53+127
*q^52+56*q^55+75*q^54+42*q^56, 1+27962*q^25+29868*q^22+28808*q^24+30009*q^21+
29448*q^23+12*q+430*q^59+q^78+537*q^58+658*q^57+347*q^60+273*q^61+217*q^62+131*
q^64+168*q^63+77*q^66+13724*q^36+15006*q^35+20511*q^31+16354*q^34+17715*q^33+
19118*q^32+99*q^65+3221*q^48+9138*q^40+10180*q^39+11305*q^38+12477*q^37+7249*q^
42+8149*q^41+66*q^2+24581*q^28+21909*q^30+25813*q^27+23260*q^29+26959*q^26+231*
q^3+605*q^4+22*q^70+30*q^69+42*q^68+15*q^71+11*q^72+7*q^73+3*q^75+5*q^74+q^77+
1297*q^5+2389*q^6+56*q^67+3921*q^7+5889*q^8+8223*q^9+6402*q^43+3727*q^47+4307*q
^46+4934*q^45+5640*q^44+10829*q^10+13588*q^11+16371*q^12+19078*q^13+21591*q^14+
23837*q^15+25761*q^16+2757*q^49+27338*q^17+28542*q^18+29382*q^19+29861*q^20+
2358*q^50+1997*q^51+1413*q^53+1689*q^52+2*q^76+977*q^55+1183*q^54+808*q^56, 1+
77*q^79+96392*q^25+95073*q^22+96757*q^24+92889*q^21+96331*q^23+13*q+5196*q^59+
101*q^78+6048*q^58+7014*q^57+4440*q^60+3782*q^61+3200*q^62+2263*q^64+2701*q^63+
1568*q^66+2*q^89+42*q^81+56*q^80+30*q^82+22*q^83+62789*q^36+66834*q^35+82245*q^
31+70857*q^34+74795*q^33+78616*q^32+1890*q^65+3*q^88+5*q^87+22037*q^48+47033*q^
40+50849*q^39+54763*q^38+58754*q^37+39788*q^42+43339*q^41+78*q^2+91404*q^28+
85634*q^30+93631*q^27+88701*q^29+95331*q^26+298*q^3+847*q^4+702*q^70+868*q^69+
1061*q^68+567*q^71+452*q^72+361*q^73+223*q^75+283*q^74+133*q^77+1958*q^5+3872*q
^6+1297*q^67+6786*q^7+10826*q^8+15998*q^9+36394*q^43+24547*q^47+27237*q^46+
30110*q^45+33160*q^44+q^91+22205*q^10+29267*q^11+36924*q^12+15*q^84+7*q^86+11*q
^85+q^90+44917*q^13+52949*q^14+60747*q^15+68086*q^16+19713*q^49+74779*q^17+
80687*q^18+85706*q^19+89785*q^20+17560*q^50+15591*q^51+12139*q^53+13781*q^52+
172*q^76+9305*q^55+10646*q^54+8092*q^56, 1+2046*q^79+311126*q^25+284948*q^22+
304914*q^24+271150*q^21+296205*q^23+14*q+41256*q^59+2466*q^78+46185*q^58+51579*
q^57+36708*q^60+32586*q^61+28804*q^62+22309*q^64+25407*q^63+17046*q^66+227*q^89
+1381*q^81+1680*q^80+1121*q^82+912*q^83+257471*q^36+268246*q^35+303177*q^31+
278345*q^34+287671*q^33+295990*q^32+19546*q^65+2*q^103+q^104+289*q^88+371*q^87+
121150*q^48+210874*q^40+222835*q^39+234633*q^38+246251*q^37+187037*q^42+198938*
q^41+91*q^2+315815*q^28+308983*q^30+316429*q^27+313274*q^29+314915*q^26+377*q^3
+1157*q^4+9534*q^70+11097*q^69+12842*q^68+8177*q^71+6971*q^72+5933*q^73+4235*q^
75+5015*q^74+2970*q^77+2872*q^5+6073*q^6+14835*q^67+11330*q^7+19151*q^8+29875*q
^9+175338*q^43+131254*q^47+141727*q^46+152622*q^45+163811*q^44+135*q^91+43617*q
^10+60273*q^11+79499*q^12+732*q^84+466*q^86+589*q^85+174*q^90+77*q^93+101*q^92+
56*q^94+42*q^95+7*q^100+11*q^99+30*q^96+15*q^98+22*q^97+100812*q^13+123598*q^14
+147160*q^15+170831*q^16+111531*q^49+5*q^101+3*q^102+193963*q^17+216001*q^18+
236451*q^19+254938*q^20+102324*q^50+93644*q^51+77689*q^53+85401*q^52+3548*q^76+
63688*q^55+70427*q^54+57383*q^56+q^105]


         The number of permutations avoiding, {[1, 3, 2]}, is given by

[1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440,

    9694845]

      The number of EVEN permutations avoiding, {[1, 3, 2]},  is given by

[1, 1, 3, 7, 22, 66, 217, 715, 2438, 8398, 29414, 104006, 371516, 1337220,

    4847637]

       The number of ODD permutations avoiding, {[1, 3, 2]},  is given by

[0, 1, 2, 7, 20, 66, 212, 715, 2424, 8398, 29372, 104006, 371384, 1337220,

    4847208]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 66, 212, 715, 2438, 8398, 29372, 104006, 371516,

    1337220, 4847208]

 ODD:, [0, 1, 3, 7, 20, 66, 217, 715, 2424, 8398, 29414, 104006, 371384,

    1337220, 4847637]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.809523810, 8.984848485,

    12.90442890, 17.58531469, 23.04154669, 29.28494880, 36.32565577,

    44.17248043, 52.83317809, 62.31464232, 72.62305359]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.322130541, 3.087081706,

    3.919077000, 4.813231443, 5.765569794, 6.772782178, 7.832059987,

    8.940982365, 10.09743523, 11.29955195, 12.54566883]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.39229, 9.53007, 15.3592, 23.1672,

    33.2418, 45.8706, 61.3412, 79.9412, 101.958, 127.680, 157.394]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4531129845,

    -0.4956475433, -0.5255510953, -0.5477705263, -0.5649492236, -0.5786237560,

    -0.5897844021, -0.5990656802, -0.6069091099, -0.6136155295, -0.6194273240]

Kurtosis: , [Float(undefined), 1.000000000, 1.955621302, 2.384495064,

    2.630998684, 2.791640199, 2.904677428, 2.988658002, 3.053517896,

    3.105124715, 3.147172309, 3.182082192, 3.211583160, 3.236769084,

    3.258576659]

                                end of this data



                         Out of a total of , 2, cases

                      2, were successful and , 0, failed

                               Success Rate: , 1.

                             Here are the failures

                                       {}