Single 4-pattern

             There all together, 7, different equivalence classes

    For the equivalence class of patterns, {{[1, 2, 3, 4]}, {[4, 3, 2, 1]}}

             the member , {[1, 2, 3, 4]}, has a scheme of depth , 4

                                  here it is:

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

    [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {4}], [[2, 1], {}, {1}],

    [[2, 3, 1], {}, {}], [[3, 4, 1, 2], {}, {1, 2}], [[3, 4, 2, 1], {}, {3}],

    [[1, 2, 3], {[0, 0, 0, 1]}, {3}], [[1, 3, 2], {}, {2}],

    [[2, 4, 1, 3], {}, {1, 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+6*q^3+5*q^4+3*q^5+q^6, 3*q^2+11*q^3+18*q^4+
22*q^5+20*q^6+15*q^7+9*q^8+4*q^9+q^10, q^3+10*q^4+26*q^5+52*q^6+77*q^7+91*q^8+
87*q^9+71*q^10+49*q^11+29*q^12+14*q^13+5*q^14+q^15, 3*q^5+15*q^6+48*q^7+106*q^8
+185*q^9+283*q^10+368*q^11+413*q^12+401*q^13+341*q^14+255*q^15+169*q^16+98*q^17
+49*q^18+20*q^19+6*q^20+q^21, 3*q^7+15*q^8+56*q^9+140*q^10+299*q^11+542*q^12+
867*q^13+1237*q^14+1606*q^15+1892*q^16+2018*q^17+1938*q^18+1680*q^19+1316*q^20+
933*q^21+597*q^22+343*q^23+174*q^24+76*q^25+27*q^26+7*q^27+q^28, q^9+10*q^10+38
*q^11+118*q^12+297*q^13+634*q^14+1198*q^15+2053*q^16+3233*q^17+4689*q^18+6315*q
^19+7916*q^20+9284*q^21+10167*q^22+10356*q^23+9767*q^24+8507*q^25+6832*q^26+
5057*q^27+3444*q^28+2151*q^29+1224*q^30+628*q^31+285*q^32+111*q^33+35*q^34+8*q^
35+q^36, 3*q^12+15*q^13+60*q^14+174*q^15+447*q^16+988*q^17+1982*q^18+3633*q^19+
6185*q^20+9826*q^21+14720*q^22+20832*q^23+27944*q^24+35563*q^25+43007*q^26+
49431*q^27+54020*q^28+56032*q^29+54987*q^30+50849*q^31+44159*q^32+35922*q^33+
27320*q^34+19389*q^35+12810*q^36+7849*q^37+4435*q^38+2291*q^39+1068*q^40+440*q^
41+155*q^42+44*q^43+9*q^44+q^45, 3*q^15+15*q^16+60*q^17+182*q^18+479*q^19+1120*
q^20+2390*q^21+4686*q^22+8568*q^23+14758*q^24+24071*q^25+37361*q^26+55354*q^27+
78560*q^28+107014*q^29+140170*q^30+176631*q^31+214234*q^32+250079*q^33+280933*q
^34+303516*q^35+315065*q^36+313676*q^37+298729*q^38+271267*q^39+234127*q^40+
191522*q^41+148141*q^42+108119*q^43+74289*q^44+47924*q^45+28921*q^46+16245*q^47
+8434*q^48+4007*q^49+1717*q^50+649*q^51+209*q^52+54*q^53+10*q^54+q^55, q^18+10*
q^19+38*q^20+130*q^21+363*q^22+912*q^23+2062*q^24+4348*q^25+8534*q^26+15827*q^
27+27821*q^28+46705*q^29+75094*q^30+116144*q^31+173129*q^32+249362*q^33+347500*
q^34+469272*q^35+614649*q^36+781587*q^37+965241*q^38+1157984*q^39+1349285*q^40+
1526582*q^41+1676226*q^42+1785051*q^43+1841833*q^44+1838982*q^45+1773712*q^46+
1648936*q^47+1473703*q^48+1262733*q^49+1034594*q^50+808625*q^51+601581*q^52+
425091*q^53+284649*q^54+180134*q^55+107366*q^56+60009*q^57+31269*q^58+15071*q^
59+6646*q^60+2640*q^61+923*q^62+274*q^63+65*q^64+11*q^65+q^66, 3*q^22+15*q^23+
60*q^24+186*q^25+513*q^26+1260*q^27+2862*q^28+6046*q^29+12051*q^30+22798*q^31+
41158*q^32+71215*q^33+118620*q^34+190773*q^35+296948*q^36+448326*q^37+657648*q^
38+938788*q^39+1305639*q^40+1770915*q^41+2344241*q^42+3030634*q^43+3828137*q^44
+4726363*q^45+5704727*q^46+6731992*q^47+7765959*q^48+8755395*q^49+9642762*q^50+
10369051*q^51+10878989*q^52+11127062*q^53+11082877*q^54+10735893*q^55+10098328*
q^56+9205871*q^57+8116128*q^58+6903994*q^59+5653482*q^60+4446648*q^61+3352291*q
^62+2417485*q^63+1664198*q^64+1091174*q^65+679693*q^66+400972*q^67+223164*q^68+
116603*q^69+56832*q^70+25620*q^71+10559*q^72+3914*q^73+1274*q^74+351*q^75+77*q^
76+12*q^77+q^78, 3*q^26+15*q^27+60*q^28+186*q^29+521*q^30+1292*q^31+2992*q^32+
6446*q^33+13161*q^34+25533*q^35+47501*q^36+84891*q^37+146469*q^38+244466*q^39+
395962*q^40+623429*q^41+956206*q^42+1430487*q^43+2090335*q^44+2986587*q^45+
4176596*q^46+5721139*q^47+7682095*q^48+10116840*q^49+13073567*q^50+16583537*q^
51+20655001*q^52+25265016*q^53+30354397*q^54+35821925*q^55+41523054*q^56+
47269904*q^57+52837060*q^58+57970471*q^59+62402862*q^60+65872235*q^61+68143853*
q^62+69032145*q^63+68421228*q^64+66281063*q^65+62677730*q^66+57775430*q^67+
51828768*q^68+45165638*q^69+38160144*q^70+31196745*q^71+24629000*q^72+18740809*
q^73+13718971*q^74+9643534*q^75+6496668*q^76+4185721*q^77+2572984*q^78+1504774*
q^79+834445*q^80+436910*q^81+214857*q^82+98560*q^83+41796*q^84+16187*q^85+5629*
q^86+1715*q^87+441*q^88+90*q^89+13*q^90+q^91, q^30+10*q^31+38*q^32+130*q^33+375
*q^34+978*q^35+2332*q^36+5218*q^37+10998*q^38+22100*q^39+42502*q^40+78685*q^41+
140686*q^42+243767*q^43+410209*q^44+671926*q^45+1073286*q^46+1674532*q^47+
2555307*q^48+3818363*q^49+5593041*q^50+8038004*q^51+11343005*q^52+15728355*q^53
+21442752*q^54+28757070*q^55+37955729*q^56+49322650*q^57+63124739*q^58+79589039
*q^59+98878828*q^60+121064905*q^61+146099392*q^62+173788303*q^63+203771371*q^64
+235505634*q^65+268262098*q^66+301131297*q^67+333046017*q^68+362817327*q^69+
389189728*q^70+410909827*q^71+426809015*q^72+435890665*q^73+437416585*q^74+
430982979*q^75+416578466*q^76+394616696*q^77+365937279*q^78+331771064*q^79+
293667291*q^80+253386860*q^81+212766965*q^82+173567745*q^83+137316899*q^84+
105177104*q^85+77861639*q^86+55615771*q^87+38264744*q^88+25313004*q^89+16068771
*q^90+9767014*q^91+5669944*q^92+3134211*q^93+1643734*q^94+814223*q^95+378810*q^
96+164339*q^97+65859*q^98+24075*q^99+7889*q^100+2260*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, 3*q^5+5*q^4+6*q^3+5*q^2+3*q+1, 3*q^8+11*q^7+18*q^6+22
*q^5+20*q^4+15*q^3+9*q^2+4*q+1, q^12+10*q^11+26*q^10+52*q^9+77*q^8+91*q^7+87*q^
6+71*q^5+49*q^4+29*q^3+14*q^2+5*q+1, 3*q^16+15*q^15+48*q^14+106*q^13+185*q^12+
283*q^11+368*q^10+413*q^9+401*q^8+341*q^7+255*q^6+169*q^5+98*q^4+49*q^3+20*q^2+
6*q+1, 3*q^21+15*q^20+56*q^19+140*q^18+299*q^17+542*q^16+867*q^15+1237*q^14+
1606*q^13+1892*q^12+2018*q^11+1938*q^10+1680*q^9+1316*q^8+933*q^7+597*q^6+343*q
^5+174*q^4+76*q^3+27*q^2+7*q+1, 1+38*q^25+634*q^22+118*q^24+1198*q^21+297*q^23+
8*q+35*q^2+10*q^26+111*q^3+285*q^4+628*q^5+1224*q^6+2151*q^7+3444*q^8+5057*q^9+
6832*q^10+8507*q^11+9767*q^12+10356*q^13+10167*q^14+9284*q^15+7916*q^16+6315*q^
17+4689*q^18+3233*q^19+2053*q^20+q^27, 1+6185*q^25+20832*q^22+9826*q^24+27944*q
^21+14720*q^23+9*q+174*q^30+447*q^29+3*q^33+44*q^2+3633*q^26+155*q^3+440*q^4+
1068*q^5+2291*q^6+15*q^32+4435*q^7+7849*q^8+12810*q^9+19389*q^10+27320*q^11+
35922*q^12+44159*q^13+50849*q^14+54987*q^15+56032*q^16+988*q^28+54020*q^17+
49431*q^18+43007*q^19+35563*q^20+1982*q^27+60*q^31, 1+140170*q^25+250079*q^22+
176631*q^24+280933*q^21+214234*q^23+10*q+24071*q^30+182*q^37+37361*q^29+15*q^39
+2390*q^34+60*q^38+4686*q^33+54*q^2+107014*q^26+209*q^3+649*q^4+1717*q^5+4007*q
^6+8568*q^32+8434*q^7+16245*q^8+28921*q^9+47924*q^10+74289*q^11+108119*q^12+
148141*q^13+191522*q^14+234127*q^15+271267*q^16+55354*q^28+298729*q^17+313676*q
^18+315065*q^19+303516*q^20+78560*q^27+3*q^40+14758*q^31+1120*q^35+479*q^36, 1+
1526582*q^25+2062*q^42+1841833*q^22+1676226*q^24+1838982*q^21+1785051*q^23+11*q
+363*q^44+614649*q^30+130*q^45+46705*q^37+781587*q^29+38*q^46+15827*q^39+4348*q
^41+173129*q^34+27821*q^38+249362*q^33+65*q^2+1349285*q^26+274*q^3+923*q^4+2640
*q^5+6646*q^6+347500*q^32+15071*q^7+31269*q^8+60009*q^9+107366*q^10+180134*q^11
+284649*q^12+425091*q^13+601581*q^14+808625*q^15+1034594*q^16+912*q^43+965241*q
^28+1262733*q^17+1473703*q^18+1648936*q^19+1773712*q^20+1157984*q^27+8534*q^40+
469272*q^31+116144*q^35+q^48+75094*q^36+10*q^47, 1+11127062*q^25+296948*q^42+
10098328*q^22+11082877*q^24+9205871*q^21+10735893*q^23+12*q+1260*q^51+118620*q^
44+186*q^53+7765959*q^30+513*q^52+15*q^55+60*q^54+71215*q^45+1770915*q^37+
8755395*q^29+41158*q^46+938788*q^39+448326*q^41+3828137*q^34+2862*q^50+1305639*
q^38+4726363*q^33+3*q^56+77*q^2+10878989*q^26+351*q^3+1274*q^4+3914*q^5+10559*q
^6+5704727*q^32+25620*q^7+56832*q^8+116603*q^9+223164*q^10+400972*q^11+679693*q
^12+1091174*q^13+1664198*q^14+2417485*q^15+3352291*q^16+190773*q^43+6046*q^49+
9642762*q^28+4446648*q^17+5653482*q^18+6903994*q^19+8116128*q^20+10369051*q^27+
657648*q^40+6731992*q^31+3030634*q^35+12051*q^48+2344241*q^36+22798*q^47, 1+
62677730*q^25+10116840*q^42+45165638*q^22+57775430*q^24+38160144*q^21+51828768*
q^23+13*q+395962*q^51+5721139*q^44+146469*q^53+13161*q^57+65872235*q^30+244466*
q^52+6446*q^58+47501*q^55+1292*q^60+84891*q^54+4176596*q^45+30354397*q^37+
68143853*q^29+2986587*q^46+3*q^65+20655001*q^39+521*q^61+13073567*q^41+47269904
*q^34+623429*q^50+15*q^64+25265016*q^38+52837060*q^33+25533*q^56+90*q^2+
66281063*q^26+441*q^3+1715*q^4+5629*q^5+16187*q^6+57970471*q^32+41796*q^7+98560
*q^8+214857*q^9+186*q^62+436910*q^10+834445*q^11+1504774*q^12+2572984*q^13+
4185721*q^14+6496668*q^15+9643534*q^16+7682095*q^43+956206*q^49+69032145*q^28+
13718971*q^17+18740809*q^18+24629000*q^19+31196745*q^20+68421228*q^27+16583537*
q^40+60*q^63+62402862*q^31+41523054*q^35+1430487*q^48+2992*q^59+35821925*q^36+
2090335*q^47, 1+293667291*q^25+173788303*q^42+173567745*q^22+253386860*q^24+
137316899*q^21+212766965*q^23+14*q+21442752*q^51+121064905*q^44+11343005*q^53+
2555307*q^57+430982979*q^30+15728355*q^52+1674532*q^58+5593041*q^55+671926*q^60
+8038004*q^54+98878828*q^45+375*q^71+10*q^74+333046017*q^37+10998*q^67+
416578466*q^29+79589039*q^46+42502*q^65+2332*q^69+268262098*q^39+410209*q^61+
203771371*q^41+410909827*q^34+28757070*q^50+78685*q^64+301131297*q^38+426809015
*q^33+3818363*q^56+104*q^2+5218*q^68+331771064*q^26+545*q^3+2260*q^4+7889*q^5+
24075*q^6+435890665*q^32+65859*q^7+164339*q^8+378810*q^9+243767*q^62+814223*q^
10+1643734*q^11+3134211*q^12+5669944*q^13+9767014*q^14+16068771*q^15+25313004*q
^16+22100*q^66+978*q^70+146099392*q^43+37955729*q^49+394616696*q^28+38264744*q^
17+55615771*q^18+77861639*q^19+105177104*q^20+365937279*q^27+130*q^72+235505634
*q^40+38*q^73+140686*q^63+437416585*q^31+q^75+389189728*q^35+49322650*q^48+
1073286*q^59+362817327*q^36+63124739*q^47]


        The number of permutations avoiding, {[1, 2, 3, 4]}, is given by

[1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705,

    167078577, 1148208090, 8026793118]

     The number of EVEN permutations avoiding, {[1, 2, 3, 4]},  is given by

[1, 1, 3, 11, 51, 258, 1382, 7879, 47175, 293311, 1881661, 12396285, 83539221,

    574104369, 4013396883]

     The number of ODD permutations avoiding, {[1, 2, 3, 4]},  is given by

[0, 1, 3, 12, 52, 255, 1379, 7888, 47184, 293279, 1881629, 12396420, 83539356,

    574103721, 4013396235]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 51, 255, 1379, 7879, 47175, 293279, 1881629, 12396285,

    83539221, 574103721, 4013396235]

 ODD:, [0, 1, 3, 12, 52, 258, 1382, 7888, 47184, 293311, 1881661, 12396420,

    83539356, 574104369, 4013396883]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.130434783, 5.475728155, 8.575048733,

    12.44947483, 17.11359168, 22.57902267, 28.85556351, 35.95167447,

    43.87477034, 52.63142150, 62.22750302, 72.66830918]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.361084856, 1.750698392, 2.158152694,

    2.596434307, 3.068854847, 3.575162332, 4.114019106, 4.683861794,

    5.283178900, 5.910589898, 6.564857216, 7.244876437]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.85255, 3.06494, 4.65762, 6.74147, 9.41787,

    12.7818, 16.9252, 21.9386, 27.9120, 34.9351, 43.0974, 52.4882]

Skewness: , [Float(undefined), 0., 0., 0.1771898400, 0.2087338158, 0.1604679439,

    0.08888935841, 0.01915779345, -0.04103329638, -0.09084499636, -0.1315612499,

    -0.1648536337, -0.1922347420, -0.2149412758, -0.2339466875]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.245929890,

    2.502688671, 2.672980523, 2.764953570, 2.815383161, 2.847159458,

    2.870774714, 2.890512627, 2.908007835, 2.923845481, 2.938339329,

    2.951632349]

                                end of this data



For the equivalence class of patterns,

    {{[1, 2, 4, 3]}, {[3, 4, 2, 1]}, {[2, 1, 3, 4]}, {[4, 3, 1, 2]}}

             the member , {[1, 2, 4, 3]}, has a scheme of depth , 4

                                  here it is:

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

    [[2, 3, 1], {}, {}], [[3, 4, 1, 2], {}, {1, 2}],

    [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {4}],

    [[3, 4, 2, 1], {}, {3}], [[1, 3, 2], {}, {2}], [[2, 4, 1, 3], {}, {1, 2}]}

                 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+5*q^4+3*q^5+q^6, 1+2*q+5*q^2+10*q^3
+16*q^4+20*q^5+20*q^6+15*q^7+9*q^8+4*q^9+q^10, 1+2*q+5*q^2+10*q^3+20*q^4+33*q^5
+50*q^6+67*q^7+79*q^8+80*q^9+68*q^10+49*q^11+29*q^12+14*q^13+5*q^14+q^15, 1+2*q
+5*q^2+10*q^3+20*q^4+37*q^5+62*q^6+98*q^7+149*q^8+207*q^9+267*q^10+319*q^11+349
*q^12+347*q^13+308*q^14+241*q^15+165*q^16+98*q^17+49*q^18+20*q^19+6*q^20+q^21,
1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+66*q^6+110*q^7+177*q^8+273*q^9+405*q^10+576*q^
11+784*q^12+1016*q^13+1254*q^14+1464*q^15+1612*q^16+1669*q^17+1612*q^18+1438*q^
19+1173*q^20+866*q^21+574*q^22+338*q^23+174*q^24+76*q^25+27*q^26+7*q^27+q^28, 1
+2*q+5*q^2+10*q^3+20*q^4+37*q^5+66*q^6+114*q^7+189*q^8+301*q^9+467*q^10+704*q^
11+1035*q^12+1476*q^13+2046*q^14+2754*q^15+3598*q^16+4553*q^17+5569*q^18+6567*q
^19+7448*q^20+8104*q^21+8433*q^22+8364*q^23+7867*q^24+6973*q^25+5780*q^26+4444*
q^27+3146*q^28+2034*q^29+1190*q^30+622*q^31+285*q^32+111*q^33+35*q^34+8*q^35+q^
36, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+66*q^6+114*q^7+193*q^8+313*q^9+495*q^10+
766*q^11+1159*q^12+1716*q^13+2490*q^14+3537*q^15+4926*q^16+6729*q^17+9006*q^18+
11812*q^19+15167*q^20+19039*q^21+23353*q^22+27956*q^23+32616*q^24+37027*q^25+
40826*q^26+43631*q^27+45108*q^28+44997*q^29+43183*q^30+39726*q^31+34870*q^32+
29045*q^33+22819*q^34+16802*q^35+11520*q^36+7305*q^37+4249*q^38+2244*q^39+1061*
q^40+440*q^41+155*q^42+44*q^43+9*q^44+q^45, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+66
*q^6+114*q^7+193*q^8+317*q^9+507*q^10+794*q^11+1221*q^12+1840*q^13+2726*q^14+
3971*q^15+5696*q^16+8043*q^17+11184*q^18+15321*q^19+20687*q^20+27529*q^21+36099
*q^22+46641*q^23+59353*q^24+74353*q^25+91652*q^26+111086*q^27+132293*q^28+
154676*q^29+177382*q^30+199331*q^31+219248*q^32+235731*q^33+247394*q^34+253024*
q^35+251739*q^36+243144*q^37+227428*q^38+205412*q^39+178531*q^40+148709*q^41+
118165*q^42+89125*q^43+63473*q^44+42449*q^45+26499*q^46+15335*q^47+8157*q^48+
3945*q^49+1709*q^50+649*q^51+209*q^52+54*q^53+10*q^54+q^55, 1+2*q+5*q^2+10*q^3+
20*q^4+37*q^5+66*q^6+114*q^7+193*q^8+317*q^9+511*q^10+806*q^11+1249*q^12+1902*q
^13+2850*q^14+4207*q^15+6126*q^16+8803*q^17+12488*q^18+17495*q^19+24213*q^20+
33122*q^21+44794*q^22+59906*q^23+79230*q^24+103641*q^25+134083*q^26+171555*q^27
+217048*q^28+271499*q^29+335688*q^30+410134*q^31+494962*q^32+589772*q^33+693500
*q^34+804311*q^35+919526*q^36+1035579*q^37+1148091*q^38+1251986*q^39+1341769*q^
40+1411855*q^41+1457026*q^42+1472926*q^43+1456605*q^44+1406964*q^45+1325086*q^
46+1214354*q^47+1080346*q^48+930491*q^49+773438*q^50+618241*q^51+473376*q^52+
345733*q^53+239789*q^54+157185*q^55+96879*q^56+55810*q^57+29840*q^58+14678*q^59
+6567*q^60+2631*q^61+923*q^62+274*q^63+65*q^64+11*q^65+q^66, 1+2*q+5*q^2+10*q^3
+20*q^4+37*q^5+66*q^6+114*q^7+193*q^8+317*q^9+511*q^10+810*q^11+1261*q^12+1930*
q^13+2912*q^14+4331*q^15+6362*q^16+9233*q^17+13244*q^18+18789*q^19+26377*q^20+
36648*q^21+50420*q^22+68709*q^23+92758*q^24+124090*q^25+164534*q^26+216250*q^27
+281770*q^28+364004*q^29+466227*q^30+592099*q^31+745572*q^32+930794*q^33+
1151999*q^34+1413285*q^35+1718350*q^36+2070224*q^37+2470834*q^38+2920610*q^39+
3418050*q^40+3959218*q^41+4537355*q^42+5142579*q^43+5761680*q^44+6378180*q^45+
6972663*q^46+7523271*q^47+8006641*q^48+8399070*q^49+8677963*q^50+8823558*q^51+
8820675*q^52+8660359*q^53+8341331*q^54+7870909*q^55+7265275*q^56+6549008*q^57+
5753727*q^58+4915976*q^59+4074426*q^60+3266601*q^61+2525499*q^62+1876523*q^63+
1335205*q^64+906276*q^65+584389*q^66+356377*q^67+204488*q^68+109746*q^69+54694*
q^70+25083*q^71+10461*q^72+3904*q^73+1274*q^74+351*q^75+77*q^76+12*q^77+q^78, 1
+2*q+5*q^2+10*q^3+20*q^4+37*q^5+66*q^6+114*q^7+193*q^8+317*q^9+511*q^10+810*q^
11+1265*q^12+1942*q^13+2940*q^14+4393*q^15+6486*q^16+9469*q^17+13674*q^18+19545
*q^19+27667*q^20+38802*q^21+53936*q^22+74335*q^23+101598*q^24+137743*q^25+
185288*q^26+247359*q^27+327776*q^28+431195*q^29+563200*q^30+730466*q^31+940865*
q^32+1203588*q^33+1529242*q^34+1929952*q^35+2419341*q^36+3012544*q^37+3726064*q
^38+4577570*q^39+5585515*q^40+6768687*q^41+8145467*q^42+9733038*q^43+11546262*q
^44+13596432*q^45+15889846*q^46+18426272*q^47+21197261*q^48+24184585*q^49+
27358767*q^50+30677875*q^51+34086837*q^52+37517295*q^53+40888238*q^54+44107626*
q^55+47074935*q^56+49684722*q^57+51831253*q^58+53413813*q^59+54342743*q^60+
54545792*q^61+53974306*q^62+52608755*q^63+50462964*q^64+47586421*q^65+44064325*
q^66+40014936*q^67+35584050*q^68+30936825*q^69+26247330*q^70+21686582*q^71+
17410144*q^72+13546518*q^73+10187779*q^74+7383802*q^75+5141204*q^76+3427501*q^
77+2179971*q^78+1317555*q^79+753372*q^80+405472*q^81+204174*q^82+95482*q^83+
41084*q^84+16068*q^85+5618*q^86+1715*q^87+441*q^88+90*q^89+13*q^90+q^91, 1+2*q+
5*q^2+10*q^3+20*q^4+37*q^5+66*q^6+114*q^7+193*q^8+317*q^9+511*q^10+810*q^11+
1265*q^12+1946*q^13+2952*q^14+4421*q^15+6548*q^16+9593*q^17+13910*q^18+19975*q^
19+28423*q^20+40092*q^21+56086*q^22+77841*q^23+107214*q^24+146583*q^25+198978*q
^26+268242*q^27+359206*q^28+477898*q^29+631784*q^30+830063*q^31+1083974*q^32+
1407163*q^33+1816073*q^34+2330383*q^35+2973449*q^36+3772810*q^37+4760651*q^38+
5974327*q^39+7456765*q^40+9256871*q^41+11429791*q^42+14037085*q^43+17146637*q^
44+20832389*q^45+25173650*q^46+30254132*q^47+36160375*q^48+42979736*q^49+
50797681*q^50+59694507*q^51+69741281*q^52+80995206*q^53+93494308*q^54+107251678
*q^55+122249341*q^56+138432037*q^57+155701251*q^58+173909863*q^59+192857854*q^
60+212289564*q^61+231893152*q^62+251302587*q^63+270102764*q^64+287837936*q^65+
304023803*q^66+318163104*q^67+329764584*q^68+338364583*q^69+343550613*q^70+
344985620*q^71+342431639*q^72+335771306*q^73+325025301*q^74+310363947*q^75+
292111073*q^76+270738860*q^77+246852877*q^78+221167219*q^79+194470300*q^80+
167583034*q^81+141311945*q^82+116400794*q^83+93484990*q^84+73053236*q^85+
55420897*q^86+40718615*q^87+28898308*q^88+19756557*q^89+12972587*q^90+8155234*q
^91+4891380*q^92+2788253*q^93+1503931*q^94+763644*q^95+362790*q^96+160045*q^97+
64938*q^98+23933*q^99+7877*q^100+2260*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, q^6+2*q^5+5*q^4+6*q^3+5*q^2+3*q+1, q^10+2*q^9+5*q^8+
10*q^7+16*q^6+20*q^5+20*q^4+15*q^3+9*q^2+4*q+1, q^15+2*q^14+5*q^13+10*q^12+20*q
^11+33*q^10+50*q^9+67*q^8+79*q^7+80*q^6+68*q^5+49*q^4+29*q^3+14*q^2+5*q+1, q^21
+2*q^20+5*q^19+10*q^18+20*q^17+37*q^16+62*q^15+98*q^14+149*q^13+207*q^12+267*q^
11+319*q^10+349*q^9+347*q^8+308*q^7+241*q^6+165*q^5+98*q^4+49*q^3+20*q^2+6*q+1,
1+10*q^25+66*q^22+20*q^24+110*q^21+37*q^23+7*q+27*q^2+5*q^26+76*q^3+174*q^4+338
*q^5+574*q^6+866*q^7+1173*q^8+1438*q^9+1612*q^10+1669*q^11+1612*q^12+1464*q^13+
1254*q^14+1016*q^15+784*q^16+q^28+576*q^17+405*q^18+273*q^19+177*q^20+2*q^27, 1
+704*q^25+2046*q^22+1035*q^24+2754*q^21+1476*q^23+8*q+66*q^30+114*q^29+5*q^34+
10*q^33+35*q^2+467*q^26+111*q^3+285*q^4+622*q^5+1190*q^6+20*q^32+2034*q^7+3146*
q^8+4444*q^9+5780*q^10+6973*q^11+7867*q^12+8364*q^13+8433*q^14+8104*q^15+7448*q
^16+189*q^28+6567*q^17+5569*q^18+4553*q^19+3598*q^20+301*q^27+37*q^31+2*q^35+q^
36, 1+15167*q^25+10*q^42+27956*q^22+19039*q^24+32616*q^21+23353*q^23+9*q+2*q^44
+3537*q^30+q^45+193*q^37+4926*q^29+66*q^39+20*q^41+766*q^34+114*q^38+1159*q^33+
44*q^2+11812*q^26+155*q^3+440*q^4+1061*q^5+2244*q^6+1716*q^32+4249*q^7+7305*q^8
+11520*q^9+16802*q^10+22819*q^11+29045*q^12+34870*q^13+39726*q^14+43183*q^15+
44997*q^16+5*q^43+6729*q^28+45108*q^17+43631*q^18+40826*q^19+37027*q^20+9006*q^
27+37*q^40+2490*q^31+495*q^35+313*q^36, 1+177382*q^25+1840*q^42+235731*q^22+
199331*q^24+247394*q^21+219248*q^23+10*q+20*q^51+794*q^44+5*q^53+74353*q^30+10*
q^52+q^55+2*q^54+507*q^45+11184*q^37+91652*q^29+317*q^46+5696*q^39+2726*q^41+
27529*q^34+37*q^50+8043*q^38+36099*q^33+54*q^2+154676*q^26+209*q^3+649*q^4+1709
*q^5+3945*q^6+46641*q^32+8157*q^7+15335*q^8+26499*q^9+42449*q^10+63473*q^11+
89125*q^12+118165*q^13+148709*q^14+178531*q^15+205412*q^16+1221*q^43+66*q^49+
111086*q^28+227428*q^17+243144*q^18+251739*q^19+253024*q^20+132293*q^27+3971*q^
40+59353*q^31+20687*q^35+114*q^48+15321*q^36+193*q^47, 1+1411855*q^25+79230*q^
42+1456605*q^22+1457026*q^24+1406964*q^21+1472926*q^23+11*q+4207*q^51+44794*q^
44+1902*q^53+317*q^57+919526*q^30+2850*q^52+193*q^58+806*q^55+66*q^60+1249*q^54
+33122*q^45+271499*q^37+1035579*q^29+24213*q^46+2*q^65+171555*q^39+37*q^61+
103641*q^41+494962*q^34+6126*q^50+5*q^64+217048*q^38+589772*q^33+511*q^56+65*q^
2+1341769*q^26+274*q^3+923*q^4+2631*q^5+6567*q^6+693500*q^32+14678*q^7+29840*q^
8+55810*q^9+20*q^62+96879*q^10+157185*q^11+239789*q^12+345733*q^13+473376*q^14+
618241*q^15+773438*q^16+q^66+59906*q^43+8803*q^49+1148091*q^28+930491*q^17+
1080346*q^18+1214354*q^19+1325086*q^20+1251986*q^27+134083*q^40+10*q^63+804311*
q^31+410134*q^35+12488*q^48+114*q^59+335688*q^36+17495*q^47, 1+8660359*q^25+
1718350*q^42+7265275*q^22+8341331*q^24+6549008*q^21+7870909*q^23+12*q+216250*q^
51+1151999*q^44+124090*q^53+36648*q^57+8006641*q^30+164534*q^52+26377*q^58+
68709*q^55+13244*q^60+92758*q^54+930794*q^45+114*q^71+20*q^74+3959218*q^37+810*
q^67+8399070*q^29+745572*q^46+1930*q^65+317*q^69+2920610*q^39+9233*q^61+2070224
*q^41+5761680*q^34+281770*q^50+2912*q^64+3418050*q^38+6378180*q^33+50420*q^56+
77*q^2+511*q^68+8820675*q^26+351*q^3+1274*q^4+3904*q^5+10461*q^6+6972663*q^32+
25083*q^7+54694*q^8+109746*q^9+6362*q^62+204488*q^10+356377*q^11+584389*q^12+5*
q^76+906276*q^13+1335205*q^14+1876523*q^15+2525499*q^16+1261*q^66+193*q^70+2*q^
77+q^78+1413285*q^43+364004*q^49+8677963*q^28+3266601*q^17+4074426*q^18+4915976
*q^19+5753727*q^20+8823558*q^27+66*q^72+2470834*q^40+37*q^73+4331*q^63+7523271*
q^31+10*q^75+5142579*q^35+466227*q^48+18789*q^59+4537355*q^36+592099*q^47, 1+
44064325*q^25+24184585*q^42+30936825*q^22+2*q^90+40014936*q^24+26247330*q^21+
35584050*q^23+13*q+5585515*q^51+18426272*q^44+3726064*q^53+1529242*q^57+
54545792*q^30+4577570*q^52+1203588*q^58+2419341*q^55+730466*q^60+q^91+3012544*q
^54+15889846*q^45+27667*q^71+9469*q^74+40888238*q^37+101598*q^67+53974306*q^29+
13596432*q^46+185288*q^65+53936*q^69+34086837*q^39+563200*q^61+27358767*q^41+
49684722*q^34+1265*q^79+6768687*q^50+247359*q^64+37517295*q^38+51831253*q^33+
1929952*q^56+90*q^2+74335*q^68+47586421*q^26+441*q^3+1715*q^4+5618*q^5+16068*q^
6+53413813*q^32+41084*q^7+95482*q^8+204174*q^9+431195*q^62+405472*q^10+753372*q
^11+1317555*q^12+4393*q^76+2179971*q^13+3427501*q^14+5141204*q^15+7383802*q^16+
137743*q^66+38802*q^70+2940*q^77+1942*q^78+21197261*q^43+8145467*q^49+52608755*
q^28+10187779*q^17+13546518*q^18+17410144*q^19+21686582*q^20+50462964*q^27+
19545*q^72+30677875*q^40+13674*q^73+327776*q^63+54342743*q^31+6486*q^75+
47074935*q^35+810*q^80+9733038*q^48+193*q^83+511*q^81+66*q^85+37*q^86+20*q^87+
10*q^88+5*q^89+114*q^84+317*q^82+940865*q^59+44107626*q^36+11546262*q^47, 1+
194470300*q^25+251302587*q^42+116400794*q^22+4421*q^90+167583034*q^24+93484990*
q^21+141311945*q^23+14*q+193*q^97+93494308*q^51+212289564*q^44+69741281*q^53+
36160375*q^57+310363947*q^30+810*q^94+80995206*q^52+30254132*q^58+50797681*q^55
+20832389*q^60+2952*q^91+59694507*q^54+192857854*q^45+1816073*q^71+830063*q^74+
317*q^96+114*q^98+329764584*q^37+4760651*q^67+292111073*q^29+173909863*q^46+
1265*q^93+66*q^99+7456765*q^65+2973449*q^69+304023803*q^39+17146637*q^61+
270102764*q^41+344985620*q^34+198978*q^79+107251678*q^50+9256871*q^64+37*q^100+
318163104*q^38+342431639*q^33+42979736*q^56+104*q^2+3772810*q^68+221167219*q^26
+545*q^3+2260*q^4+7877*q^5+23933*q^6+335771306*q^32+64938*q^7+160045*q^8+362790
*q^9+511*q^95+14037085*q^62+20*q^101+763644*q^10+1503931*q^11+2788253*q^12+
477898*q^76+4891380*q^13+8155234*q^14+12972587*q^15+19756557*q^16+5974327*q^66+
2330383*q^70+359206*q^77+268242*q^78+231893152*q^43+10*q^102+122249341*q^49+
270738860*q^28+28898308*q^17+40718615*q^18+55420897*q^19+73053236*q^20+
246852877*q^27+1407163*q^72+287837936*q^40+1083974*q^73+5*q^103+11429791*q^63+
325025301*q^31+631784*q^75+343550613*q^35+2*q^104+146583*q^80+138432037*q^48+q^
105+56086*q^83+107214*q^81+28423*q^85+19975*q^86+13910*q^87+9593*q^88+6548*q^89
+40092*q^84+77841*q^82+25173650*q^59+338364583*q^36+1946*q^92+155701251*q^47]


        The number of permutations avoiding, {[1, 2, 4, 3]}, is given by

[1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705,

    167078577, 1148208090, 8026793118]

     The number of EVEN permutations avoiding, {[1, 2, 4, 3]},  is given by

[1, 1, 3, 12, 52, 257, 1381, 7885, 47181, 293297, 1881647, 12396354, 83539290,

    574104037, 4013396551]

     The number of ODD permutations avoiding, {[1, 2, 4, 3]},  is given by

[0, 1, 3, 11, 51, 256, 1380, 7882, 47178, 293293, 1881643, 12396351, 83539287,

    574104053, 4013396567]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 52, 256, 1380, 7885, 47181, 293293, 1881643, 12396354,

    83539290, 574104053, 4013396567]

 ODD:, [0, 1, 3, 11, 51, 257, 1381, 7882, 47178, 293297, 1881647, 12396351,

    83539287, 574104037, 4013396551]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.086956522, 5.330097087, 8.269005848,

    11.92828685, 16.32587049, 21.47617079, 27.39132784, 34.08181538,

    41.55680895, 49.82443938, 58.89197905, 68.76598379]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.442010778, 1.962702464, 2.526613584,

    3.135119662, 3.786893755, 4.479857229, 5.211902460, 5.981099163,

    6.785729405, 7.624272615, 8.495379377, 9.397845576]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.07940, 3.85220, 6.38378, 9.82898, 14.3406,

    20.0691, 27.1639, 35.7735, 46.0461, 58.1295, 72.1715, 88.3195]

Skewness: , [Float(undefined), 0., 0., -0.06512211045, -0.1497608398,

    -0.2227244116, -0.2776274559, -0.3174322220, -0.3463347581, -0.3676292180,

    -0.3836380313, -0.3958921682, -0.4054535543, -0.4130368996, -0.4191416200]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.527831120,

    2.780164652, 2.919663102, 3.002292300, 3.055649981, 3.092767061,

    3.120030242, 3.140840818, 3.157189517, 3.170309881, 3.181066132,

    3.189940295]

                                end of this data



                         Out of a total of , 7, cases

                      2, were successful and , 5, failed

                             Success Rate: , 0.286

                             Here are the failures

{{{[1, 4, 3, 2]}, {[2, 3, 4, 1]}, {[3, 2, 1, 4]}, {[4, 1, 2, 3]}},

    {{[1, 3, 2, 4]}, {[4, 2, 3, 1]}}, {{[2, 1, 4, 3]}, {[3, 4, 1, 2]}}, {

    {[1, 4, 2, 3]}, {[2, 3, 1, 4]}, {[2, 4, 3, 1]}, {[1, 3, 4, 2]},

    {[3, 2, 4, 1]}, {[4, 1, 3, 2]}, {[3, 1, 2, 4]}, {[4, 2, 1, 3]}},

    {{[2, 4, 1, 3]}, {[3, 1, 4, 2]}}}