Pair of a 3-pattern and a 5-pattern

             There all together, 42, different equivalence classes

For the equivalence class of patterns, {{[1, 2, 3], [3, 2, 1, 5, 4]},

    {[3, 2, 1], [4, 5, 1, 2, 3]}, {[3, 2, 1], [3, 4, 5, 1, 2]},

    {[1, 2, 3], [2, 1, 5, 4, 3]}}

      the member , {[1, 2, 3], [3, 2, 1, 5, 4]}, has a scheme of depth , 3

                                  here it is:

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

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

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

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

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

[1, 1+q, 2*q+2*q^2+q^3, q^2+4*q^3+5*q^4+3*q^5+q^6, q^4+4*q^5+10*q^6+12*q^7+9*q^
8+4*q^9+q^10, q^7+5*q^8+14*q^9+26*q^10+31*q^11+25*q^12+14*q^13+5*q^14+q^15, q^
11+6*q^12+20*q^13+44*q^14+71*q^15+82*q^16+70*q^17+44*q^18+20*q^19+6*q^20+q^21,
q^16+7*q^17+27*q^18+70*q^19+134*q^20+197*q^21+223*q^22+196*q^23+134*q^24+70*q^
25+27*q^26+7*q^27+q^28, q^22+8*q^23+35*q^24+104*q^25+231*q^26+400*q^27+554*q^28
+616*q^29+553*q^30+400*q^31+231*q^32+104*q^33+35*q^34+8*q^35+q^36, q^29+9*q^30+
44*q^31+147*q^32+370*q^33+735*q^34+1184*q^35+1570*q^36+1723*q^37+1569*q^38+1184
*q^39+735*q^40+370*q^41+147*q^42+44*q^43+9*q^44+q^45, q^37+10*q^38+54*q^39+200*
q^40+561*q^41+1252*q^42+2289*q^43+3488*q^44+4477*q^45+4862*q^46+4476*q^47+3488*
q^48+2289*q^49+1252*q^50+561*q^51+200*q^52+54*q^53+10*q^54+q^55, q^46+11*q^47+
65*q^48+264*q^49+815*q^50+2013*q^51+4102*q^52+7029*q^53+10253*q^54+12827*q^55+
13815*q^56+12826*q^57+10253*q^58+7029*q^59+4102*q^60+2013*q^61+815*q^62+264*q^
63+65*q^64+11*q^65+q^66, q^56+12*q^57+77*q^58+340*q^59+1144*q^60+3092*q^61+6930
*q^62+13144*q^63+21384*q^64+30108*q^65+36895*q^66+39468*q^67+36894*q^68+30108*q
^69+21384*q^70+13144*q^71+6930*q^72+3092*q^73+1144*q^74+340*q^75+77*q^76+12*q^
77+q^78, q^67+13*q^68+90*q^69+429*q^70+1561*q^71+4576*q^72+11166*q^73+23166*q^
74+41458*q^75+64636*q^76+88386*q^77+106471*q^78+113257*q^79+106470*q^80+88386*q
^81+64636*q^82+41458*q^83+23166*q^84+11166*q^85+4576*q^86+1561*q^87+429*q^88+90
*q^89+13*q^90+q^91, q^79+14*q^80+104*q^81+532*q^82+2080*q^83+6566*q^84+17303*q^
85+38908*q^86+75790*q^87+129260*q^88+194480*q^89+259492*q^90+308114*q^91+326198
*q^92+308113*q^93+259492*q^94+194480*q^95+129260*q^96+75790*q^97+38908*q^98+
17303*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, 1+3*q+5*q^2+4*q^3+q^4, q^6+4*q^5+10*q^4+12*q^3+9*q^2+4*q+
1, q^8+5*q^7+14*q^6+26*q^5+31*q^4+25*q^3+14*q^2+5*q+1, q^10+6*q^9+20*q^8+44*q^7
+71*q^6+82*q^5+70*q^4+44*q^3+20*q^2+6*q+1, q^12+7*q^11+27*q^10+70*q^9+134*q^8+
197*q^7+223*q^6+196*q^5+134*q^4+70*q^3+27*q^2+7*q+1, q^14+8*q^13+35*q^12+104*q^
11+231*q^10+400*q^9+554*q^8+616*q^7+553*q^6+400*q^5+231*q^4+104*q^3+35*q^2+8*q+
1, q^16+9*q^15+44*q^14+147*q^13+370*q^12+735*q^11+1184*q^10+1570*q^9+1723*q^8+
1569*q^7+1184*q^6+735*q^5+370*q^4+147*q^3+44*q^2+9*q+1, q^18+10*q^17+54*q^16+
200*q^15+561*q^14+1252*q^13+2289*q^12+3488*q^11+4477*q^10+4862*q^9+4476*q^8+
3488*q^7+2289*q^6+1252*q^5+561*q^4+200*q^3+54*q^2+10*q+1, q^20+11*q^19+65*q^18+
264*q^17+815*q^16+2013*q^15+4102*q^14+7029*q^13+10253*q^12+12827*q^11+13815*q^
10+12826*q^9+10253*q^8+7029*q^7+4102*q^6+2013*q^5+815*q^4+264*q^3+65*q^2+11*q+1
, q^22+12*q^21+77*q^20+340*q^19+1144*q^18+3092*q^17+6930*q^16+13144*q^15+21384*
q^14+30108*q^13+36895*q^12+39468*q^11+36894*q^10+30108*q^9+21384*q^8+13144*q^7+
6930*q^6+3092*q^5+1144*q^4+340*q^3+77*q^2+12*q+1, 1+90*q^22+q^24+429*q^21+13*q^
23+13*q+90*q^2+429*q^3+1561*q^4+4576*q^5+11166*q^6+23166*q^7+41458*q^8+64636*q^
9+88386*q^10+106470*q^11+113257*q^12+106471*q^13+88386*q^14+64636*q^15+41458*q^
16+23166*q^17+11166*q^18+4576*q^19+1561*q^20, 1+14*q^25+2080*q^22+104*q^24+6566
*q^21+532*q^23+14*q+104*q^2+q^26+532*q^3+2080*q^4+6566*q^5+17303*q^6+38908*q^7+
75790*q^8+129260*q^9+194480*q^10+259492*q^11+308113*q^12+326198*q^13+308114*q^
14+259492*q^15+194480*q^16+129260*q^17+75790*q^18+38908*q^19+17303*q^20]


 The number of permutations avoiding, {[1, 2, 3], [3, 2, 1, 5, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 2, 3], [3, 2, 1, 5, 4]},

     is given by

[1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

The number of ODD permutations avoiding, {[1, 2, 3], [3, 2, 1, 5, 4]},

     is given by

[0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

 ODD:, [0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.800000000, 3.928571429, 6.975609756, 10.99180328,

    15.99726027, 21.99908592, 28.99969521, 36.99989839, 45.99996613,

    55.99998871, 66.99999624, 78.99999875, 91.99999958]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.297047395, 1.528397326,

    1.732048641, 1.914774435, 2.081617172, 2.236045256, 2.380466658,

    2.516607740, 2.645749888, 2.768874093, 2.886751153]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.68233, 2.33600, 2.99999, 3.66636,

    4.33313, 4.99990, 5.66662, 6.33331, 6.99999, 7.66666, 8.33333]

Skewness: , [Float(undefined), 0., 0.3436215967, 0.1429807726, 0.04524164240,

    0.01379421457, 0.004216602547, 0.001310494397, 0.0004102032965,

                                                                        -5
    0.0001252235634, 0.00004448003447, 0.00001254827062, 0.5399504043 10  , 0.,

    0.]

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

    2.664161122, 2.735419550, 2.773540712, 2.800755247, 2.822308051,

    2.839973598, 2.854675738, 2.867057136, 2.877559242, 2.886589141,

    2.894402315]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [3, 4, 5, 2, 1]},

    {[1, 3, 2], [5, 4, 1, 2, 3]}, {[1, 3, 2], [3, 4, 5, 2, 1]},

    {[3, 1, 2], [3, 2, 1, 4, 5]}, {[3, 1, 2], [1, 2, 5, 4, 3]},

    {[2, 3, 1], [1, 2, 5, 4, 3]}, {[2, 1, 3], [5, 4, 1, 2, 3]},

    {[2, 3, 1], [3, 2, 1, 4, 5]}}

      the member , {[2, 1, 3], [3, 4, 5, 2, 1]}, has a scheme of depth , 5

                                  here it is:

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

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

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

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

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

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

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

    [1, 4, 5, 3, 2],

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

    [[2, 1], {[0, 0, 1]}, {1}], [[2, 4, 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, 0, 0, 0, 0, 0]}, {4}], [[1, 3, 5, 2, 4], {[0, 0, 0, 0, 0, 0]}, {4}],

    [[1, 3, 4, 2, 5], {[0, 0, 0, 0, 0, 0]}, {4}], [[1, 4, 5, 2, 3],

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

    [[1, 2, 3], {[1, 1, 0, 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+6*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+10*q^7+13
*q^8+15*q^9+14*q^10+13*q^11+12*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+12*q^7+17*q^8+20*q^9+25*q^10+26*q^11+31*q^12+32*q^13+35*q^14+31*q^
15+27*q^16+25*q^17+22*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+14*q^7+19*q^8+24*q^9+30*q^10+35*q^11+45*q^12+50*q^13+58*q^14+64*q^15+68*q^
16+73*q^17+76*q^18+79*q^19+76*q^20+68*q^21+58*q^22+52*q^23+47*q^24+37*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+14*q^7+21*q^8+26*q^9+34*q^
10+40*q^11+54*q^12+62*q^13+77*q^14+88*q^15+102*q^16+113*q^17+130*q^18+143*q^19+
160*q^20+163*q^21+172*q^22+174*q^23+180*q^24+184*q^25+180*q^26+166*q^27+146*q^
28+126*q^29+110*q^30+99*q^31+84*q^32+58*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+14*q^7+21*q^8+28*q^9+36*q^10+44*q^11+59*q^12+71*q^13+89*q
^14+105*q^15+127*q^16+148*q^17+173*q^18+197*q^19+228*q^20+255*q^21+280*q^22+305
*q^23+334*q^24+364*q^25+390*q^26+406*q^27+420*q^28+426*q^29+422*q^30+425*q^31+
428*q^32+424*q^33+400*q^34+359*q^35+314*q^36+272*q^37+236*q^38+209*q^39+183*q^
40+142*q^41+86*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+14*
q^7+21*q^8+28*q^9+38*q^10+46*q^11+63*q^12+76*q^13+98*q^14+117*q^15+144*q^16+171
*q^17+209*q^18+241*q^19+285*q^20+325*q^21+376*q^22+420*q^23+478*q^24+534*q^25+
600*q^26+651*q^27+713*q^28+762*q^29+818*q^30+866*q^31+924*q^32+971*q^33+1014*q^
34+1036*q^35+1043*q^36+1039*q^37+1029*q^38+1012*q^39+1000*q^40+987*q^41+948*q^
42+873*q^43+774*q^44+675*q^45+586*q^46+508*q^47+445*q^48+392*q^49+325*q^50+228*
q^51+122*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+14*q^7+
21*q^8+28*q^9+38*q^10+48*q^11+65*q^12+80*q^13+103*q^14+126*q^15+156*q^16+188*q^
17+232*q^18+275*q^19+330*q^20+383*q^21+449*q^22+518*q^23+599*q^24+685*q^25+780*
q^26+877*q^27+988*q^28+1094*q^29+1203*q^30+1319*q^31+1454*q^32+1580*q^33+1712*q
^34+1837*q^35+1952*q^36+2059*q^37+2154*q^38+2251*q^39+2350*q^40+2442*q^41+2516*
q^42+2568*q^43+2584*q^44+2569*q^45+2525*q^46+2469*q^47+2413*q^48+2358*q^49+2299
*q^50+2217*q^51+2080*q^52+1885*q^53+1664*q^54+1451*q^55+1261*q^56+1094*q^57+953
*q^58+837*q^59+717*q^60+553*q^61+350*q^62+167*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+14*q^7+21*q^8+28*q^9+38*q^10+48*q^11+67*q^12+82*q^
13+107*q^14+131*q^15+165*q^16+200*q^17+249*q^18+298*q^19+364*q^20+426*q^21+508*
q^22+592*q^23+700*q^24+808*q^25+937*q^26+1066*q^27+1224*q^28+1382*q^29+1563*q^
30+1742*q^31+1959*q^32+2167*q^33+2410*q^34+2647*q^35+2899*q^36+3144*q^37+3402*q
^38+3660*q^39+3949*q^40+4220*q^41+4502*q^42+4774*q^43+5039*q^44+5257*q^45+5463*
q^46+5647*q^47+5824*q^48+5983*q^49+6135*q^50+6270*q^51+6361*q^52+6385*q^53+6347
*q^54+6246*q^55+6095*q^56+5916*q^57+5724*q^58+5544*q^59+5375*q^60+5174*q^61+
4891*q^62+4506*q^63+4046*q^64+3568*q^65+3117*q^66+2712*q^67+2355*q^68+2047*q^69
+1790*q^70+1554*q^71+1270*q^72+903*q^73+517*q^74+222*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+14*q^7+21*q^8+28*q^9+38*q^10+48*q^11+67*q^12
+84*q^13+109*q^14+135*q^15+170*q^16+209*q^17+261*q^18+315*q^19+387*q^20+460*q^
21+551*q^22+649*q^23+775*q^24+910*q^25+1063*q^26+1225*q^27+1419*q^28+1627*q^29+
1863*q^30+2115*q^31+2407*q^32+2708*q^33+3046*q^34+3405*q^35+3796*q^36+4205*q^37
+4633*q^38+5080*q^39+5583*q^40+6101*q^41+6646*q^42+7202*q^43+7779*q^44+8353*q^
45+8922*q^46+9502*q^47+10093*q^48+10689*q^49+11287*q^50+11884*q^51+12479*q^52+
13032*q^53+13516*q^54+13946*q^55+14340*q^56+14675*q^57+14940*q^58+15188*q^59+
15430*q^60+15635*q^61+15750*q^62+15758*q^63+15646*q^64+15403*q^65+15042*q^66+
14595*q^67+14092*q^68+13561*q^69+13032*q^70+12534*q^71+12040*q^72+11441*q^73+
10649*q^74+9687*q^75+8652*q^76+7635*q^77+6687*q^78+5829*q^79+5067*q^80+4402*q^
81+3837*q^82+3344*q^83+2824*q^84+2173*q^85+1420*q^86+739*q^87+288*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+14*q^7+21*q^8+28*q^9+38*q^10+
48*q^11+67*q^12+84*q^13+111*q^14+137*q^15+174*q^16+214*q^17+270*q^18+327*q^19+
404*q^20+483*q^21+585*q^22+692*q^23+832*q^24+983*q^25+1166*q^26+1352*q^27+1581*
q^28+1824*q^29+2114*q^30+2424*q^31+2792*q^32+3171*q^33+3612*q^34+4074*q^35+4601
*q^36+5159*q^37+5782*q^38+6423*q^39+7151*q^40+7909*q^41+8748*q^42+9622*q^43+
10573*q^44+11538*q^45+12566*q^46+13613*q^47+14738*q^48+15905*q^49+17139*q^50+
18390*q^51+19696*q^52+21012*q^53+22360*q^54+23666*q^55+24985*q^56+26278*q^57+
27567*q^58+28825*q^59+30108*q^60+31389*q^61+32631*q^62+33780*q^63+34819*q^64+
35736*q^65+36545*q^66+37200*q^67+37718*q^68+38126*q^69+38443*q^70+38687*q^71+
38879*q^72+38975*q^73+38875*q^74+38504*q^75+37862*q^76+36996*q^77+35935*q^78+
34710*q^79+33375*q^80+31986*q^81+30595*q^82+29251*q^83+27953*q^84+26587*q^85+
24957*q^86+22964*q^87+20726*q^88+18460*q^89+16310*q^90+14324*q^91+12516*q^92+
10896*q^93+9469*q^94+8239*q^95+7181*q^96+6168*q^97+4997*q^98+3593*q^99+2159*q^
100+1027*q^101+366*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+q^2+q^3, 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+6*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
+10*q^8+13*q^7+15*q^6+14*q^5+13*q^4+12*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+12*q^14+17*q^13+20*q^12+25*q^11+26*q^10+31*q^9+32*q^8+
35*q^7+31*q^6+27*q^5+25*q^4+22*q^3+15*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+14*q^
21+7*q^23+7*q+21*q^2+2*q^26+37*q^3+47*q^4+52*q^5+58*q^6+68*q^7+76*q^8+79*q^9+76
*q^10+73*q^11+68*q^12+64*q^13+58*q^14+50*q^15+45*q^16+q^28+35*q^17+30*q^18+24*q
^19+19*q^20+q^27, 1+40*q^25+77*q^22+54*q^24+88*q^21+62*q^23+8*q+11*q^30+14*q^29
+2*q^34+3*q^33+28*q^2+34*q^26+58*q^3+84*q^4+99*q^5+110*q^6+5*q^32+126*q^7+146*q
^8+166*q^9+180*q^10+184*q^11+180*q^12+174*q^13+172*q^14+163*q^15+160*q^16+21*q^
28+143*q^17+130*q^18+113*q^19+102*q^20+26*q^27+7*q^31+q^35+q^36, 1+228*q^25+3*q
^42+305*q^22+255*q^24+334*q^21+280*q^23+9*q+q^44+105*q^30+q^45+21*q^37+127*q^29
+11*q^39+5*q^41+44*q^34+14*q^38+59*q^33+36*q^2+197*q^26+86*q^3+142*q^4+183*q^5+
209*q^6+71*q^32+236*q^7+272*q^8+314*q^9+359*q^10+400*q^11+424*q^12+428*q^13+425
*q^14+422*q^15+426*q^16+2*q^43+148*q^28+420*q^17+406*q^18+390*q^19+364*q^20+173
*q^27+7*q^40+89*q^31+36*q^35+28*q^36, 1+818*q^25+76*q^42+971*q^22+866*q^24+1014
*q^21+924*q^23+10*q+5*q^51+46*q^44+2*q^53+534*q^30+3*q^52+q^55+q^54+38*q^45+209
*q^37+600*q^29+28*q^46+144*q^39+98*q^41+325*q^34+7*q^50+171*q^38+376*q^33+45*q^
2+762*q^26+122*q^3+228*q^4+325*q^5+392*q^6+420*q^32+445*q^7+508*q^8+586*q^9+675
*q^10+774*q^11+873*q^12+948*q^13+987*q^14+1000*q^15+1012*q^16+63*q^43+11*q^49+
651*q^28+1029*q^17+1039*q^18+1043*q^19+1036*q^20+713*q^27+117*q^40+478*q^31+285
*q^35+14*q^48+241*q^36+21*q^47, 1+2442*q^25+599*q^42+2584*q^22+2516*q^24+2569*q
^21+2568*q^23+11*q+126*q^51+449*q^44+80*q^53+28*q^57+1952*q^30+103*q^52+21*q^58
+48*q^55+11*q^60+65*q^54+383*q^45+1094*q^37+2059*q^29+330*q^46+q^65+877*q^39+7*
q^61+685*q^41+1454*q^34+156*q^50+2*q^64+988*q^38+1580*q^33+38*q^56+55*q^2+2350*
q^26+167*q^3+350*q^4+553*q^5+717*q^6+1712*q^32+837*q^7+953*q^8+1094*q^9+5*q^62+
1261*q^10+1451*q^11+1664*q^12+1885*q^13+2080*q^14+2217*q^15+2299*q^16+q^66+518*
q^43+188*q^49+2154*q^28+2358*q^17+2413*q^18+2469*q^19+2525*q^20+2251*q^27+780*q
^40+3*q^63+1837*q^31+1319*q^35+232*q^48+14*q^59+1203*q^36+275*q^47, 1+6385*q^25
+2899*q^42+6095*q^22+6347*q^24+5916*q^21+6246*q^23+12*q+1066*q^51+2410*q^44+808
*q^53+426*q^57+5824*q^30+937*q^52+364*q^58+592*q^55+249*q^60+700*q^54+2167*q^45
+14*q^71+5*q^74+4220*q^37+48*q^67+5983*q^29+1959*q^46+82*q^65+28*q^69+3660*q^39
+200*q^61+3144*q^41+5039*q^34+1224*q^50+107*q^64+3949*q^38+5257*q^33+508*q^56+
66*q^2+38*q^68+6361*q^26+222*q^3+517*q^4+903*q^5+1270*q^6+5463*q^32+1554*q^7+
1790*q^8+2047*q^9+165*q^62+2355*q^10+2712*q^11+3117*q^12+2*q^76+3568*q^13+4046*
q^14+4506*q^15+4891*q^16+67*q^66+21*q^70+q^77+q^78+2647*q^43+1382*q^49+6135*q^
28+5174*q^17+5375*q^18+5544*q^19+5724*q^20+6270*q^27+11*q^72+3402*q^40+7*q^73+
131*q^63+5647*q^31+3*q^75+4774*q^35+1563*q^48+298*q^59+4502*q^36+1742*q^47, 1+
15042*q^25+10689*q^42+13561*q^22+q^90+14595*q^24+13032*q^21+14092*q^23+13*q+
5583*q^51+9502*q^44+4633*q^53+3046*q^57+15635*q^30+5080*q^52+2708*q^58+3796*q^
55+2115*q^60+q^91+4205*q^54+8922*q^45+387*q^71+209*q^74+13516*q^37+775*q^67+
15750*q^29+8353*q^46+1063*q^65+551*q^69+12479*q^39+1863*q^61+11287*q^41+14675*q
^34+67*q^79+6101*q^50+1225*q^64+13032*q^38+14940*q^33+3405*q^56+78*q^2+649*q^68
+15403*q^26+288*q^3+739*q^4+1420*q^5+2173*q^6+15188*q^32+2824*q^7+3344*q^8+3837
*q^9+1627*q^62+4402*q^10+5067*q^11+5829*q^12+135*q^76+6687*q^13+7635*q^14+8652*
q^15+9687*q^16+910*q^66+460*q^70+109*q^77+84*q^78+10093*q^43+6646*q^49+15758*q^
28+10649*q^17+11441*q^18+12040*q^19+12534*q^20+15646*q^27+315*q^72+11884*q^40+
261*q^73+1419*q^63+15430*q^31+170*q^75+14340*q^35+48*q^80+7202*q^48+21*q^83+38*
q^81+11*q^85+7*q^86+5*q^87+3*q^88+2*q^89+14*q^84+28*q^82+2407*q^59+13946*q^36+
7779*q^47, 1+33375*q^25+33780*q^42+29251*q^22+137*q^90+31986*q^24+27953*q^21+
30595*q^23+14*q+21*q^97+22360*q^51+31389*q^44+19696*q^53+14738*q^57+38504*q^30+
48*q^94+21012*q^52+13613*q^58+17139*q^55+11538*q^60+111*q^91+18390*q^54+30108*q
^45+3612*q^71+2424*q^74+28*q^96+14*q^98+37718*q^37+5782*q^67+37862*q^29+28825*q
^46+67*q^93+11*q^99+7151*q^65+4601*q^69+36545*q^39+10573*q^61+34819*q^41+38687*
q^34+1166*q^79+23666*q^50+7909*q^64+7*q^100+37200*q^38+38879*q^33+15905*q^56+91
*q^2+5159*q^68+34710*q^26+366*q^3+1027*q^4+2159*q^5+3593*q^6+38975*q^32+4997*q^
7+6168*q^8+7181*q^9+38*q^95+9622*q^62+5*q^101+8239*q^10+9469*q^11+10896*q^12+
1824*q^76+12516*q^13+14324*q^14+16310*q^15+18460*q^16+6423*q^66+4074*q^70+1581*
q^77+1352*q^78+32631*q^43+3*q^102+24985*q^49+36996*q^28+20726*q^17+22964*q^18+
24957*q^19+26587*q^20+35935*q^27+3171*q^72+35736*q^40+2792*q^73+2*q^103+8748*q^
63+38875*q^31+2114*q^75+38443*q^35+q^104+983*q^80+26278*q^48+q^105+585*q^83+832
*q^81+404*q^85+327*q^86+270*q^87+214*q^88+174*q^89+483*q^84+692*q^82+12566*q^59
+38126*q^36+84*q^92+27567*q^47]


 The number of permutations avoiding, {[2, 1, 3], [3, 4, 5, 2, 1]}, is given by

[1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017, 190885, 533294,

    1484021]

The number of EVEN permutations avoiding, {[2, 1, 3], [3, 4, 5, 2, 1]},

     is given by

[1, 1, 3, 7, 22, 61, 182, 518, 1498, 4254, 12082, 34023, 95499, 266679, 742125]

The number of ODD permutations avoiding, {[2, 1, 3], [3, 4, 5, 2, 1]},

     is given by

[0, 1, 2, 7, 19, 60, 173, 514, 1475, 4242, 12029, 33994, 95386, 266615, 741896]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 60, 173, 518, 1498, 4242, 12029, 34023, 95499, 266615,

    741896]

 ODD:, [0, 1, 3, 7, 19, 61, 182, 514, 1475, 4254, 12082, 33994, 95386, 266679,

    742125]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.780487805, 8.876033058,

    12.65633803, 17.13372093, 22.31718803, 28.21257062, 34.82348306,

    42.15216784, 50.20005763, 58.96810952, 68.45699488]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.342733395, 3.168987762,

    4.108415698, 5.153040281, 6.293630536, 7.522080629, 8.831828359,

    10.21762625, 11.67521067, 13.20103029, 14.79205503]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.48840, 10.0425, 16.8791, 26.5538,

    39.6098, 56.5817, 78.0012, 104.400, 136.311, 174.267, 218.805]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4180400731,

    -0.4177996242, -0.4117243206, -0.4062668078, -0.4018420464, -0.3977964717,

    -0.3936488011, -0.3891993245, -0.3844181537, -0.3793818758, -0.3741484366]

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

    2.578362051, 2.657612294, 2.697824371, 2.727539763, 2.753980593,

    2.778163731, 2.799848104, 2.818926983, 2.835508457, 2.849867833,

    2.862249660]

                                end of this data



For the equivalence class of patterns, {{[2, 3, 1], [1, 5, 4, 2, 3]},

    {[2, 3, 1], [4, 3, 1, 2, 5]}, {[2, 1, 3], [5, 1, 2, 4, 3]},

    {[3, 1, 2], [3, 4, 2, 1, 5]}, {[2, 1, 3], [2, 3, 5, 4, 1]},

    {[1, 3, 2], [3, 2, 4, 5, 1]}, {[1, 3, 2], [5, 2, 1, 3, 4]},

    {[3, 1, 2], [1, 4, 5, 3, 2]}}

      the member , {[1, 3, 2], [5, 2, 1, 3, 4]}, has a scheme of depth , 3

                                  here it is:

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

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

    [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {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+
4*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+6*q^5+8*q^6+9*q^7+13*
q^8+13*q^9+14*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+6
*q^5+10*q^6+10*q^7+15*q^8+16*q^9+23*q^10+25*q^11+28*q^12+30*q^13+36*q^14+35*q^
15+37*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+6*q^5+10*
q^6+12*q^7+16*q^8+18*q^9+26*q^10+30*q^11+38*q^12+42*q^13+48*q^14+57*q^15+68*q^
16+72*q^17+78*q^18+79*q^19+86*q^20+92*q^21+87*q^22+82*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+6*q^5+10*q^6+12*q^7+18*q^8+19*q^9+28*q^
10+33*q^11+43*q^12+48*q^13+62*q^14+71*q^15+86*q^16+93*q^17+113*q^18+123*q^19+
145*q^20+152*q^21+174*q^22+192*q^23+199*q^24+203*q^25+212*q^26+214*q^27+225*q^
28+219*q^29+195*q^30+162*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+6*q^5+10*q^6+12*q^7+18*q^8+21*q^9+29*q^10+35*q^11+46*q^12+53*q^13+68
*q^14+81*q^15+102*q^16+113*q^17+134*q^18+154*q^19+183*q^20+205*q^21+240*q^22+
273*q^23+296*q^24+319*q^25+359*q^26+395*q^27+441*q^28+467*q^29+493*q^30+520*q^
31+536*q^32+532*q^33+544*q^34+554*q^35+551*q^36+541*q^37+485*q^38+399*q^39+295*
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+6*q^5+10*q^6+
12*q^7+18*q^8+21*q^9+31*q^10+36*q^11+48*q^12+56*q^13+73*q^14+87*q^15+112*q^16+
125*q^17+156*q^18+177*q^19+214*q^20+243*q^21+291*q^22+332*q^23+378*q^24+420*q^
25+486*q^26+533*q^27+615*q^28+667*q^29+755*q^30+817*q^31+890*q^32+947*q^33+1022
*q^34+1097*q^35+1203*q^36+1277*q^37+1339*q^38+1366*q^39+1382*q^40+1403*q^41+
1404*q^42+1393*q^43+1411*q^44+1390*q^45+1328*q^46+1209*q^47+1003*q^48+758*q^49+
505*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+6*q^5+10
*q^6+12*q^7+18*q^8+21*q^9+31*q^10+38*q^11+49*q^12+58*q^13+76*q^14+92*q^15+118*q
^16+135*q^17+168*q^18+195*q^19+239*q^20+276*q^21+329*q^22+383*q^23+439*q^24+499
*q^25+583*q^26+661*q^27+766*q^28+851*q^29+967*q^30+1074*q^31+1194*q^32+1309*q^
33+1460*q^34+1610*q^35+1786*q^36+1947*q^37+2111*q^38+2265*q^39+2427*q^40+2594*q
^41+2747*q^42+2910*q^43+3109*q^44+3268*q^45+3470*q^46+3605*q^47+3671*q^48+3690*
q^49+3666*q^50+3644*q^51+3649*q^52+3612*q^53+3571*q^54+3511*q^55+3316*q^56+2999
*q^57+2535*q^58+1951*q^59+1356*q^60+823*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+6*q^5+10*q^6+12*q^7+18*q^8+21*q^9+31*q^10+38*q^11+
51*q^12+59*q^13+78*q^14+95*q^15+123*q^16+141*q^17+178*q^18+207*q^19+257*q^20+
297*q^21+364*q^22+423*q^23+490*q^24+560*q^25+664*q^26+759*q^27+894*q^28+998*q^
29+1157*q^30+1298*q^31+1471*q^32+1627*q^33+1845*q^34+2060*q^35+2318*q^36+2557*q
^37+2852*q^38+3110*q^39+3423*q^40+3707*q^41+4041*q^42+4375*q^43+4785*q^44+5162*
q^45+5646*q^46+6020*q^47+6408*q^48+6733*q^49+7108*q^50+7457*q^51+7870*q^52+8259
*q^53+8670*q^54+9083*q^55+9434*q^56+9693*q^57+9886*q^58+9887*q^59+9780*q^60+
9651*q^61+9508*q^62+9409*q^63+9326*q^64+9121*q^65+8837*q^66+8339*q^67+7501*q^68
+6384*q^69+5027*q^70+3598*q^71+2309*q^72+1288*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+6*q^5+10*q^6+12*q^7+18*q^8+21*q^9+31*q^10+38*
q^11+51*q^12+61*q^13+79*q^14+97*q^15+126*q^16+146*q^17+184*q^18+217*q^19+269*q^
20+315*q^21+385*q^22+454*q^23+532*q^24+613*q^25+725*q^26+840*q^27+994*q^28+1127
*q^29+1308*q^30+1488*q^31+1696*q^32+1909*q^33+2181*q^34+2469*q^35+2796*q^36+
3113*q^37+3499*q^38+3882*q^39+4326*q^40+4768*q^41+5262*q^42+5799*q^43+6409*q^44
+7016*q^45+7732*q^46+8410*q^47+9133*q^48+9830*q^49+10612*q^50+11411*q^51+12286*
q^52+13160*q^53+14121*q^54+15130*q^55+16149*q^56+17153*q^57+18150*q^58+19015*q^
59+19805*q^60+20553*q^61+21402*q^62+22319*q^63+23262*q^64+24190*q^65+25099*q^66
+25801*q^67+26367*q^68+26570*q^69+26538*q^70+26281*q^71+25755*q^72+25219*q^73+
24779*q^74+24336*q^75+23931*q^76+23349*q^77+22369*q^78+20969*q^79+18889*q^80+
16119*q^81+12896*q^82+9493*q^83+6337*q^84+3773*q^85+1948*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+6*q^5+10*q^6+12*q^7+18*q^8+21*q^9+
31*q^10+38*q^11+51*q^12+61*q^13+81*q^14+98*q^15+128*q^16+149*q^17+189*q^18+223*
q^19+279*q^20+327*q^21+403*q^22+475*q^23+563*q^24+651*q^25+780*q^26+903*q^27+
1075*q^28+1227*q^29+1439*q^30+1640*q^31+1890*q^32+2138*q^33+2468*q^34+2805*q^35
+3216*q^36+3613*q^37+4089*q^38+4560*q^39+5137*q^40+5714*q^41+6395*q^42+7089*q^
43+7929*q^44+8752*q^45+9753*q^46+10705*q^47+11785*q^48+12848*q^49+14060*q^50+
15283*q^51+16707*q^52+18127*q^53+19749*q^54+21402*q^55+23198*q^56+24989*q^57+
26927*q^58+28712*q^59+30653*q^60+32539*q^61+34685*q^62+36850*q^63+39271*q^64+
41660*q^65+44219*q^66+46631*q^67+49108*q^68+51330*q^69+53549*q^70+55526*q^71+
57420*q^72+59205*q^73+61077*q^74+63008*q^75+65222*q^76+67329*q^77+69195*q^78+
70780*q^79+71709*q^80+71982*q^81+71613*q^82+70521*q^83+69146*q^84+67532*q^85+
65799*q^86+64336*q^87+63002*q^88+61472*q^89+59622*q^90+56874*q^91+52919*q^92+
47643*q^93+40868*q^94+33034*q^95+24873*q^96+17173*q^97+10726*q^98+5953*q^99+
2861*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+q^2+q^3, 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+4*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+6*q^10+8*q^9
+9*q^8+13*q^7+13*q^6+14*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+6*q^16+10*q^15+10*q^14+15*q^13+16*q^12+23*q^11+25*q^10+28*q^9+30*q^8+36
*q^7+35*q^6+37*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+10*q^22+5*q^24+12*q^21+
6*q^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+82*q^5+87*q^6+92*q^7+86*q^8+79*q^9+78*q^
10+72*q^11+68*q^12+57*q^13+48*q^14+42*q^15+38*q^16+q^28+30*q^17+26*q^18+18*q^19
+16*q^20+q^27, 1+33*q^25+62*q^22+43*q^24+71*q^21+48*q^23+8*q+10*q^30+12*q^29+2*
q^34+3*q^33+28*q^2+28*q^26+63*q^3+112*q^4+162*q^5+195*q^6+5*q^32+219*q^7+225*q^
8+214*q^9+212*q^10+203*q^11+199*q^12+192*q^13+174*q^14+152*q^15+145*q^16+18*q^
28+123*q^17+113*q^18+93*q^19+86*q^20+19*q^27+6*q^31+q^35+q^36, 1+183*q^25+3*q^
42+273*q^22+205*q^24+296*q^21+240*q^23+9*q+q^44+81*q^30+q^45+18*q^37+102*q^29+
10*q^39+5*q^41+35*q^34+12*q^38+46*q^33+36*q^2+154*q^26+92*q^3+182*q^4+295*q^5+
399*q^6+53*q^32+485*q^7+541*q^8+551*q^9+554*q^10+544*q^11+532*q^12+536*q^13+520
*q^14+493*q^15+467*q^16+2*q^43+113*q^28+441*q^17+395*q^18+359*q^19+319*q^20+134
*q^27+6*q^40+68*q^31+29*q^35+21*q^36, 1+755*q^25+56*q^42+947*q^22+817*q^24+1022
*q^21+890*q^23+10*q+5*q^51+36*q^44+2*q^53+420*q^30+3*q^52+q^55+q^54+31*q^45+156
*q^37+486*q^29+21*q^46+112*q^39+73*q^41+243*q^34+6*q^50+125*q^38+291*q^33+45*q^
2+667*q^26+129*q^3+282*q^4+505*q^5+758*q^6+332*q^32+1003*q^7+1209*q^8+1328*q^9+
1390*q^10+1411*q^11+1393*q^12+1404*q^13+1403*q^14+1382*q^15+1366*q^16+48*q^43+
10*q^49+533*q^28+1339*q^17+1277*q^18+1203*q^19+1097*q^20+615*q^27+87*q^40+378*q
^31+214*q^35+12*q^48+177*q^36+18*q^47, 1+2594*q^25+439*q^42+3109*q^22+2747*q^24
+3268*q^21+2910*q^23+11*q+92*q^51+329*q^44+58*q^53+21*q^57+1786*q^30+76*q^52+18
*q^58+38*q^55+10*q^60+49*q^54+276*q^45+851*q^37+1947*q^29+239*q^46+q^65+661*q^
39+6*q^61+499*q^41+1194*q^34+118*q^50+2*q^64+766*q^38+1309*q^33+31*q^56+55*q^2+
2427*q^26+175*q^3+420*q^4+823*q^5+1356*q^6+1460*q^32+1951*q^7+2535*q^8+2999*q^9
+5*q^62+3316*q^10+3511*q^11+3571*q^12+3612*q^13+3649*q^14+3644*q^15+3666*q^16+q
^66+383*q^43+135*q^49+2111*q^28+3690*q^17+3671*q^18+3605*q^19+3470*q^20+2265*q^
27+583*q^40+3*q^63+1610*q^31+1074*q^35+168*q^48+12*q^59+967*q^36+195*q^47, 1+
8259*q^25+2318*q^42+9434*q^22+8670*q^24+9693*q^21+9083*q^23+12*q+759*q^51+1845*
q^44+560*q^53+297*q^57+6408*q^30+664*q^52+257*q^58+423*q^55+178*q^60+490*q^54+
1627*q^45+12*q^71+5*q^74+3707*q^37+38*q^67+6733*q^29+1471*q^46+59*q^65+21*q^69+
3110*q^39+141*q^61+2557*q^41+4785*q^34+894*q^50+78*q^64+3423*q^38+5162*q^33+364
*q^56+66*q^2+31*q^68+7870*q^26+231*q^3+605*q^4+1288*q^5+2309*q^6+5646*q^32+3598
*q^7+5027*q^8+6384*q^9+123*q^62+7501*q^10+8339*q^11+8837*q^12+2*q^76+9121*q^13+
9326*q^14+9409*q^15+9508*q^16+51*q^66+18*q^70+q^77+q^78+2060*q^43+998*q^49+7108
*q^28+9651*q^17+9780*q^18+9887*q^19+9886*q^20+7457*q^27+10*q^72+2852*q^40+6*q^
73+95*q^63+6020*q^31+3*q^75+4375*q^35+1157*q^48+207*q^59+4041*q^36+1298*q^47, 1
+25099*q^25+9830*q^42+26570*q^22+q^90+25801*q^24+26538*q^21+26367*q^23+13*q+
4326*q^51+8410*q^44+3499*q^53+2181*q^57+20553*q^30+3882*q^52+1909*q^58+2796*q^
55+1488*q^60+q^91+3113*q^54+7732*q^45+269*q^71+146*q^74+14121*q^37+532*q^67+
21402*q^29+7016*q^46+725*q^65+385*q^69+12286*q^39+1308*q^61+10612*q^41+17153*q^
34+51*q^79+4768*q^50+840*q^64+13160*q^38+18150*q^33+2469*q^56+78*q^2+454*q^68+
24190*q^26+298*q^3+847*q^4+1948*q^5+3773*q^6+19015*q^32+6337*q^7+9493*q^8+12896
*q^9+1127*q^62+16119*q^10+18889*q^11+20969*q^12+97*q^76+22369*q^13+23349*q^14+
23931*q^15+24336*q^16+613*q^66+315*q^70+79*q^77+61*q^78+9133*q^43+5262*q^49+
22319*q^28+24779*q^17+25219*q^18+25755*q^19+26281*q^20+23262*q^27+217*q^72+
11411*q^40+184*q^73+994*q^63+19805*q^31+126*q^75+16149*q^35+38*q^80+5799*q^48+
18*q^83+31*q^81+10*q^85+6*q^86+5*q^87+3*q^88+2*q^89+12*q^84+21*q^82+1696*q^59+
15130*q^36+6409*q^47, 1+71709*q^25+36850*q^42+70521*q^22+98*q^90+71982*q^24+
69146*q^21+71613*q^23+14*q+18*q^97+19749*q^51+32539*q^44+16707*q^53+11785*q^57+
63008*q^30+38*q^94+18127*q^52+10705*q^58+14060*q^55+8752*q^60+81*q^91+15283*q^
54+30653*q^45+2468*q^71+1640*q^74+21*q^96+12*q^98+49108*q^37+4089*q^67+65222*q^
29+28712*q^46+51*q^93+10*q^99+5137*q^65+3216*q^69+44219*q^39+7929*q^61+39271*q^
41+55526*q^34+780*q^79+21402*q^50+5714*q^64+6*q^100+46631*q^38+57420*q^33+12848
*q^56+91*q^2+3613*q^68+70780*q^26+377*q^3+1157*q^4+2861*q^5+5953*q^6+59205*q^32
+10726*q^7+17173*q^8+24873*q^9+31*q^95+7089*q^62+5*q^101+33034*q^10+40868*q^11+
47643*q^12+1227*q^76+52919*q^13+56874*q^14+59622*q^15+61472*q^16+4560*q^66+2805
*q^70+1075*q^77+903*q^78+34685*q^43+3*q^102+23198*q^49+67329*q^28+63002*q^17+
64336*q^18+65799*q^19+67532*q^20+69195*q^27+2138*q^72+41660*q^40+1890*q^73+2*q^
103+6395*q^63+61077*q^31+1439*q^75+53549*q^35+q^104+651*q^80+24989*q^48+q^105+
403*q^83+563*q^81+279*q^85+223*q^86+189*q^87+149*q^88+128*q^89+327*q^84+475*q^
82+9753*q^59+51330*q^36+61*q^92+26927*q^47]


 The number of permutations avoiding, {[1, 3, 2], [5, 2, 1, 3, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 2, 1, 3, 4]},

     is given by

[1, 1, 3, 7, 22, 62, 188, 552, 1659, 4939, 14821, 44345, 133040, 398760,

    1196286]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 2, 1, 3, 4]},

     is given by

[0, 1, 2, 7, 19, 60, 177, 542, 1622, 4903, 14704, 44229, 132681, 398402,

    1195199]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 60, 177, 552, 1659, 4903, 14704, 44345, 133040, 398402,

    1195199]

 ODD:, [0, 1, 3, 7, 19, 62, 188, 542, 1622, 4939, 14821, 44229, 132681, 398760,

    1196286]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.829268293, 9.065573770,

    13.09863014, 17.94698355, 23.62023773, 30.12304410, 37.45754445,

    45.62468670, 54.62487722, 64.45828577, 75.12498176]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.346792715, 3.164189235,

    4.070760864, 5.056438783, 6.114542557, 7.240897936, 8.432709254,

    9.687804263, 11.00425031, 12.38020234, 13.81386415]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.50744, 10.0121, 16.5711, 25.5676,

    37.3876, 52.4306, 71.1106, 93.8536, 121.094, 153.269, 190.823]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4739251896,

    -0.5475630158, -0.6068606886, -0.6515354388, -0.6834100967, -0.7049935414,

    -0.7186948247, -0.7264850196, -0.7298840942, -0.7300611087, -0.7278251243]

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

    2.598851773, 2.758699907, 2.896875725, 3.013589508, 3.107274715,

    3.179367888, 3.233105314, 3.272146879, 3.299816038, 3.318897353,

    3.331381916]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [3, 5, 4, 1, 2]},

    {[1, 3, 2], [4, 3, 5, 1, 2]}, {[1, 3, 2], [4, 5, 2, 1, 3]},

    {[3, 1, 2], [2, 3, 1, 5, 4]}, {[2, 3, 1], [3, 1, 2, 5, 4]},

    {[2, 1, 3], [4, 5, 1, 3, 2]}, {[3, 1, 2], [2, 1, 4, 5, 3]},

    {[2, 3, 1], [2, 1, 5, 3, 4]}}

      the member , {[1, 3, 2], [4, 3, 5, 1, 2]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

                 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+6*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+10*q^7+14
*q^8+15*q^9+13*q^10+15*q^11+11*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+12*q^7+18*q^8+21*q^9+25*q^10+29*q^11+32*q^12+33*q^13+37*q^14+32*q^
15+28*q^16+27*q^17+19*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+14*q^7+20*q^8+25*q^9+31*q^10+39*q^11+48*q^12+55*q^13+62*q^14+71*q^15+75*q^
16+82*q^17+83*q^18+84*q^19+79*q^20+77*q^21+60*q^22+54*q^23+45*q^24+31*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+14*q^7+22*q^8+27*q^9+35*q^
10+45*q^11+58*q^12+69*q^13+86*q^14+99*q^15+116*q^16+133*q^17+149*q^18+166*q^19+
183*q^20+192*q^21+197*q^22+207*q^23+208*q^24+205*q^25+197*q^26+179*q^27+160*q^
28+143*q^29+108*q^30+96*q^31+72*q^32+48*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+14*q^7+22*q^8+29*q^9+37*q^10+49*q^11+64*q^12+79*q^13+100*
q^14+121*q^15+146*q^16+177*q^17+204*q^18+237*q^19+275*q^20+312*q^21+341*q^22+
384*q^23+416*q^24+448*q^25+478*q^26+497*q^27+515*q^28+532*q^29+517*q^30+514*q^
31+505*q^32+482*q^33+442*q^34+402*q^35+343*q^36+300*q^37+250*q^38+190*q^39+161*
q^40+112*q^41+71*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+
14*q^7+22*q^8+29*q^9+39*q^10+51*q^11+68*q^12+85*q^13+110*q^14+135*q^15+168*q^16
+205*q^17+250*q^18+295*q^19+350*q^20+411*q^21+471*q^22+544*q^23+622*q^24+695*q^
25+780*q^26+858*q^27+936*q^28+1022*q^29+1092*q^30+1162*q^31+1223*q^32+1286*q^33
+1318*q^34+1345*q^35+1354*q^36+1345*q^37+1333*q^38+1294*q^39+1223*q^40+1167*q^
41+1089*q^42+988*q^43+864*q^44+755*q^45+624*q^46+537*q^47+424*q^48+326*q^49+259
*q^50+170*q^51+101*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+14*q^7+22*q^8+29*q^9+39*q^10+53*q^11+70*q^12+89*q^13+116*q^14+145*q^15+182*q^
16+227*q^17+278*q^18+339*q^19+410*q^20+489*q^21+574*q^22+681*q^23+794*q^24+917*
q^25+1054*q^26+1199*q^27+1356*q^28+1531*q^29+1692*q^30+1877*q^31+2066*q^32+2251
*q^33+2438*q^34+2627*q^35+2791*q^36+2960*q^37+3110*q^38+3242*q^39+3357*q^40+
3449*q^41+3508*q^42+3548*q^43+3548*q^44+3499*q^45+3418*q^46+3334*q^47+3185*q^48
+3045*q^49+2848*q^50+2616*q^51+2400*q^52+2154*q^53+1869*q^54+1600*q^55+1356*q^
56+1105*q^57+929*q^58+708*q^59+544*q^60+404*q^61+252*q^62+139*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+14*q^7+22*q^8+29*q^9+39*q^10+53*q^
11+72*q^12+91*q^13+120*q^14+151*q^15+192*q^16+241*q^17+300*q^18+367*q^19+454*q^
20+547*q^21+654*q^22+787*q^23+935*q^24+1096*q^25+1288*q^26+1491*q^27+1724*q^28+
1988*q^29+2258*q^30+2564*q^31+2898*q^32+3243*q^33+3606*q^34+4010*q^35+4412*q^36
+4828*q^37+5266*q^38+5696*q^39+6137*q^40+6584*q^41+6988*q^42+7410*q^43+7803*q^
44+8148*q^45+8463*q^46+8766*q^47+8976*q^48+9173*q^49+9302*q^50+9350*q^51+9346*q
^52+9290*q^53+9118*q^54+8914*q^55+8626*q^56+8241*q^57+7849*q^58+7419*q^59+6899*
q^60+6408*q^61+5801*q^62+5196*q^63+4608*q^64+4019*q^65+3391*q^66+2872*q^67+2372
*q^68+1922*q^69+1567*q^70+1169*q^71+883*q^72+615*q^73+365*q^74+186*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+14*q^7+22*q^8+29*q^9+39*q^10+
53*q^11+72*q^12+93*q^13+122*q^14+155*q^15+198*q^16+251*q^17+314*q^18+389*q^19+
482*q^20+591*q^21+712*q^22+865*q^23+1043*q^24+1240*q^25+1471*q^26+1732*q^27+
2028*q^28+2374*q^29+2744*q^30+3167*q^31+3640*q^32+4160*q^33+4712*q^34+5334*q^35
+6005*q^36+6720*q^37+7489*q^38+8307*q^39+9165*q^40+10090*q^41+11034*q^42+12008*
q^43+13030*q^44+14060*q^45+15078*q^46+16140*q^47+17182*q^48+18207*q^49+19203*q^
50+20158*q^51+21047*q^52+21918*q^53+22660*q^54+23306*q^55+23904*q^56+24340*q^57
+24655*q^58+24885*q^59+24964*q^60+24902*q^61+24678*q^62+24305*q^63+23766*q^64+
23132*q^65+22318*q^66+21370*q^67+20347*q^68+19206*q^69+17947*q^70+16704*q^71+
15400*q^72+14013*q^73+12652*q^74+11171*q^75+9787*q^76+8482*q^77+7243*q^78+6019*
q^79+5048*q^80+4083*q^81+3290*q^82+2595*q^83+1907*q^84+1397*q^85+917*q^86+517*q
^87+243*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+14*q^7+22
*q^8+29*q^9+39*q^10+53*q^11+72*q^12+93*q^13+124*q^14+157*q^15+202*q^16+257*q^17
+324*q^18+403*q^19+504*q^20+619*q^21+756*q^22+923*q^23+1121*q^24+1346*q^25+1617
*q^26+1918*q^27+2273*q^28+2685*q^29+3142*q^30+3671*q^31+4272*q^32+4941*q^33+
5684*q^34+6523*q^35+7441*q^36+8466*q^37+9596*q^38+10808*q^39+12139*q^40+13589*q
^41+15137*q^42+16812*q^43+18590*q^44+20478*q^45+22466*q^46+24581*q^47+26750*q^
48+29050*q^49+31417*q^50+33826*q^51+36302*q^52+38841*q^53+41354*q^54+43903*q^55
+46433*q^56+48880*q^57+51306*q^58+53650*q^59+55840*q^60+57986*q^61+59911*q^62+
61649*q^63+63189*q^64+64523*q^65+65539*q^66+66391*q^67+66873*q^68+67071*q^69+
67009*q^70+66596*q^71+65874*q^72+64918*q^73+63582*q^74+61894*q^75+59906*q^76+
57681*q^77+55172*q^78+52498*q^79+49599*q^80+46524*q^81+43424*q^82+40206*q^83+
36822*q^84+33653*q^85+30341*q^86+27063*q^87+23861*q^88+20700*q^89+17826*q^90+
15235*q^91+12787*q^92+10544*q^93+8731*q^94+6953*q^95+5544*q^96+4238*q^97+3067*q
^98+2160*q^99+1342*q^100+717*q^101+311*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+q^2+q^3, 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+6*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
+10*q^8+14*q^7+15*q^6+13*q^5+15*q^4+11*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+12*q^14+18*q^13+21*q^12+25*q^11+29*q^10+32*q^9+33*q^8+
37*q^7+32*q^6+28*q^5+27*q^4+19*q^3+15*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+14*q^
21+7*q^23+7*q+21*q^2+2*q^26+31*q^3+45*q^4+54*q^5+60*q^6+77*q^7+79*q^8+84*q^9+83
*q^10+82*q^11+75*q^12+71*q^13+62*q^14+55*q^15+48*q^16+q^28+39*q^17+31*q^18+25*q
^19+20*q^20+q^27, 1+45*q^25+86*q^22+58*q^24+99*q^21+69*q^23+8*q+11*q^30+14*q^29
+2*q^34+3*q^33+28*q^2+35*q^26+48*q^3+72*q^4+96*q^5+108*q^6+5*q^32+143*q^7+160*q
^8+179*q^9+197*q^10+205*q^11+208*q^12+207*q^13+197*q^14+192*q^15+183*q^16+22*q^
28+166*q^17+149*q^18+133*q^19+116*q^20+27*q^27+7*q^31+q^35+q^36, 1+275*q^25+3*q
^42+384*q^22+312*q^24+416*q^21+341*q^23+9*q+q^44+121*q^30+q^45+22*q^37+146*q^29
+11*q^39+5*q^41+49*q^34+14*q^38+64*q^33+36*q^2+237*q^26+71*q^3+112*q^4+161*q^5+
190*q^6+79*q^32+250*q^7+300*q^8+343*q^9+402*q^10+442*q^11+482*q^12+505*q^13+514
*q^14+517*q^15+532*q^16+2*q^43+177*q^28+515*q^17+497*q^18+478*q^19+448*q^20+204
*q^27+7*q^40+100*q^31+37*q^35+29*q^36, 1+1092*q^25+85*q^42+1286*q^22+1162*q^24+
1318*q^21+1223*q^23+10*q+5*q^51+51*q^44+2*q^53+695*q^30+3*q^52+q^55+q^54+39*q^
45+250*q^37+780*q^29+29*q^46+168*q^39+110*q^41+411*q^34+7*q^50+205*q^38+471*q^
33+45*q^2+1022*q^26+101*q^3+170*q^4+259*q^5+326*q^6+544*q^32+424*q^7+537*q^8+
624*q^9+755*q^10+864*q^11+988*q^12+1089*q^13+1167*q^14+1223*q^15+1294*q^16+68*q
^43+11*q^49+858*q^28+1333*q^17+1345*q^18+1354*q^19+1345*q^20+936*q^27+135*q^40+
622*q^31+350*q^35+14*q^48+295*q^36+22*q^47, 1+3449*q^25+794*q^42+3548*q^22+3508
*q^24+3499*q^21+3548*q^23+11*q+145*q^51+574*q^44+89*q^53+29*q^57+2791*q^30+116*
q^52+22*q^58+53*q^55+11*q^60+70*q^54+489*q^45+1531*q^37+2960*q^29+410*q^46+q^65
+1199*q^39+7*q^61+917*q^41+2066*q^34+182*q^50+2*q^64+1356*q^38+2251*q^33+39*q^
56+55*q^2+3357*q^26+139*q^3+252*q^4+404*q^5+544*q^6+2438*q^32+708*q^7+929*q^8+
1105*q^9+5*q^62+1356*q^10+1600*q^11+1869*q^12+2154*q^13+2400*q^14+2616*q^15+
2848*q^16+q^66+681*q^43+227*q^49+3110*q^28+3045*q^17+3185*q^18+3334*q^19+3418*q
^20+3242*q^27+1054*q^40+3*q^63+2627*q^31+1877*q^35+278*q^48+14*q^59+1692*q^36+
339*q^47, 1+9290*q^25+4412*q^42+8626*q^22+9118*q^24+8241*q^21+8914*q^23+12*q+
1491*q^51+3606*q^44+1096*q^53+547*q^57+8976*q^30+1288*q^52+454*q^58+787*q^55+
300*q^60+935*q^54+3243*q^45+14*q^71+5*q^74+6584*q^37+53*q^67+9173*q^29+2898*q^
46+91*q^65+29*q^69+5696*q^39+241*q^61+4828*q^41+7803*q^34+1724*q^50+120*q^64+
6137*q^38+8148*q^33+654*q^56+66*q^2+39*q^68+9346*q^26+186*q^3+365*q^4+615*q^5+
883*q^6+8463*q^32+1169*q^7+1567*q^8+1922*q^9+192*q^62+2372*q^10+2872*q^11+3391*
q^12+2*q^76+4019*q^13+4608*q^14+5196*q^15+5801*q^16+72*q^66+22*q^70+q^77+q^78+
4010*q^43+1988*q^49+9302*q^28+6408*q^17+6899*q^18+7419*q^19+7849*q^20+9350*q^27
+11*q^72+5266*q^40+7*q^73+151*q^63+8766*q^31+3*q^75+7410*q^35+2258*q^48+367*q^
59+6988*q^36+2564*q^47, 1+22318*q^25+18207*q^42+19206*q^22+q^90+21370*q^24+
17947*q^21+20347*q^23+13*q+9165*q^51+16140*q^44+7489*q^53+4712*q^57+24902*q^30+
8307*q^52+4160*q^58+6005*q^55+3167*q^60+q^91+6720*q^54+15078*q^45+482*q^71+251*
q^74+22660*q^37+1043*q^67+24678*q^29+14060*q^46+1471*q^65+712*q^69+21047*q^39+
2744*q^61+19203*q^41+24340*q^34+72*q^79+10090*q^50+1732*q^64+21918*q^38+24655*q
^33+5334*q^56+78*q^2+865*q^68+23132*q^26+243*q^3+517*q^4+917*q^5+1397*q^6+24885
*q^32+1907*q^7+2595*q^8+3290*q^9+2374*q^62+4083*q^10+5048*q^11+6019*q^12+155*q^
76+7243*q^13+8482*q^14+9787*q^15+11171*q^16+1240*q^66+591*q^70+122*q^77+93*q^78
+17182*q^43+11034*q^49+24305*q^28+12652*q^17+14013*q^18+15400*q^19+16704*q^20+
23766*q^27+389*q^72+20158*q^40+314*q^73+2028*q^63+24964*q^31+198*q^75+23904*q^
35+53*q^80+12008*q^48+22*q^83+39*q^81+11*q^85+7*q^86+5*q^87+3*q^88+2*q^89+14*q^
84+29*q^82+3640*q^59+23306*q^36+13030*q^47, 1+49599*q^25+61649*q^42+40206*q^22+
157*q^90+46524*q^24+36822*q^21+43424*q^23+14*q+22*q^97+41354*q^51+57986*q^44+
36302*q^53+26750*q^57+61894*q^30+53*q^94+38841*q^52+24581*q^58+31417*q^55+20478
*q^60+124*q^91+33826*q^54+55840*q^45+5684*q^71+3671*q^74+29*q^96+14*q^98+66873*
q^37+9596*q^67+59906*q^29+53650*q^46+72*q^93+11*q^99+12139*q^65+7441*q^69+65539
*q^39+18590*q^61+63189*q^41+66596*q^34+1617*q^79+43903*q^50+13589*q^64+7*q^100+
66391*q^38+65874*q^33+29050*q^56+91*q^2+8466*q^68+52498*q^26+311*q^3+717*q^4+
1342*q^5+2160*q^6+64918*q^32+3067*q^7+4238*q^8+5544*q^9+39*q^95+16812*q^62+5*q^
101+6953*q^10+8731*q^11+10544*q^12+2685*q^76+12787*q^13+15235*q^14+17826*q^15+
20700*q^16+10808*q^66+6523*q^70+2273*q^77+1918*q^78+59911*q^43+3*q^102+46433*q^
49+57681*q^28+23861*q^17+27063*q^18+30341*q^19+33653*q^20+55172*q^27+4941*q^72+
64523*q^40+4272*q^73+2*q^103+15137*q^63+63582*q^31+3142*q^75+67009*q^35+q^104+
1346*q^80+48880*q^48+q^105+756*q^83+1121*q^81+504*q^85+403*q^86+324*q^87+257*q^
88+202*q^89+619*q^84+923*q^82+22466*q^59+67071*q^36+93*q^92+51306*q^47]


 The number of permutations avoiding, {[1, 3, 2], [4, 3, 5, 1, 2]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 3, 2], [4, 3, 5, 1, 2]},

     is given by

[1, 1, 3, 7, 22, 60, 184, 544, 1639, 4915, 14749, 44281, 132820, 398590,

    1195666]

The number of ODD permutations avoiding, {[1, 3, 2], [4, 3, 5, 1, 2]},

     is given by

[0, 1, 2, 7, 19, 62, 181, 550, 1642, 4927, 14776, 44293, 132901, 398572,

    1195819]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 62, 181, 544, 1639, 4927, 14776, 44281, 132820, 398572,

    1195819]

 ODD:, [0, 1, 3, 7, 19, 60, 184, 550, 1642, 4915, 14749, 44293, 132901, 398590,

    1195666]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.780487805, 8.868852459,

    12.62191781, 17.04021938, 22.12435233, 27.87472059, 34.29154953,

    41.37495202, 49.12498071, 57.54165904, 66.62499702]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.342733395, 3.154364189,

    4.057067116, 5.042980745, 6.106024133, 7.241589246, 8.446005909,

    9.716170096, 11.04934181, 12.44304630, 13.89502555]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.48840, 9.95001, 16.4598, 25.4317,

    37.2835, 52.4406, 71.3350, 94.4040, 122.088, 154.829, 193.072]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4180400731,

    -0.4168425707, -0.4069032589, -0.3949647839, -0.3826642832, -0.3703720952,

    -0.3582830163, -0.3465179134, -0.3351768591, -0.3243235942, -0.3139936430]

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

    2.578362051, 2.676644199, 2.735952593, 2.776529849, 2.805887345,

    2.827604824, 2.843899336, 2.856358943, 2.866093594, 2.873925609,

    2.880344024]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [2, 3, 4, 5, 1]},

    {[2, 3, 1], [1, 5, 4, 3, 2]}, {[1, 3, 2], [5, 1, 2, 3, 4]},

    {[2, 1, 3], [5, 1, 2, 3, 4]}, {[1, 3, 2], [2, 3, 4, 5, 1]},

    {[3, 1, 2], [1, 5, 4, 3, 2]}, {[2, 3, 1], [4, 3, 2, 1, 5]},

    {[3, 1, 2], [4, 3, 2, 1, 5]}}

      the member , {[2, 1, 3], [2, 3, 4, 5, 1]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

    [[1, 2, 3], {[1, 0, 0, 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+4*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+4*q^4+6*q^5+8*q^6+9*q^7+11*
q^8+14*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+4*q^4+6
*q^5+9*q^6+10*q^7+13*q^8+17*q^9+21*q^10+24*q^11+26*q^12+30*q^13+36*q^14+40*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+4*q^4+6*q^5+9*q^6
+11*q^7+14*q^8+19*q^9+24*q^10+28*q^11+33*q^12+40*q^13+48*q^14+57*q^15+63*q^16+
67*q^17+72*q^18+81*q^19+92*q^20+100*q^21+98*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q^8+20*q^9+26*q^10+
31*q^11+37*q^12+46*q^13+58*q^14+69*q^15+79*q^16+89*q^17+101*q^18+117*q^19+137*q
^20+156*q^21+169*q^22+177*q^23+183*q^24+195*q^25+215*q^26+237*q^27+250*q^28+244
*q^29+214*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+4
*q^4+6*q^5+9*q^6+11*q^7+15*q^8+21*q^9+27*q^10+33*q^11+40*q^12+50*q^13+64*q^14+
78*q^15+91*q^16+105*q^17+123*q^18+146*q^19+172*q^20+200*q^21+226*q^22+249*q^23+
271*q^24+301*q^25+341*q^26+386*q^27+428*q^28+460*q^29+475*q^30+482*q^31+494*q^
32+522*q^33+566*q^34+610*q^35+631*q^36+610*q^37+539*q^38+428*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q
^8+21*q^9+28*q^10+34*q^11+42*q^12+53*q^13+68*q^14+84*q^15+100*q^16+116*q^17+139
*q^18+168*q^19+201*q^20+236*q^21+270*q^22+305*q^23+345*q^24+392*q^25+450*q^26+
516*q^27+584*q^28+648*q^29+702*q^30+753*q^31+811*q^32+886*q^33+980*q^34+1085*q^
35+1182*q^36+1253*q^37+1288*q^38+1295*q^39+1296*q^40+1322*q^41+1388*q^42+1483*q
^43+1570*q^44+1601*q^45+1536*q^46+1363*q^47+1101*q^48+799*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q
^8+21*q^9+28*q^10+35*q^11+43*q^12+55*q^13+71*q^14+88*q^15+106*q^16+125*q^17+150
*q^18+183*q^19+223*q^20+265*q^21+306*q^22+350*q^23+402*q^24+466*q^25+543*q^26+
628*q^27+717*q^28+811*q^29+902*q^30+994*q^31+1097*q^32+1222*q^33+1372*q^34+1539
*q^35+1708*q^36+1866*q^37+2002*q^38+2122*q^39+2242*q^40+2386*q^41+2573*q^42+
2801*q^43+3044*q^44+3266*q^45+3429*q^46+3509*q^47+3510*q^48+3474*q^49+3460*q^50
+3520*q^51+3672*q^52+3875*q^53+4044*q^54+4076*q^55+3890*q^56+3461*q^57+2835*q^
58+2111*q^59+1411*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q^8+21*q^9+28*q^10+35*q^11+44*q^12+56*q^13
+73*q^14+91*q^15+110*q^16+131*q^17+159*q^18+194*q^19+238*q^20+286*q^21+335*q^22
+386*q^23+447*q^24+524*q^25+618*q^26+722*q^27+832*q^28+947*q^29+1068*q^30+1200*
q^31+1351*q^32+1526*q^33+1727*q^34+1952*q^35+2193*q^36+2437*q^37+2674*q^38+2907
*q^39+3154*q^40+3437*q^41+3766*q^42+4147*q^43+4564*q^44+4989*q^45+5389*q^46+
5737*q^47+6029*q^48+6297*q^49+6587*q^50+6950*q^51+7410*q^52+7953*q^53+8523*q^54
+9036*q^55+9407*q^56+9584*q^57+9566*q^58+9416*q^59+9249*q^60+9195*q^61+9339*q^
62+9680*q^63+10108*q^64+10425*q^65+10409*q^66+9891*q^67+8823*q^68+7306*q^69+
5558*q^70+3841*q^71+2380*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q^8+21*q^9+28*q^10+35*q^11+44*q^12+
57*q^13+74*q^14+93*q^15+113*q^16+135*q^17+165*q^18+203*q^19+249*q^20+301*q^21+
356*q^22+414*q^23+483*q^24+569*q^25+676*q^26+798*q^27+927*q^28+1063*q^29+1208*q
^30+1370*q^31+1560*q^32+1786*q^33+2043*q^34+2325*q^35+2627*q^36+2946*q^37+3279*
q^38+3628*q^39+4002*q^40+4421*q^41+4902*q^42+5452*q^43+6057*q^44+6693*q^45+7333
*q^46+7960*q^47+8571*q^48+9190*q^49+9853*q^50+10604*q^51+11469*q^52+12449*q^53+
13505*q^54+14571*q^55+15575*q^56+16459*q^57+17204*q^58+17844*q^59+18462*q^60+
19172*q^61+20071*q^62+21193*q^63+22488*q^64+23819*q^65+25000*q^66+25847*q^67+
26233*q^68+26138*q^69+25669*q^70+25040*q^71+24524*q^72+24369*q^73+24704*q^74+
25456*q^75+26338*q^76+26898*q^77+26649*q^78+25233*q^79+22559*q^80+18852*q^81+
14594*q^82+10368*q^83+6687*q^84+3862*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+4*q^4+6*q^5+9*q^6+11*q^7+15*q^8+21*q^9+28*q^10+35
*q^11+44*q^12+57*q^13+75*q^14+94*q^15+115*q^16+138*q^17+169*q^18+209*q^19+258*q
^20+312*q^21+371*q^22+435*q^23+511*q^24+604*q^25+721*q^26+856*q^27+1003*q^28+
1159*q^29+1325*q^30+1511*q^31+1734*q^32+2000*q^33+2307*q^34+2647*q^35+3012*q^36
+3397*q^37+3809*q^38+4258*q^39+4758*q^40+5320*q^41+5954*q^42+6666*q^43+7450*q^
44+8294*q^45+9183*q^46+10100*q^47+11035*q^48+12011*q^49+13070*q^50+14258*q^51+
15595*q^52+17077*q^53+18680*q^54+20365*q^55+22075*q^56+23757*q^57+25383*q^58+
26977*q^59+28608*q^60+30374*q^61+32364*q^62+34629*q^63+37147*q^64+39824*q^65+
42510*q^66+45047*q^67+47294*q^68+49188*q^69+50768*q^70+52176*q^71+53632*q^72+
55374*q^73+57578*q^74+60286*q^75+63352*q^76+66455*q^77+69169*q^78+71076*q^79+
71889*q^80+71534*q^81+70186*q^82+68250*q^83+66283*q^84+64869*q^85+64458*q^86+
65201*q^87+66819*q^88+68584*q^89+69460*q^90+68371*q^91+64550*q^92+57821*q^93+
48699*q^94+38256*q^95+27808*q^96+18537*q^97+11210*q^98+6062*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+q^2+q^3, 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+4
*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+4*q^11+6*q^10+8*q^9
+9*q^8+11*q^7+14*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
+4*q^17+6*q^16+9*q^15+10*q^14+13*q^13+17*q^12+21*q^11+24*q^10+26*q^9+30*q^8+36*
q^7+40*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+9*q^22+4*q^24+11*q^21+6*
q^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+86*q^5+98*q^6+100*q^7+92*q^8+81*q^9+72*q^
10+67*q^11+63*q^12+57*q^13+48*q^14+40*q^15+33*q^16+q^28+28*q^17+24*q^18+19*q^19
+14*q^20+q^27, 1+31*q^25+58*q^22+37*q^24+69*q^21+46*q^23+8*q+9*q^30+11*q^29+2*q
^34+3*q^33+28*q^2+26*q^26+63*q^3+112*q^4+167*q^5+214*q^6+4*q^32+244*q^7+250*q^8
+237*q^9+215*q^10+195*q^11+183*q^12+177*q^13+169*q^14+156*q^15+137*q^16+15*q^28
+117*q^17+101*q^18+89*q^19+79*q^20+20*q^27+6*q^31+q^35+q^36, 1+172*q^25+3*q^42+
249*q^22+200*q^24+271*q^21+226*q^23+9*q+q^44+78*q^30+q^45+15*q^37+91*q^29+9*q^
39+4*q^41+33*q^34+11*q^38+40*q^33+36*q^2+146*q^26+92*q^3+182*q^4+301*q^5+428*q^
6+50*q^32+539*q^7+610*q^8+631*q^9+610*q^10+566*q^11+522*q^12+494*q^13+482*q^14+
475*q^15+460*q^16+2*q^43+105*q^28+428*q^17+386*q^18+341*q^19+301*q^20+123*q^27+
6*q^40+64*q^31+27*q^35+21*q^36, 1+702*q^25+53*q^42+886*q^22+753*q^24+980*q^21+
811*q^23+10*q+4*q^51+34*q^44+2*q^53+392*q^30+3*q^52+q^55+q^54+28*q^45+139*q^37+
450*q^29+21*q^46+100*q^39+68*q^41+236*q^34+6*q^50+116*q^38+270*q^33+45*q^2+648*
q^26+129*q^3+282*q^4+512*q^5+799*q^6+305*q^32+1101*q^7+1363*q^8+1536*q^9+1601*q
^10+1570*q^11+1483*q^12+1388*q^13+1322*q^14+1296*q^15+1295*q^16+42*q^43+9*q^49+
516*q^28+1288*q^17+1253*q^18+1182*q^19+1085*q^20+584*q^27+84*q^40+345*q^31+201*
q^35+11*q^48+168*q^36+15*q^47, 1+2386*q^25+402*q^42+3044*q^22+2573*q^24+3266*q^
21+2801*q^23+11*q+88*q^51+306*q^44+55*q^53+21*q^57+1708*q^30+71*q^52+15*q^58+35
*q^55+9*q^60+43*q^54+265*q^45+811*q^37+1866*q^29+223*q^46+q^65+628*q^39+6*q^61+
466*q^41+1097*q^34+106*q^50+2*q^64+717*q^38+1222*q^33+28*q^56+55*q^2+2242*q^26+
175*q^3+420*q^4+831*q^5+1411*q^6+1372*q^32+2111*q^7+2835*q^8+3461*q^9+4*q^62+
3890*q^10+4076*q^11+4044*q^12+3875*q^13+3672*q^14+3520*q^15+3460*q^16+q^66+350*
q^43+125*q^49+2002*q^28+3474*q^17+3510*q^18+3509*q^19+3429*q^20+2122*q^27+543*q
^40+3*q^63+1539*q^31+994*q^35+150*q^48+11*q^59+902*q^36+183*q^47, 1+7953*q^25+
2193*q^42+9407*q^22+8523*q^24+9584*q^21+9036*q^23+12*q+722*q^51+1727*q^44+524*q
^53+286*q^57+6029*q^30+618*q^52+238*q^58+386*q^55+159*q^60+447*q^54+1526*q^45+
11*q^71+4*q^74+3437*q^37+35*q^67+6297*q^29+1351*q^46+56*q^65+21*q^69+2907*q^39+
131*q^61+2437*q^41+4564*q^34+832*q^50+73*q^64+3154*q^38+4989*q^33+335*q^56+66*q
^2+28*q^68+7410*q^26+231*q^3+605*q^4+1297*q^5+2380*q^6+5389*q^32+3841*q^7+5558*
q^8+7306*q^9+110*q^62+8823*q^10+9891*q^11+10409*q^12+2*q^76+10425*q^13+10108*q^
14+9680*q^15+9339*q^16+44*q^66+15*q^70+q^77+q^78+1952*q^43+947*q^49+6587*q^28+
9195*q^17+9249*q^18+9416*q^19+9566*q^20+6950*q^27+9*q^72+2674*q^40+6*q^73+91*q^
63+5737*q^31+3*q^75+4147*q^35+1068*q^48+194*q^59+3766*q^36+1200*q^47, 1+25000*q
^25+9190*q^42+26138*q^22+q^90+25847*q^24+25669*q^21+26233*q^23+13*q+4002*q^51+
7960*q^44+3279*q^53+2043*q^57+19172*q^30+3628*q^52+1786*q^58+2627*q^55+1370*q^
60+q^91+2946*q^54+7333*q^45+249*q^71+135*q^74+13505*q^37+483*q^67+20071*q^29+
6693*q^46+676*q^65+356*q^69+11469*q^39+1208*q^61+9853*q^41+16459*q^34+44*q^79+
4421*q^50+798*q^64+12449*q^38+17204*q^33+2325*q^56+78*q^2+414*q^68+23819*q^26+
298*q^3+847*q^4+1958*q^5+3862*q^6+17844*q^32+6687*q^7+10368*q^8+14594*q^9+1063*
q^62+18852*q^10+22559*q^11+25233*q^12+93*q^76+26649*q^13+26898*q^14+26338*q^15+
25456*q^16+569*q^66+301*q^70+74*q^77+57*q^78+8571*q^43+4902*q^49+21193*q^28+
24704*q^17+24369*q^18+24524*q^19+25040*q^20+22488*q^27+203*q^72+10604*q^40+165*
q^73+927*q^63+18462*q^31+113*q^75+15575*q^35+35*q^80+5452*q^48+15*q^83+28*q^81+
9*q^85+6*q^86+4*q^87+3*q^88+2*q^89+11*q^84+21*q^82+1560*q^59+14571*q^36+6057*q^
47, 1+71889*q^25+34629*q^42+68250*q^22+94*q^90+71534*q^24+66283*q^21+70186*q^23
+14*q+15*q^97+18680*q^51+30374*q^44+15595*q^53+11035*q^57+60286*q^30+35*q^94+
17077*q^52+10100*q^58+13070*q^55+8294*q^60+75*q^91+14258*q^54+28608*q^45+2307*q
^71+1511*q^74+21*q^96+11*q^98+47294*q^37+3809*q^67+63352*q^29+26977*q^46+44*q^
93+9*q^99+4758*q^65+3012*q^69+42510*q^39+7450*q^61+37147*q^41+52176*q^34+721*q^
79+20365*q^50+5320*q^64+6*q^100+45047*q^38+53632*q^33+12011*q^56+91*q^2+3397*q^
68+71076*q^26+377*q^3+1157*q^4+2872*q^5+6062*q^6+55374*q^32+11210*q^7+18537*q^8
+27808*q^9+28*q^95+6666*q^62+4*q^101+38256*q^10+48699*q^11+57821*q^12+1159*q^76
+64550*q^13+68371*q^14+69460*q^15+68584*q^16+4258*q^66+2647*q^70+1003*q^77+856*
q^78+32364*q^43+3*q^102+22075*q^49+66455*q^28+66819*q^17+65201*q^18+64458*q^19+
64869*q^20+69169*q^27+2000*q^72+39824*q^40+1734*q^73+2*q^103+5954*q^63+57578*q^
31+1325*q^75+50768*q^35+q^104+604*q^80+23757*q^48+q^105+371*q^83+511*q^81+258*q
^85+209*q^86+169*q^87+138*q^88+115*q^89+312*q^84+435*q^82+9183*q^59+49188*q^36+
57*q^92+25383*q^47]


 The number of permutations avoiding, {[2, 1, 3], [2, 3, 4, 5, 1]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [2, 3, 4, 5, 1]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[2, 1, 3], [2, 3, 4, 5, 1]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.853658537, 9.139344262,

    13.23835616, 18.16179159, 23.91466016, 30.49918716, 37.91634208,

    46.16653871, 55.24995014, 65.16664743, 75.91665931]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.332809210, 3.129226515,

    4.017176482, 4.991616635, 6.047059805, 7.178455751, 8.381390143,

    9.652043725, 10.98709768, 12.38364401, 13.83911283]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.44200, 9.79206, 16.1377, 24.9162,

    36.5669, 51.5302, 70.2477, 93.1619, 120.716, 153.355, 191.521]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5032000763,

    -0.6010069452, -0.6730520263, -0.7217571521, -0.7523760618, -0.7698742491,

    -0.7781415014, -0.7800145127, -0.7775141181, -0.7720410279, -0.7646047799]

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

    2.673308354, 2.881030569, 3.031655581, 3.139797927, 3.216234550,

    3.269185744, 3.304956721, 3.328279218, 3.342665466, 3.350587528,

    3.353980802]

                                end of this data



For the equivalence class of patterns, {{[3, 1, 2], [3, 2, 4, 1, 5]},

    {[2, 1, 3], [2, 4, 5, 3, 1]}, {[2, 1, 3], [5, 1, 4, 2, 3]},

    {[2, 3, 1], [4, 2, 1, 3, 5]}, {[1, 3, 2], [5, 3, 1, 2, 4]},

    {[1, 3, 2], [3, 4, 2, 5, 1]}, {[2, 3, 1], [1, 5, 2, 4, 3]},

    {[3, 1, 2], [1, 3, 5, 4, 2]}}

      the member , {[3, 1, 2], [3, 2, 4, 1, 5]}, has a scheme of depth , 5

                                  here it is:

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

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

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

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

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

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

    [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {1, 2}],

    [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 0, 0]}, {3}], [[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], [0, 1, 0, 0, 0, 0]}, {3}],

    [[2, 1, 5, 3, 4], {[0, 0, 0, 0, 0, 0]}, {3}], [[2, 1, 5, 4, 3],

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

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

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

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 1+4*q+6*q^2+7*q^3+6*q^
4+5*q^5+5*q^6+3*q^7+2*q^8+q^9+q^10, 1+5*q+10*q^2+14*q^3+15*q^4+13*q^5+14*q^6+13
*q^7+10*q^8+8*q^9+7*q^10+5*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+15*q^2+25*q^3+32
*q^4+32*q^5+33*q^6+34*q^7+32*q^8+30*q^9+27*q^10+22*q^11+19*q^12+16*q^13+12*q^14
+10*q^15+7*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+21*q^2+41*q^3+61*q^4+71*q
^5+76*q^6+80*q^7+81*q^8+82*q^9+81*q^10+77*q^11+68*q^12+60*q^13+53*q^14+48*q^15+
42*q^16+34*q^17+27*q^18+22*q^19+18*q^20+14*q^21+10*q^22+7*q^23+5*q^24+3*q^25+2*
q^26+q^27+q^28, 1+8*q+28*q^2+63*q^3+107*q^4+143*q^5+166*q^6+182*q^7+191*q^8+199
*q^9+206*q^10+210*q^11+207*q^12+196*q^13+178*q^14+165*q^15+153*q^16+137*q^17+
124*q^18+110*q^19+94*q^20+81*q^21+69*q^22+58*q^23+49*q^24+39*q^25+30*q^26+24*q^
27+20*q^28+14*q^29+10*q^30+7*q^31+5*q^32+3*q^33+2*q^34+q^35+q^36, 1+9*q+36*q^2+
92*q^3+176*q^4+266*q^5+340*q^6+396*q^7+434*q^8+464*q^9+490*q^10+513*q^11+534*q^
12+542*q^13+529*q^14+510*q^15+487*q^16+453*q^17+422*q^18+392*q^19+363*q^20+335*
q^21+302*q^22+265*q^23+236*q^24+208*q^25+180*q^26+158*q^27+138*q^28+115*q^29+95
*q^30+79*q^31+65*q^32+54*q^33+42*q^34+32*q^35+26*q^36+20*q^37+14*q^38+10*q^39+7
*q^40+5*q^41+3*q^42+2*q^43+q^44+q^45, 1+10*q+45*q^2+129*q^3+275*q^4+464*q^5+654
*q^6+819*q^7+947*q^8+1048*q^9+1133*q^10+1206*q^11+1279*q^12+1344*q^13+1383*q^14
+1399*q^15+1395*q^16+1360*q^17+1305*q^18+1242*q^19+1176*q^20+1114*q^21+1056*q^
22+986*q^23+911*q^24+835*q^25+759*q^26+685*q^27+619*q^28+557*q^29+498*q^30+445*
q^31+390*q^32+338*q^33+293*q^34+252*q^35+216*q^36+186*q^37+157*q^38+129*q^39+
105*q^40+86*q^41+70*q^42+57*q^43+44*q^44+34*q^45+26*q^46+20*q^47+14*q^48+10*q^
49+7*q^50+5*q^51+3*q^52+2*q^53+q^54+q^55, 1+11*q+55*q^2+175*q^3+412*q^4+768*q^5
+1189*q^6+1609*q^7+1978*q^8+2286*q^9+2548*q^10+2772*q^11+2981*q^12+3183*q^13+
3359*q^14+3505*q^15+3628*q^16+3690*q^17+3683*q^18+3623*q^19+3520*q^20+3407*q^21
+3286*q^22+3149*q^23+3009*q^24+2860*q^25+2698*q^26+2530*q^27+2354*q^28+2174*q^
29+2000*q^30+1836*q^31+1678*q^32+1537*q^33+1399*q^34+1259*q^35+1130*q^36+1012*q
^37+896*q^38+791*q^39+696*q^40+611*q^41+537*q^42+465*q^43+396*q^44+339*q^45+288
*q^46+242*q^47+205*q^48+171*q^49+139*q^50+112*q^51+91*q^52+73*q^53+59*q^54+46*q
^55+34*q^56+26*q^57+20*q^58+14*q^59+10*q^60+7*q^61+5*q^62+3*q^63+2*q^64+q^65+q^
66, 1+12*q+66*q^2+231*q^3+596*q^4+1217*q^5+2058*q^6+3011*q^7+3952*q^8+4804*q^9+
5557*q^10+6216*q^11+6813*q^12+7379*q^13+7905*q^14+8389*q^15+8853*q^16+9266*q^17
+9578*q^18+9749*q^19+9768*q^20+9695*q^21+9554*q^22+9348*q^23+9106*q^24+8829*q^
25+8520*q^26+8201*q^27+7868*q^28+7499*q^29+7099*q^30+6676*q^31+6235*q^32+5823*q
^33+5427*q^34+5037*q^35+4670*q^36+4319*q^37+3975*q^38+3643*q^39+3317*q^40+3006*
q^41+2721*q^42+2454*q^43+2203*q^44+1983*q^45+1776*q^46+1574*q^47+1395*q^48+1230
*q^49+1074*q^50+935*q^51+813*q^52+705*q^53+613*q^54+525*q^55+442*q^56+373*q^57+
314*q^58+261*q^59+219*q^60+181*q^61+146*q^62+117*q^63+94*q^64+75*q^65+61*q^66+
46*q^67+34*q^68+26*q^69+20*q^70+14*q^71+10*q^72+7*q^73+5*q^74+3*q^75+2*q^76+q^
77+q^78, 1+13*q+78*q^2+298*q^3+837*q^4+1859*q^5+3414*q^6+5390*q^7+7564*q^8+9716
*q^9+11730*q^10+13562*q^11+15232*q^12+16795*q^13+18264*q^14+19642*q^15+20970*q^
16+22267*q^17+23497*q^18+24561*q^19+25344*q^20+25842*q^21+26082*q^22+26085*q^23
+25904*q^24+25583*q^25+25130*q^26+24589*q^27+23990*q^28+23334*q^29+22631*q^30+
21852*q^31+20955*q^32+19983*q^33+18990*q^34+17972*q^35+16959*q^36+15980*q^37+
15021*q^38+14080*q^39+13172*q^40+12296*q^41+11442*q^42+10604*q^43+9777*q^44+
8986*q^45+8257*q^46+7560*q^47+6907*q^48+6304*q^49+5734*q^50+5196*q^51+4694*q^52
+4223*q^53+3788*q^54+3389*q^55+3016*q^56+2675*q^57+2378*q^58+2103*q^59+1848*q^
60+1621*q^61+1413*q^62+1222*q^63+1054*q^64+908*q^65+783*q^66+673*q^67+569*q^68+
476*q^69+399*q^70+333*q^71+275*q^72+229*q^73+188*q^74+151*q^75+120*q^76+96*q^77
+77*q^78+61*q^79+46*q^80+34*q^81+26*q^82+20*q^83+14*q^84+10*q^85+7*q^86+5*q^87+
3*q^88+2*q^89+q^90+q^91, 1+14*q+91*q^2+377*q^3+1146*q^4+2752*q^5+5459*q^6+9272*
q^7+13906*q^8+18920*q^9+23941*q^10+28739*q^11+33232*q^12+37460*q^13+41460*q^14+
45246*q^15+48869*q^16+52406*q^17+55910*q^18+59333*q^19+62468*q^20+65165*q^21+
67343*q^22+68892*q^23+69841*q^24+70276*q^25+70229*q^26+69827*q^27+69157*q^28+
68232*q^29+67126*q^30+65872*q^31+64420*q^32+62757*q^33+60896*q^34+58816*q^35+
56570*q^36+54217*q^37+51784*q^38+49336*q^39+46894*q^40+44477*q^41+42125*q^42+
39834*q^43+37587*q^44+35379*q^45+33200*q^46+31033*q^47+28942*q^48+26931*q^49+
25002*q^50+23180*q^51+21447*q^52+19796*q^53+18250*q^54+16805*q^55+15423*q^56+
14105*q^57+12861*q^58+11688*q^59+10615*q^60+9618*q^61+8689*q^62+7838*q^63+7052*
q^64+6326*q^65+5665*q^66+5057*q^67+4490*q^68+3975*q^69+3506*q^70+3084*q^71+2716
*q^72+2385*q^73+2079*q^74+1808*q^75+1563*q^76+1342*q^77+1151*q^78+986*q^79+841*
q^80+717*q^81+603*q^82+502*q^83+418*q^84+347*q^85+285*q^86+236*q^87+193*q^88+
154*q^89+122*q^90+98*q^91+77*q^92+61*q^93+46*q^94+34*q^95+26*q^96+20*q^97+14*q^
98+10*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+q+2*q^2+q^3, 1+q+2*q^2+3*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+6*q^8+7*q^7
+6*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1, q^15+5*q^14+10*q^13+14*q^12+15*q^11+13*q^10
+14*q^9+13*q^8+10*q^7+8*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+15*q^19+25
*q^18+32*q^17+32*q^16+33*q^15+34*q^14+32*q^13+30*q^12+27*q^11+22*q^10+19*q^9+16
*q^8+12*q^7+10*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+41*q^25+76*q^22+61*q^24+80*q^
21+71*q^23+q+2*q^2+21*q^26+3*q^3+5*q^4+7*q^5+10*q^6+14*q^7+18*q^8+22*q^9+27*q^
10+34*q^11+42*q^12+48*q^13+53*q^14+60*q^15+68*q^16+q^28+77*q^17+81*q^18+82*q^19
+81*q^20+7*q^27, 1+210*q^25+178*q^22+207*q^24+165*q^21+196*q^23+q+166*q^30+182*
q^29+28*q^34+63*q^33+2*q^2+206*q^26+3*q^3+5*q^4+7*q^5+10*q^6+107*q^32+14*q^7+20
*q^8+24*q^9+30*q^10+39*q^11+49*q^12+58*q^13+69*q^14+81*q^15+94*q^16+191*q^28+
110*q^17+124*q^18+137*q^19+153*q^20+199*q^27+143*q^31+8*q^35+q^36, 1+363*q^25+
92*q^42+265*q^22+335*q^24+236*q^21+302*q^23+q+9*q^44+510*q^30+q^45+434*q^37+487
*q^29+340*q^39+176*q^41+513*q^34+396*q^38+534*q^33+2*q^2+392*q^26+3*q^3+5*q^4+7
*q^5+10*q^6+542*q^32+14*q^7+20*q^8+26*q^9+32*q^10+42*q^11+54*q^12+65*q^13+79*q^
14+95*q^15+115*q^16+36*q^43+453*q^28+138*q^17+158*q^18+180*q^19+208*q^20+422*q^
27+266*q^40+529*q^31+490*q^35+464*q^36, 1+498*q^25+1344*q^42+338*q^22+445*q^24+
293*q^21+390*q^23+q+275*q^51+1206*q^44+45*q^53+835*q^30+129*q^52+q^55+10*q^54+
1133*q^45+1305*q^37+759*q^29+1048*q^46+1395*q^39+1383*q^41+1114*q^34+464*q^50+
1360*q^38+1056*q^33+2*q^2+557*q^26+3*q^3+5*q^4+7*q^5+10*q^6+986*q^32+14*q^7+20*
q^8+26*q^9+34*q^10+44*q^11+57*q^12+70*q^13+86*q^14+105*q^15+129*q^16+1279*q^43+
654*q^49+685*q^28+157*q^17+186*q^18+216*q^19+252*q^20+619*q^27+1399*q^40+911*q^
31+1176*q^35+819*q^48+1242*q^36+947*q^47, 1+611*q^25+3009*q^42+396*q^22+537*q^
24+339*q^21+465*q^23+q+3505*q^51+3286*q^44+3183*q^53+2286*q^57+1130*q^30+3359*q
^52+1978*q^58+2772*q^55+1189*q^60+2981*q^54+3407*q^45+2174*q^37+1012*q^29+3520*
q^46+11*q^65+2530*q^39+768*q^61+2860*q^41+1678*q^34+3628*q^50+55*q^64+2354*q^38
+1537*q^33+2548*q^56+2*q^2+696*q^26+3*q^3+5*q^4+7*q^5+10*q^6+1399*q^32+14*q^7+
20*q^8+26*q^9+412*q^62+34*q^10+46*q^11+59*q^12+73*q^13+91*q^14+112*q^15+139*q^
16+q^66+3149*q^43+3690*q^49+896*q^28+171*q^17+205*q^18+242*q^19+288*q^20+791*q^
27+2698*q^40+175*q^63+1259*q^31+1836*q^35+3683*q^48+1609*q^59+2000*q^36+3623*q^
47, 1+705*q^25+4670*q^42+442*q^22+613*q^24+373*q^21+525*q^23+q+8201*q^51+5427*q
^44+8829*q^53+9695*q^57+1395*q^30+8520*q^52+9768*q^58+9348*q^55+9578*q^60+9106*
q^54+5823*q^45+3011*q^71+596*q^74+3006*q^37+6216*q^67+1230*q^29+6235*q^46+7379*
q^65+4804*q^69+3643*q^39+9266*q^61+4319*q^41+2203*q^34+7868*q^50+7905*q^64+3317
*q^38+1983*q^33+9554*q^56+2*q^2+5557*q^68+813*q^26+3*q^3+5*q^4+7*q^5+10*q^6+
1776*q^32+14*q^7+20*q^8+26*q^9+8853*q^62+34*q^10+46*q^11+61*q^12+66*q^76+75*q^
13+94*q^14+117*q^15+146*q^16+6813*q^66+3952*q^70+12*q^77+q^78+5037*q^43+7499*q^
49+1074*q^28+181*q^17+219*q^18+261*q^19+314*q^20+935*q^27+2058*q^72+3975*q^40+
1217*q^73+8389*q^63+1574*q^31+231*q^75+2454*q^35+7099*q^48+9749*q^59+2721*q^36+
6676*q^47, 1+783*q^25+6304*q^42+476*q^22+13*q^90+673*q^24+399*q^21+569*q^23+q+
13172*q^51+7560*q^44+15021*q^53+18990*q^57+1621*q^30+14080*q^52+19983*q^58+
16959*q^55+21852*q^60+q^91+15980*q^54+8257*q^45+25344*q^71+22267*q^74+3788*q^37
+25904*q^67+1413*q^29+8986*q^46+25130*q^65+26082*q^69+4694*q^39+22631*q^61+5734
*q^41+2675*q^34+15232*q^79+12296*q^50+24589*q^64+4223*q^38+2378*q^33+17972*q^56
+2*q^2+26085*q^68+908*q^26+3*q^3+5*q^4+7*q^5+10*q^6+2103*q^32+14*q^7+20*q^8+26*
q^9+23334*q^62+34*q^10+46*q^11+61*q^12+19642*q^76+77*q^13+96*q^14+120*q^15+151*
q^16+25583*q^66+25842*q^70+18264*q^77+16795*q^78+6907*q^43+11442*q^49+1222*q^28
+188*q^17+229*q^18+275*q^19+333*q^20+1054*q^27+24561*q^72+5196*q^40+23497*q^73+
23990*q^63+1848*q^31+20970*q^75+3016*q^35+13562*q^80+10604*q^48+7564*q^83+11730
*q^81+3414*q^85+1859*q^86+837*q^87+298*q^88+78*q^89+5390*q^84+9716*q^82+20955*q
^59+3389*q^36+9777*q^47, 1+841*q^25+7838*q^42+502*q^22+45246*q^90+717*q^24+418*
q^21+603*q^23+q+13906*q^97+18250*q^51+9618*q^44+21447*q^53+28942*q^57+1808*q^30
+28739*q^94+19796*q^52+31033*q^58+25002*q^55+35379*q^60+41460*q^91+23180*q^54+
10615*q^45+60896*q^71+65872*q^74+18920*q^96+9272*q^98+4490*q^37+51784*q^67+1563
*q^29+11688*q^46+33232*q^93+5459*q^99+46894*q^65+56570*q^69+5665*q^39+37587*q^
61+7052*q^41+3084*q^34+70229*q^79+16805*q^50+44477*q^64+2752*q^100+5057*q^38+
2716*q^33+26931*q^56+2*q^2+54217*q^68+986*q^26+3*q^3+5*q^4+7*q^5+10*q^6+2385*q^
32+14*q^7+20*q^8+26*q^9+23941*q^95+39834*q^62+1146*q^101+34*q^10+46*q^11+61*q^
12+68232*q^76+77*q^13+98*q^14+122*q^15+154*q^16+49336*q^66+58816*q^70+69157*q^
77+69827*q^78+8689*q^43+377*q^102+15423*q^49+1342*q^28+193*q^17+236*q^18+285*q^
19+347*q^20+1151*q^27+62757*q^72+6326*q^40+64420*q^73+91*q^103+42125*q^63+2079*
q^31+67126*q^75+3506*q^35+14*q^104+70276*q^80+14105*q^48+q^105+67343*q^83+69841
*q^81+62468*q^85+59333*q^86+55910*q^87+52406*q^88+48869*q^89+65165*q^84+68892*q
^82+33200*q^59+3975*q^36+37460*q^92+12861*q^47]


 The number of permutations avoiding, {[3, 1, 2], [3, 2, 4, 1, 5]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[3, 1, 2], [3, 2, 4, 1, 5]},

     is given by

[1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

The number of ODD permutations avoiding, {[3, 1, 2], [3, 2, 4, 1, 5]},

     is given by

[0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

 ODD:, [0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.400000000, 2.642857143, 4.195121951, 6.024590164,

    8.101369863, 10.40127971, 12.90643097, 15.60424710, 18.48613040,

    21.54628898, 24.78084909, 28.18722794, 31.76370540]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.350085750, 3.178986112,

    4.106543164, 5.120025404, 6.208121487, 7.361920999, 8.574853594,

    9.842282418, 11.16103686, 12.52899468, 13.94474654]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.52290, 10.1060, 16.8637, 26.2147,

    38.5408, 54.1979, 73.5281, 96.8705, 124.569, 156.976, 194.456]

Skewness: , [Float(undefined), 0., 0.2715454176, 0.3874842230, 0.4418810426,

    0.4781412394, 0.5094299837, 0.5379152345, 0.5632349328, 0.5849036937,

    0.6027397955, 0.6168759452, 0.6276264485, 0.6354076022, 0.6406639660]

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

    2.565727201, 2.669682760, 2.755657699, 2.837136527, 2.914931757,

    2.987272100, 3.052540785, 3.109849783, 3.158898207, 3.199976192,

    3.233638178]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [4, 5, 3, 1, 2]},

    {[1, 3, 2], [4, 5, 3, 1, 2]}, {[3, 1, 2], [2, 1, 3, 5, 4]},

    {[2, 3, 1], [2, 1, 3, 5, 4]}}

      the member , {[1, 3, 2], [4, 5, 3, 1, 2]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

                 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+5*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+13
*q^8+16*q^9+14*q^10+15*q^11+11*q^12+7*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+17*q^8+22*q^9+26*q^10+30*q^11+32*q^12+35*q^13+34*q^14+34*q^
15+27*q^16+23*q^17+15*q^18+9*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+19*q^8+26*q^9+32*q^10+40*q^11+48*q^12+58*q^13+61*q^14+72*q^15+77*q^16
+80*q^17+81*q^18+78*q^19+74*q^20+64*q^21+57*q^22+40*q^23+31*q^24+19*q^25+11*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+21*q^8+28*q^9+36*q^10
+46*q^11+58*q^12+72*q^13+85*q^14+103*q^15+116*q^16+134*q^17+148*q^18+163*q^19+
175*q^20+185*q^21+186*q^22+191*q^23+187*q^24+175*q^25+163*q^26+143*q^27+124*q^
28+100*q^29+80*q^30+53*q^31+39*q^32+23*q^33+13*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+21*q^8+30*q^9+38*q^10+50*q^11+64*q^12+82*q^13+99*q^
14+125*q^15+148*q^16+177*q^17+205*q^18+237*q^19+268*q^20+304*q^21+335*q^22+362*
q^23+394*q^24+411*q^25+433*q^26+442*q^27+450*q^28+443*q^29+436*q^30+410*q^31+
387*q^32+351*q^33+309*q^34+268*q^35+221*q^36+181*q^37+136*q^38+103*q^39+66*q^40
+47*q^41+27*q^42+15*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
+21*q^8+30*q^9+40*q^10+52*q^11+68*q^12+88*q^13+109*q^14+139*q^15+170*q^16+207*q
^17+249*q^18+298*q^19+345*q^20+406*q^21+464*q^22+527*q^23+595*q^24+664*q^25+728
*q^26+793*q^27+856*q^28+909*q^29+964*q^30+1006*q^31+1035*q^32+1052*q^33+1066*q^
34+1050*q^35+1037*q^36+999*q^37+951*q^38+890*q^39+821*q^40+733*q^41+650*q^42+
562*q^43+470*q^44+388*q^45+307*q^46+238*q^47+172*q^48+126*q^49+79*q^50+55*q^51+
31*q^52+17*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+21*q^8+
30*q^9+40*q^10+54*q^11+70*q^12+92*q^13+115*q^14+149*q^15+184*q^16+229*q^17+279*
q^18+340*q^19+407*q^20+487*q^21+569*q^22+665*q^23+772*q^24+883*q^25+1007*q^26+
1134*q^27+1271*q^28+1404*q^29+1552*q^30+1684*q^31+1829*q^32+1956*q^33+2086*q^34
+2199*q^35+2305*q^36+2383*q^37+2453*q^38+2501*q^39+2521*q^40+2527*q^41+2500*q^
42+2449*q^43+2368*q^44+2280*q^45+2146*q^46+2017*q^47+1852*q^48+1686*q^49+1506*q
^50+1327*q^51+1136*q^52+968*q^53+804*q^54+648*q^55+517*q^56+393*q^57+295*q^58+
208*q^59+149*q^60+92*q^61+63*q^62+35*q^63+19*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+21*q^8+30*q^9+40*q^10+54*q^11+72*q^12+94*q^13+119*q^
14+155*q^15+194*q^16+243*q^17+301*q^18+370*q^19+449*q^20+547*q^21+651*q^22+774*
q^23+913*q^24+1069*q^25+1238*q^26+1433*q^27+1640*q^28+1861*q^29+2109*q^30+2363*
q^31+2635*q^32+2916*q^33+3204*q^34+3497*q^35+3807*q^36+4096*q^37+4392*q^38+4673
*q^39+4942*q^40+5182*q^41+5418*q^42+5604*q^43+5768*q^44+5898*q^45+5985*q^46+
6030*q^47+6032*q^48+5982*q^49+5889*q^50+5759*q^51+5567*q^52+5350*q^53+5084*q^54
+4787*q^55+4453*q^56+4119*q^57+3741*q^58+3375*q^59+2994*q^60+2625*q^61+2265*q^
62+1926*q^63+1602*q^64+1321*q^65+1065*q^66+836*q^67+646*q^68+479*q^69+352*q^70+
244*q^71+172*q^72+105*q^73+71*q^74+39*q^75+21*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+21*q^8+30*q^9+40*q^10+54*q^11+72*q^12+96*q^13+121*q
^14+159*q^15+200*q^16+253*q^17+315*q^18+392*q^19+479*q^20+589*q^21+711*q^22+854
*q^23+1023*q^24+1214*q^25+1427*q^26+1673*q^27+1951*q^28+2250*q^29+2596*q^30+
2964*q^31+3372*q^32+3811*q^33+4292*q^34+4785*q^35+5333*q^36+5890*q^37+6482*q^38
+7088*q^39+7723*q^40+8352*q^41+8999*q^42+9645*q^43+10266*q^44+10889*q^45+11476*
q^46+12031*q^47+12551*q^48+13037*q^49+13432*q^50+13803*q^51+14075*q^52+14288*q^
53+14410*q^54+14475*q^55+14411*q^56+14298*q^57+14085*q^58+13782*q^59+13401*q^60
+12943*q^61+12400*q^62+11802*q^63+11153*q^64+10429*q^65+9700*q^66+8923*q^67+
8139*q^68+7342*q^69+6580*q^70+5796*q^71+5077*q^72+4375*q^73+3729*q^74+3130*q^75
+2596*q^76+2107*q^77+1695*q^78+1337*q^79+1024*q^80+775*q^81+565*q^82+409*q^83+
280*q^84+195*q^85+118*q^86+79*q^87+43*q^88+23*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+21*q^8+30*q^9+40*q^10+54*q^11+72*q^12+96*q^13+123*q
^14+161*q^15+204*q^16+259*q^17+325*q^18+406*q^19+501*q^20+619*q^21+753*q^22+914
*q^23+1103*q^24+1322*q^25+1573*q^26+1866*q^27+2194*q^28+2570*q^29+2997*q^30+
3471*q^31+4003*q^32+4594*q^33+5247*q^34+5958*q^35+6747*q^36+7586*q^37+8506*q^38
+9482*q^39+10529*q^40+11633*q^41+12810*q^42+14025*q^43+15303*q^44+16618*q^45+
17970*q^46+19342*q^47+20740*q^48+22126*q^49+23518*q^50+24888*q^51+26214*q^52+
27500*q^53+28727*q^54+29875*q^55+30922*q^56+31889*q^57+32704*q^58+33425*q^59+
33985*q^60+34407*q^61+34667*q^62+34785*q^63+34699*q^64+34479*q^65+34069*q^66+
33503*q^67+32769*q^68+31896*q^69+30846*q^70+29690*q^71+28405*q^72+26994*q^73+
25511*q^74+23942*q^75+22316*q^76+20654*q^77+18988*q^78+17292*q^79+15656*q^80+
14033*q^81+12474*q^82+10976*q^83+9583*q^84+8250*q^85+7053*q^86+5943*q^87+4952*q
^88+4074*q^89+3309*q^90+2635*q^91+2081*q^92+1609*q^93+1212*q^94+904*q^95+651*q^
96+466*q^97+316*q^98+218*q^99+131*q^100+87*q^101+47*q^102+25*q^103+14*q^104+q^
105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+5
*q^6+5*q^5+7*q^4+7*q^3+5*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+13*q^7+16*q^6+14*q^5+15*q^4+11*q^3+7*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+17*q^13+22*q^12+26*q^11+30*q^10+32*q^9+35*q^8+34
*q^7+34*q^6+27*q^5+23*q^4+15*q^3+9*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+15*q^21+7
*q^23+7*q+11*q^2+2*q^26+19*q^3+31*q^4+40*q^5+57*q^6+64*q^7+74*q^8+78*q^9+81*q^
10+80*q^11+77*q^12+72*q^13+61*q^14+58*q^15+48*q^16+q^28+40*q^17+32*q^18+26*q^19
+19*q^20+q^27, 1+46*q^25+85*q^22+58*q^24+103*q^21+72*q^23+8*q+11*q^30+15*q^29+2
*q^34+3*q^33+13*q^2+36*q^26+23*q^3+39*q^4+53*q^5+80*q^6+5*q^32+100*q^7+124*q^8+
143*q^9+163*q^10+175*q^11+187*q^12+191*q^13+186*q^14+185*q^15+175*q^16+21*q^28+
163*q^17+148*q^18+134*q^19+116*q^20+28*q^27+7*q^31+q^35+q^36, 1+268*q^25+3*q^42
+362*q^22+304*q^24+394*q^21+335*q^23+9*q+q^44+125*q^30+q^45+21*q^37+148*q^29+11
*q^39+5*q^41+50*q^34+15*q^38+64*q^33+15*q^2+237*q^26+27*q^3+47*q^4+66*q^5+103*q
^6+82*q^32+136*q^7+181*q^8+221*q^9+268*q^10+309*q^11+351*q^12+387*q^13+410*q^14
+436*q^15+443*q^16+2*q^43+177*q^28+450*q^17+442*q^18+433*q^19+411*q^20+205*q^27
+7*q^40+99*q^31+38*q^35+30*q^36, 1+964*q^25+88*q^42+1052*q^22+1006*q^24+1066*q^
21+1035*q^23+10*q+5*q^51+52*q^44+2*q^53+664*q^30+3*q^52+q^55+q^54+40*q^45+249*q
^37+728*q^29+30*q^46+170*q^39+109*q^41+406*q^34+7*q^50+207*q^38+464*q^33+17*q^2
+909*q^26+31*q^3+55*q^4+79*q^5+126*q^6+527*q^32+172*q^7+238*q^8+307*q^9+388*q^
10+470*q^11+562*q^12+650*q^13+733*q^14+821*q^15+890*q^16+68*q^43+11*q^49+793*q^
28+951*q^17+999*q^18+1037*q^19+1050*q^20+856*q^27+139*q^40+595*q^31+345*q^35+15
*q^48+298*q^36+21*q^47, 1+2527*q^25+772*q^42+2368*q^22+2500*q^24+2280*q^21+2449
*q^23+11*q+149*q^51+569*q^44+92*q^53+30*q^57+2305*q^30+115*q^52+21*q^58+54*q^55
+11*q^60+70*q^54+487*q^45+1404*q^37+2383*q^29+407*q^46+q^65+1134*q^39+7*q^61+
883*q^41+1829*q^34+184*q^50+2*q^64+1271*q^38+1956*q^33+40*q^56+19*q^2+2521*q^26
+35*q^3+63*q^4+92*q^5+149*q^6+2086*q^32+208*q^7+295*q^8+393*q^9+5*q^62+517*q^10
+648*q^11+804*q^12+968*q^13+1136*q^14+1327*q^15+1506*q^16+q^66+665*q^43+229*q^
49+2453*q^28+1686*q^17+1852*q^18+2017*q^19+2146*q^20+2501*q^27+1007*q^40+3*q^63
+2199*q^31+1684*q^35+279*q^48+15*q^59+1552*q^36+340*q^47, 1+5350*q^25+3807*q^42
+4453*q^22+5084*q^24+4119*q^21+4787*q^23+12*q+1433*q^51+3204*q^44+1069*q^53+547
*q^57+6032*q^30+1238*q^52+449*q^58+774*q^55+301*q^60+913*q^54+2916*q^45+15*q^71
+5*q^74+5182*q^37+54*q^67+5982*q^29+2635*q^46+94*q^65+30*q^69+4673*q^39+243*q^
61+4096*q^41+5768*q^34+1640*q^50+119*q^64+4942*q^38+5898*q^33+651*q^56+21*q^2+
40*q^68+5567*q^26+39*q^3+71*q^4+105*q^5+172*q^6+5985*q^32+244*q^7+352*q^8+479*q
^9+194*q^62+646*q^10+836*q^11+1065*q^12+2*q^76+1321*q^13+1602*q^14+1926*q^15+
2265*q^16+72*q^66+21*q^70+q^77+q^78+3497*q^43+1861*q^49+5889*q^28+2625*q^17+
2994*q^18+3375*q^19+3741*q^20+5759*q^27+11*q^72+4392*q^40+7*q^73+155*q^63+6030*
q^31+3*q^75+5604*q^35+2109*q^48+370*q^59+5418*q^36+2363*q^47, 1+9700*q^25+13037
*q^42+7342*q^22+q^90+8923*q^24+6580*q^21+8139*q^23+13*q+7723*q^51+12031*q^44+
6482*q^53+4292*q^57+12943*q^30+7088*q^52+3811*q^58+5333*q^55+2964*q^60+q^91+
5890*q^54+11476*q^45+479*q^71+253*q^74+14410*q^37+1023*q^67+12400*q^29+10889*q^
46+1427*q^65+711*q^69+14075*q^39+2596*q^61+13432*q^41+14298*q^34+72*q^79+8352*q
^50+1673*q^64+14288*q^38+14085*q^33+4785*q^56+23*q^2+854*q^68+10429*q^26+43*q^3
+79*q^4+118*q^5+195*q^6+13782*q^32+280*q^7+409*q^8+565*q^9+2250*q^62+775*q^10+
1024*q^11+1337*q^12+159*q^76+1695*q^13+2107*q^14+2596*q^15+3130*q^16+1214*q^66+
589*q^70+121*q^77+96*q^78+12551*q^43+8999*q^49+11802*q^28+3729*q^17+4375*q^18+
5077*q^19+5796*q^20+11153*q^27+392*q^72+13803*q^40+315*q^73+1951*q^63+13401*q^
31+200*q^75+14411*q^35+54*q^80+9645*q^48+21*q^83+40*q^81+11*q^85+7*q^86+5*q^87+
3*q^88+2*q^89+15*q^84+30*q^82+3372*q^59+14475*q^36+10266*q^47, 1+15656*q^25+
34785*q^42+10976*q^22+161*q^90+14033*q^24+9583*q^21+12474*q^23+14*q+21*q^97+
28727*q^51+34407*q^44+26214*q^53+20740*q^57+23942*q^30+54*q^94+27500*q^52+19342
*q^58+23518*q^55+16618*q^60+123*q^91+24888*q^54+33985*q^45+5247*q^71+3471*q^74+
30*q^96+15*q^98+32769*q^37+8506*q^67+22316*q^29+33425*q^46+72*q^93+11*q^99+
10529*q^65+6747*q^69+34069*q^39+15303*q^61+34699*q^41+29690*q^34+1573*q^79+
29875*q^50+11633*q^64+7*q^100+33503*q^38+28405*q^33+22126*q^56+25*q^2+7586*q^68
+17292*q^26+47*q^3+87*q^4+131*q^5+218*q^6+26994*q^32+316*q^7+466*q^8+651*q^9+40
*q^95+14025*q^62+5*q^101+904*q^10+1212*q^11+1609*q^12+2570*q^76+2081*q^13+2635*
q^14+3309*q^15+4074*q^16+9482*q^66+5958*q^70+2194*q^77+1866*q^78+34667*q^43+3*q
^102+30922*q^49+20654*q^28+4952*q^17+5943*q^18+7053*q^19+8250*q^20+18988*q^27+
4594*q^72+34479*q^40+4003*q^73+2*q^103+12810*q^63+25511*q^31+2997*q^75+30846*q^
35+q^104+1322*q^80+31889*q^48+q^105+753*q^83+1103*q^81+501*q^85+406*q^86+325*q^
87+259*q^88+204*q^89+619*q^84+914*q^82+17970*q^59+31896*q^36+96*q^92+32704*q^47
]


 The number of permutations avoiding, {[1, 3, 2], [4, 5, 3, 1, 2]}, is given by

[1, 2, 5, 14, 41, 121, 354, 1021, 2901, 8130, 22513, 61713, 167746, 452789,

    1215197]

The number of EVEN permutations avoiding, {[1, 3, 2], [4, 5, 3, 1, 2]},

     is given by

[1, 1, 3, 7, 21, 60, 176, 511, 1451, 4065, 11257, 30856, 83872, 226395, 607599]

The number of ODD permutations avoiding, {[1, 3, 2], [4, 5, 3, 1, 2]},

     is given by

[0, 1, 2, 7, 20, 61, 178, 510, 1450, 4065, 11256, 30857, 83874, 226394, 607598]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 178, 511, 1451, 4065, 11256, 30856, 83872, 226394,

    607598]

 ODD:, [0, 1, 3, 7, 20, 60, 176, 510, 1450, 4065, 11257, 30857, 83874, 226395,

    607599]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.756097561, 8.768595041,

    12.37288136, 16.55435847, 21.30299897, 26.61143911, 32.47394839,

    38.88592355, 45.84364456, 53.34414926, 61.38515648]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.324634645, 3.103611149,

    3.969082370, 4.919751025, 5.952027683, 7.061298435, 8.242684908,

    9.491401698, 10.80290956, 12.17298002, 13.59772388]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.40393, 9.63240, 15.7536, 24.2040,

    35.4266, 49.8619, 67.9419, 90.0867, 116.703, 148.181, 184.898]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4141084804,

    -0.4039943419, -0.3807968281, -0.3545077952, -0.3289433683, -0.3052859005,

    -0.2837271963, -0.2641240358, -0.2462726563, -0.2299898097, -0.2151172818]

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

    2.623152346, 2.744561438, 2.800388610, 2.824317272, 2.834091900,

    2.837959237, 2.839712085, 2.841042141, 2.842691990, 2.845000140,

    2.847989876]

                                end of this data



For the equivalence class of patterns, {{[2, 3, 1], [1, 4, 3, 2, 5]},

    {[1, 3, 2], [5, 2, 3, 4, 1]}, {[2, 1, 3], [5, 2, 3, 4, 1]},

    {[3, 1, 2], [1, 4, 3, 2, 5]}}

      the member , {[1, 3, 2], [5, 2, 3, 4, 1]}, has a scheme of depth , 5

                                  here it is:

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

    [[5, 3, 4, 1, 2], {[0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}, {2, 3}],

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

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

    [[5, 3, 4, 2, 1], {[0, 0, 0, 1, 0, 0]}, {4}],

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

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

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

    [[5, 2, 4, 1, 3], {[0, 0, 0, 0, 0, 0]}, {4}], [[5, 2, 3, 1, 4], {

    [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, 0],

    [0, 1, 0, 0, 0, 0]}, {4}], [[4, 2, 3, 1, 5], {[0, 0, 1, 0, 0, 0],

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

    [[3, 2, 1], {}, {2}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {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+6*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+10*q^7+12
*q^8+15*q^9+15*q^10+13*q^11+12*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+12*q^7+16*q^8+19*q^9+25*q^10+25*q^11+28*q^12+32*q^13+37*q^14+34*q^
15+28*q^16+25*q^17+22*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+14*q^7+18*q^8+23*q^9+29*q^10+33*q^11+40*q^12+46*q^13+57*q^14+62*q^15+63*q^
16+69*q^17+75*q^18+83*q^19+86*q^20+76*q^21+62*q^22+53*q^23+47*q^24+37*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+14*q^7+20*q^8+25*q^9+33*q^
10+37*q^11+48*q^12+56*q^13+71*q^14+82*q^15+91*q^16+99*q^17+115*q^18+135*q^19+
154*q^20+158*q^21+164*q^22+167*q^23+180*q^24+195*q^25+204*q^26+195*q^27+168*q^
28+138*q^29+115*q^30+100*q^31+84*q^32+58*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+14*q^7+20*q^8+27*q^9+35*q^10+41*q^11+52*q^12+64*q^13+81*
q^14+94*q^15+111*q^16+127*q^17+147*q^18+173*q^19+204*q^20+226*q^21+244*q^22+263
*q^23+294*q^24+332*q^25+372*q^26+398*q^27+410*q^28+416*q^29+423*q^30+436*q^31+
459*q^32+484*q^33+481*q^34+436*q^35+370*q^36+306*q^37+253*q^38+215*q^39+184*q^
40+142*q^41+86*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+14*
q^7+20*q^8+27*q^9+37*q^10+43*q^11+56*q^12+68*q^13+89*q^14+104*q^15+123*q^16+145
*q^17+175*q^18+205*q^19+244*q^20+276*q^21+312*q^22+345*q^23+390*q^24+446*q^25+
518*q^26+570*q^27+614*q^28+654*q^29+705*q^30+760*q^31+835*q^32+918*q^33+997*q^
34+1031*q^35+1044*q^36+1055*q^37+1067*q^38+1079*q^39+1104*q^40+1141*q^41+1156*q
^42+1099*q^43+967*q^44+814*q^45+676*q^46+559*q^47+468*q^48+399*q^49+326*q^50+
228*q^51+122*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+14*q
^7+20*q^8+27*q^9+37*q^10+45*q^11+58*q^12+72*q^13+93*q^14+112*q^15+133*q^16+157*
q^17+193*q^18+231*q^19+276*q^20+316*q^21+364*q^22+413*q^23+474*q^24+546*q^25+
632*q^26+716*q^27+798*q^28+874*q^29+955*q^30+1050*q^31+1177*q^32+1314*q^33+1459
*q^34+1571*q^35+1669*q^36+1767*q^37+1872*q^38+1991*q^39+2134*q^40+2307*q^41+
2476*q^42+2593*q^43+2651*q^44+2667*q^45+2667*q^46+2672*q^47+2679*q^48+2693*q^49
+2727*q^50+2751*q^51+2681*q^52+2460*q^53+2131*q^54+1790*q^55+1490*q^56+1235*q^
57+1027*q^58+867*q^59+725*q^60+554*q^61+350*q^62+167*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+14*q^7+20*q^8+27*q^9+37*q^10+45*q^11+60*q^12
+74*q^13+97*q^14+116*q^15+141*q^16+167*q^17+205*q^18+249*q^19+302*q^20+346*q^21
+404*q^22+465*q^23+544*q^24+630*q^25+734*q^26+836*q^27+946*q^28+1058*q^29+1185*
q^30+1316*q^31+1485*q^32+1672*q^33+1885*q^34+2073*q^35+2255*q^36+2433*q^37+2638
*q^38+2869*q^39+3134*q^40+3425*q^41+3744*q^42+4037*q^43+4297*q^44+4513*q^45+
4735*q^46+4970*q^47+5236*q^48+5523*q^49+5856*q^50+6211*q^51+6515*q^52+6684*q^53
+6749*q^54+6760*q^55+6739*q^56+6693*q^57+6637*q^58+6593*q^59+6578*q^60+6570*q^
61+6452*q^62+6079*q^63+5439*q^64+4673*q^65+3931*q^66+3280*q^67+2725*q^68+2262*q
^69+1894*q^70+1592*q^71+1279*q^72+904*q^73+517*q^74+222*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+14*q^7+20*q^8+27*q^9+37*q^10+45*q^11+60*
q^12+76*q^13+99*q^14+120*q^15+145*q^16+175*q^17+215*q^18+261*q^19+320*q^20+372*
q^21+434*q^22+503*q^23+596*q^24+700*q^25+820*q^26+938*q^27+1068*q^28+1212*q^29+
1373*q^30+1548*q^31+1761*q^32+1994*q^33+2259*q^34+2519*q^35+2789*q^36+3067*q^37
+3364*q^38+3689*q^39+4076*q^40+4513*q^41+4992*q^42+5455*q^43+5905*q^44+6339*q^
45+6793*q^46+7274*q^47+7806*q^48+8388*q^49+9032*q^50+9740*q^51+10459*q^52+11078
*q^53+11615*q^54+12122*q^55+12655*q^56+13207*q^57+13751*q^58+14324*q^59+14989*q
^60+15730*q^61+16403*q^62+16861*q^63+17065*q^64+17096*q^65+17039*q^66+16911*q^
67+16714*q^68+16459*q^69+16176*q^70+15931*q^71+15742*q^72+15456*q^73+14785*q^74
+13562*q^75+11934*q^76+10213*q^77+8615*q^78+7211*q^79+6005*q^80+4987*q^81+4156*
q^82+3486*q^83+2871*q^84+2183*q^85+1421*q^86+739*q^87+288*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+14*q^7+20*q^8+27*q^9+37*q^10+45*q^11+
60*q^12+76*q^13+101*q^14+122*q^15+149*q^16+179*q^17+223*q^18+271*q^19+332*q^20+
390*q^21+460*q^22+533*q^23+634*q^24+750*q^25+890*q^26+1024*q^27+1172*q^28+1334*
q^29+1529*q^30+1742*q^31+1997*q^32+2274*q^33+2593*q^34+2907*q^35+3249*q^36+3619
*q^37+4032*q^38+4463*q^39+4960*q^40+5521*q^41+6154*q^42+6795*q^43+7457*q^44+
8117*q^45+8817*q^46+9550*q^47+10368*q^48+11276*q^49+12290*q^50+13362*q^51+14479
*q^52+15570*q^53+16655*q^54+17716*q^55+18831*q^56+20007*q^57+21245*q^58+22536*q
^59+23971*q^60+25560*q^61+27213*q^62+28720*q^63+30033*q^64+31247*q^65+32473*q^
66+33673*q^67+34870*q^68+36077*q^69+37312*q^70+38587*q^71+39976*q^72+41404*q^73
+42563*q^74+43171*q^75+43264*q^76+43067*q^77+42723*q^78+42224*q^79+41533*q^80+
40681*q^81+39722*q^82+38741*q^83+37847*q^84+36951*q^85+35595*q^86+33263*q^87+
29916*q^88+26058*q^89+22269*q^90+18840*q^91+15826*q^92+13216*q^93+10992*q^94+
9143*q^95+7642*q^96+6357*q^97+5054*q^98+3604*q^99+2160*q^100+1027*q^101+366*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+5
*q^6+5*q^5+7*q^4+6*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
+10*q^8+12*q^7+15*q^6+15*q^5+13*q^4+12*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+12*q^14+16*q^13+19*q^12+25*q^11+25*q^10+28*q^9+32*q^8+
37*q^7+34*q^6+28*q^5+25*q^4+22*q^3+15*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+14*q^
21+7*q^23+7*q+21*q^2+2*q^26+37*q^3+47*q^4+53*q^5+62*q^6+76*q^7+86*q^8+83*q^9+75
*q^10+69*q^11+63*q^12+62*q^13+57*q^14+46*q^15+40*q^16+q^28+33*q^17+29*q^18+23*q
^19+18*q^20+q^27, 1+37*q^25+71*q^22+48*q^24+82*q^21+56*q^23+8*q+11*q^30+14*q^29
+2*q^34+3*q^33+28*q^2+33*q^26+58*q^3+84*q^4+100*q^5+115*q^6+5*q^32+138*q^7+168*
q^8+195*q^9+204*q^10+195*q^11+180*q^12+167*q^13+164*q^14+158*q^15+154*q^16+20*q
^28+135*q^17+115*q^18+99*q^19+91*q^20+25*q^27+7*q^31+q^35+q^36, 1+204*q^25+3*q^
42+263*q^22+226*q^24+294*q^21+244*q^23+9*q+q^44+94*q^30+q^45+20*q^37+111*q^29+
11*q^39+5*q^41+41*q^34+14*q^38+52*q^33+36*q^2+173*q^26+86*q^3+142*q^4+184*q^5+
215*q^6+64*q^32+253*q^7+306*q^8+370*q^9+436*q^10+481*q^11+484*q^12+459*q^13+436
*q^14+423*q^15+416*q^16+2*q^43+127*q^28+410*q^17+398*q^18+372*q^19+332*q^20+147
*q^27+7*q^40+81*q^31+35*q^35+27*q^36, 1+705*q^25+68*q^42+918*q^22+760*q^24+997*
q^21+835*q^23+10*q+5*q^51+43*q^44+2*q^53+446*q^30+3*q^52+q^55+q^54+37*q^45+175*
q^37+518*q^29+27*q^46+123*q^39+89*q^41+276*q^34+7*q^50+145*q^38+312*q^33+45*q^2
+654*q^26+122*q^3+228*q^4+326*q^5+399*q^6+345*q^32+468*q^7+559*q^8+676*q^9+814*
q^10+967*q^11+1099*q^12+1156*q^13+1141*q^14+1104*q^15+1079*q^16+56*q^43+11*q^49
+570*q^28+1067*q^17+1055*q^18+1044*q^19+1031*q^20+614*q^27+104*q^40+390*q^31+
244*q^35+14*q^48+205*q^36+20*q^47, 1+2307*q^25+474*q^42+2651*q^22+2476*q^24+
2667*q^21+2593*q^23+11*q+112*q^51+364*q^44+72*q^53+27*q^57+1669*q^30+93*q^52+20
*q^58+45*q^55+11*q^60+58*q^54+316*q^45+874*q^37+1767*q^29+276*q^46+q^65+716*q^
39+7*q^61+546*q^41+1177*q^34+133*q^50+2*q^64+798*q^38+1314*q^33+37*q^56+55*q^2+
2134*q^26+167*q^3+350*q^4+554*q^5+725*q^6+1459*q^32+867*q^7+1027*q^8+1235*q^9+5
*q^62+1490*q^10+1790*q^11+2131*q^12+2460*q^13+2681*q^14+2751*q^15+2727*q^16+q^
66+413*q^43+157*q^49+1872*q^28+2693*q^17+2679*q^18+2672*q^19+2667*q^20+1991*q^
27+632*q^40+3*q^63+1571*q^31+1050*q^35+193*q^48+14*q^59+955*q^36+231*q^47, 1+
6684*q^25+2255*q^42+6739*q^22+6749*q^24+6693*q^21+6760*q^23+12*q+836*q^51+1885*
q^44+630*q^53+346*q^57+5236*q^30+734*q^52+302*q^58+465*q^55+205*q^60+544*q^54+
1672*q^45+14*q^71+5*q^74+3425*q^37+45*q^67+5523*q^29+1485*q^46+74*q^65+27*q^69+
2869*q^39+167*q^61+2433*q^41+4297*q^34+946*q^50+97*q^64+3134*q^38+4513*q^33+404
*q^56+66*q^2+37*q^68+6515*q^26+222*q^3+517*q^4+904*q^5+1279*q^6+4735*q^32+1592*
q^7+1894*q^8+2262*q^9+141*q^62+2725*q^10+3280*q^11+3931*q^12+2*q^76+4673*q^13+
5439*q^14+6079*q^15+6452*q^16+60*q^66+20*q^70+q^77+q^78+2073*q^43+1058*q^49+
5856*q^28+6570*q^17+6578*q^18+6593*q^19+6637*q^20+6211*q^27+11*q^72+2638*q^40+7
*q^73+116*q^63+4970*q^31+3*q^75+4037*q^35+1185*q^48+249*q^59+3744*q^36+1316*q^
47, 1+17039*q^25+8388*q^42+16459*q^22+q^90+16911*q^24+16176*q^21+16714*q^23+13*
q+4076*q^51+7274*q^44+3364*q^53+2259*q^57+15730*q^30+3689*q^52+1994*q^58+2789*q
^55+1548*q^60+q^91+3067*q^54+6793*q^45+320*q^71+175*q^74+11615*q^37+596*q^67+
16403*q^29+6339*q^46+820*q^65+434*q^69+10459*q^39+1373*q^61+9032*q^41+13207*q^
34+60*q^79+4513*q^50+938*q^64+11078*q^38+13751*q^33+2519*q^56+78*q^2+503*q^68+
17096*q^26+288*q^3+739*q^4+1421*q^5+2183*q^6+14324*q^32+2871*q^7+3486*q^8+4156*
q^9+1212*q^62+4987*q^10+6005*q^11+7211*q^12+120*q^76+8615*q^13+10213*q^14+11934
*q^15+13562*q^16+700*q^66+372*q^70+99*q^77+76*q^78+7806*q^43+4992*q^49+16861*q^
28+14785*q^17+15456*q^18+15742*q^19+15931*q^20+17065*q^27+261*q^72+9740*q^40+
215*q^73+1068*q^63+14989*q^31+145*q^75+12655*q^35+45*q^80+5455*q^48+20*q^83+37*
q^81+11*q^85+7*q^86+5*q^87+3*q^88+2*q^89+14*q^84+27*q^82+1761*q^59+12122*q^36+
5905*q^47, 1+41533*q^25+28720*q^42+38741*q^22+122*q^90+40681*q^24+37847*q^21+
39722*q^23+14*q+20*q^97+16655*q^51+25560*q^44+14479*q^53+10368*q^57+43171*q^30+
45*q^94+15570*q^52+9550*q^58+12290*q^55+8117*q^60+101*q^91+13362*q^54+23971*q^
45+2593*q^71+1742*q^74+27*q^96+14*q^98+34870*q^37+4032*q^67+43264*q^29+22536*q^
46+60*q^93+11*q^99+4960*q^65+3249*q^69+32473*q^39+7457*q^61+30033*q^41+38587*q^
34+890*q^79+17716*q^50+5521*q^64+7*q^100+33673*q^38+39976*q^33+11276*q^56+91*q^
2+3619*q^68+42224*q^26+366*q^3+1027*q^4+2160*q^5+3604*q^6+41404*q^32+5054*q^7+
6357*q^8+7642*q^9+37*q^95+6795*q^62+5*q^101+9143*q^10+10992*q^11+13216*q^12+
1334*q^76+15826*q^13+18840*q^14+22269*q^15+26058*q^16+4463*q^66+2907*q^70+1172*
q^77+1024*q^78+27213*q^43+3*q^102+18831*q^49+43067*q^28+29916*q^17+33263*q^18+
35595*q^19+36951*q^20+42723*q^27+2274*q^72+31247*q^40+1997*q^73+2*q^103+6154*q^
63+42563*q^31+1529*q^75+37312*q^35+q^104+750*q^80+20007*q^48+q^105+460*q^83+634
*q^81+332*q^85+271*q^86+223*q^87+179*q^88+149*q^89+390*q^84+533*q^82+8817*q^59+
36077*q^36+76*q^92+21245*q^47]


 The number of permutations avoiding, {[1, 3, 2], [5, 2, 3, 4, 1]}, is given by

[1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017, 190885, 533294,

    1484021]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 2, 3, 4, 1]},

     is given by

[1, 1, 3, 7, 22, 61, 181, 518, 1495, 4254, 12074, 34023, 95481, 266679, 742087]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 2, 3, 4, 1]},

     is given by

[0, 1, 2, 7, 19, 60, 174, 514, 1478, 4242, 12037, 33994, 95404, 266615, 741934]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 60, 174, 518, 1495, 4242, 12037, 34023, 95481, 266615,

    741934]

 ODD:, [0, 1, 3, 7, 19, 61, 181, 514, 1478, 4254, 12074, 33994, 95404, 266679,

    742087]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.780487805, 8.892561983,

    12.72676056, 17.31201550, 22.66868483, 28.81002825, 35.74455643,

    43.47783642, 52.01364172, 61.35463553, 71.50276984]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.342733395, 3.169591190,

    4.105131030, 5.135143678, 6.246119670, 7.427992253, 8.673937003,

    9.979461124, 11.34157923, 12.75823902, 14.22796165]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.48840, 10.0463, 16.8521, 26.3697,

    39.0140, 55.1751, 75.2372, 99.5896, 128.631, 162.773, 202.435]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4180400731,

    -0.4326649839, -0.4545716244, -0.4823413370, -0.5108621743, -0.5369944912,

    -0.5595378945, -0.5783919249, -0.5939345235, -0.6066598993, -0.6171086866]

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

    2.578362051, 2.664549155, 2.731589337, 2.803751981, 2.879260541,

    2.952243413, 3.018883250, 3.077508355, 3.128091741, 3.171226380,

    3.207960641]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [1, 2, 3, 5, 4]},

    {[1, 3, 2], [2, 1, 3, 4, 5]}, {[2, 3, 1], [5, 4, 3, 1, 2]},

    {[3, 1, 2], [4, 5, 3, 2, 1]}}

      the member , {[2, 1, 3], [1, 2, 3, 5, 4]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

                 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+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+2*q^3+3*q^4+5*q^5+7*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+2*q^4+2*q^5+4*q^6+3*q^7+9*q^8+
14*q^9+21*q^10+26*q^11+34*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+2*q^5+2*q^6+2*q^7+2*q^8+4*q^9+6*q^10+13*q^11+17*q^12+25*
q^13+35*q^14+55*q^15+71*q^16+88*q^17+101*q^18+114*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+2*q^6+2*q^7+2*q^8+2*q^10+3*q^
11+8*q^12+7*q^13+12*q^14+16*q^15+22*q^16+32*q^17+55*q^18+76*q^19+106*q^20+139*q
^21+185*q^22+232*q^23+271*q^24+300*q^25+318*q^26+321*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+2*q^7+2*q^8+2*q^9+2*q^12
+3*q^13+6*q^14+9*q^15+7*q^16+9*q^17+10*q^18+22*q^19+27*q^20+41*q^21+53*q^22+80*
q^23+105*q^24+153*q^25+214*q^26+303*q^27+389*q^28+496*q^29+604*q^30+721*q^31+
811*q^32+870*q^33+889*q^34+871*q^35+807*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+2*q^8+2*q^9+2*q^10+2*q^14+3*q^15+6*q
^16+7*q^17+8*q^18+3*q^19+8*q^20+10*q^21+16*q^22+24*q^23+34*q^24+37*q^25+49*q^26
+61*q^27+92*q^28+130*q^29+178*q^30+242*q^31+337*q^32+458*q^33+617*q^34+823*q^35
+1079*q^36+1357*q^37+1639*q^38+1924*q^39+2181*q^40+2384*q^41+2485*q^42+2481*q^
43+2369*q^44+2161*q^45+1864*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+2*q^9+2*q^10+2*q^11+2*q^16+3*q^17+6*q^18+7*q^
19+6*q^20+5*q^21+6*q^23+12*q^24+17*q^25+18*q^26+28*q^27+27*q^28+34*q^29+39*q^30
+54*q^31+71*q^32+107*q^33+130*q^34+182*q^35+219*q^36+297*q^37+397*q^38+566*q^39
+745*q^40+1015*q^41+1346*q^42+1796*q^43+2318*q^44+2967*q^45+3692*q^46+4485*q^47
+5240*q^48+5940*q^49+6516*q^50+6910*q^51+7047*q^52+6894*q^53+6468*q^54+5809*q^
55+4971*q^56+4024*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+2*q^10+2*q^11+2*q^12+2*q^18+3*q^19+6*q^20+7*q^21+6*q^
22+3*q^23+2*q^24+6*q^26+10*q^27+18*q^28+18*q^29+24*q^30+24*q^31+21*q^32+20*q^33
+35*q^34+53*q^35+69*q^36+76*q^37+104*q^38+127*q^39+155*q^40+188*q^41+245*q^42+
328*q^43+432*q^44+571*q^45+752*q^46+973*q^47+1267*q^48+1699*q^49+2273*q^50+3022
*q^51+3948*q^52+5138*q^53+6579*q^54+8277*q^55+10174*q^56+12246*q^57+14343*q^58+
16323*q^59+17979*q^60+19219*q^61+19882*q^62+19872*q^63+19119*q^64+17683*q^65+
15691*q^66+13320*q^67+10757*q^68+8214*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+2*q^11+2*q^12+2*q^13+2*q^20+3*q^
21+6*q^22+7*q^23+6*q^24+3*q^25+2*q^27+6*q^29+10*q^30+16*q^31+20*q^32+22*q^33+18
*q^34+20*q^35+14*q^36+16*q^37+29*q^38+50*q^39+62*q^40+78*q^41+80*q^42+96*q^43+
100*q^44+128*q^45+155*q^46+228*q^47+265*q^48+339*q^49+419*q^50+543*q^51+652*q^
52+842*q^53+1054*q^54+1402*q^55+1795*q^56+2376*q^57+3069*q^58+4025*q^59+5190*q^
60+6814*q^61+8880*q^62+11544*q^63+14739*q^64+18635*q^65+23165*q^66+28295*q^67+
33784*q^68+39454*q^69+44937*q^70+49829*q^71+53668*q^72+56100*q^73+56890*q^74+
55849*q^75+52963*q^76+48396*q^77+42535*q^78+35859*q^79+28889*q^80+22115*q^81+
15988*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+2*q^12+2*q^13+2*q^14+2*q^22+3*q^23+6*q^24+7*q^25+6*q^26+3*q^27+
2*q^30+6*q^32+10*q^33+16*q^34+18*q^35+24*q^36+18*q^37+12*q^38+10*q^39+11*q^40+
16*q^41+31*q^42+44*q^43+62*q^44+68*q^45+72*q^46+74*q^47+83*q^48+83*q^49+103*q^
50+135*q^51+192*q^52+246*q^53+294*q^54+321*q^55+381*q^56+467*q^57+552*q^58+702*
q^59+894*q^60+1112*q^61+1391*q^62+1747*q^63+2179*q^64+2766*q^65+3466*q^66+4442*
q^67+5701*q^68+7382*q^69+9494*q^70+12316*q^71+15852*q^72+20411*q^73+26134*q^74+
33441*q^75+42313*q^76+52922*q^77+65120*q^78+78922*q^79+93824*q^80+109286*q^81+
124417*q^82+138384*q^83+150040*q^84+158392*q^85+162532*q^86+162006*q^87+156640*
q^88+146634*q^89+132611*q^90+115608*q^91+96925*q^92+77899*q^93+59758*q^94+43507
*q^95+29864*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+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+2*q^12+3*q^11+5*q^10+7*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+2*q^17+2*q^16+4*q^15+3*q^14+9*q^
13+14*q^12+21*q^11+26*q^10+34*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+2*q^22+2*q^21+2*q^23+7*q+21*q^2+41*q^3+65*q^4+86*q^5+102*q^6+115*q
^7+118*q^8+114*q^9+101*q^10+88*q^11+71*q^12+55*q^13+35*q^14+25*q^15+17*q^16+q^
28+13*q^17+6*q^18+4*q^19+2*q^20, 1+3*q^25+12*q^22+8*q^24+16*q^21+7*q^23+8*q+2*q
^30+2*q^29+28*q^2+2*q^26+63*q^3+112*q^4+167*q^5+219*q^6+268*q^7+303*q^8+321*q^9
+318*q^10+300*q^11+271*q^12+232*q^13+185*q^14+139*q^15+106*q^16+2*q^28+76*q^17+
55*q^18+32*q^19+22*q^20+q^36, 1+27*q^25+80*q^22+41*q^24+105*q^21+53*q^23+9*q+9*
q^30+q^45+2*q^37+7*q^29+2*q^38+2*q^33+36*q^2+22*q^26+92*q^3+182*q^4+301*q^5+434
*q^6+3*q^32+574*q^7+704*q^8+807*q^9+871*q^10+889*q^11+870*q^12+811*q^13+721*q^
14+604*q^15+496*q^16+9*q^28+389*q^17+303*q^18+214*q^19+153*q^20+10*q^27+6*q^31+
2*q^36, 1+178*q^25+458*q^22+242*q^24+617*q^21+337*q^23+10*q+37*q^30+q^55+2*q^45
+8*q^37+49*q^29+2*q^46+6*q^39+2*q^41+10*q^34+7*q^38+16*q^33+45*q^2+130*q^26+129
*q^3+282*q^4+512*q^5+806*q^6+24*q^32+1149*q^7+1515*q^8+1864*q^9+2161*q^10+2369*
q^11+2481*q^12+2485*q^13+2384*q^14+2181*q^15+1924*q^16+61*q^28+1639*q^17+1357*q
^18+1079*q^19+823*q^20+92*q^27+3*q^40+34*q^31+8*q^35+3*q^36+2*q^47, 1+1015*q^25
+12*q^42+2318*q^22+1346*q^24+2967*q^21+1796*q^23+11*q+2*q^57+219*q^30+2*q^55+5*
q^45+34*q^37+297*q^29+6*q^46+28*q^39+17*q^41+71*q^34+2*q^50+27*q^38+107*q^33+2*
q^56+55*q^2+745*q^26+175*q^3+420*q^4+831*q^5+1419*q^6+130*q^32+2174*q^7+3065*q^
8+4024*q^9+4971*q^10+5809*q^11+6468*q^12+6894*q^13+7047*q^14+6910*q^15+6516*q^
16+q^66+6*q^43+3*q^49+397*q^28+5940*q^17+5240*q^18+4485*q^19+3692*q^20+566*q^27
+18*q^40+182*q^31+54*q^35+6*q^48+39*q^36+7*q^47, 1+5138*q^25+69*q^42+10174*q^22
+6579*q^24+12246*q^21+8277*q^23+12*q+10*q^51+35*q^44+7*q^57+1267*q^30+6*q^52+6*
q^58+3*q^55+2*q^60+2*q^54+20*q^45+188*q^37+2*q^67+1699*q^29+21*q^46+127*q^39+76
*q^41+432*q^34+18*q^50+155*q^38+571*q^33+6*q^56+66*q^2+2*q^68+3948*q^26+231*q^3
+605*q^4+1297*q^5+2389*q^6+752*q^32+3921*q^7+5889*q^8+8214*q^9+10757*q^10+13320
*q^11+15691*q^12+17683*q^13+19119*q^14+19872*q^15+19882*q^16+2*q^66+q^78+53*q^
43+18*q^49+2273*q^28+19219*q^17+17979*q^18+16323*q^19+14343*q^20+3022*q^27+104*
q^40+973*q^31+328*q^35+24*q^48+3*q^59+245*q^36+24*q^47, 1+23165*q^25+339*q^42+
39454*q^22+28295*q^24+44937*q^21+33784*q^23+13*q+62*q^51+228*q^44+29*q^53+18*q^
57+6814*q^30+50*q^52+22*q^58+14*q^55+16*q^60+q^91+16*q^54+155*q^45+2*q^71+1054*
q^37+6*q^67+8880*q^29+128*q^46+6*q^69+652*q^39+10*q^61+419*q^41+2376*q^34+2*q^
79+78*q^50+2*q^64+842*q^38+3069*q^33+20*q^56+78*q^2+7*q^68+18635*q^26+298*q^3+
847*q^4+1958*q^5+3872*q^6+4025*q^32+6786*q^7+10826*q^8+15988*q^9+6*q^62+22115*q
^10+28889*q^11+35859*q^12+42535*q^13+48396*q^14+52963*q^15+55849*q^16+3*q^66+3*
q^70+2*q^78+265*q^43+80*q^49+11544*q^28+56890*q^17+56100*q^18+53668*q^19+49829*
q^20+14739*q^27+543*q^40+5190*q^31+1795*q^35+2*q^80+96*q^48+20*q^59+1402*q^36+
100*q^47, 1+93824*q^25+1747*q^42+138384*q^22+109286*q^24+150040*q^21+124417*q^
23+14*q+294*q^51+1112*q^44+192*q^53+83*q^57+33441*q^30+246*q^52+74*q^58+103*q^
55+68*q^60+2*q^91+135*q^54+894*q^45+16*q^71+5701*q^37+12*q^67+42313*q^29+702*q^
46+2*q^93+11*q^65+24*q^69+3466*q^39+62*q^61+2179*q^41+12316*q^34+6*q^79+321*q^
50+16*q^64+4442*q^38+15852*q^33+83*q^56+91*q^2+18*q^68+78922*q^26+377*q^3+1157*
q^4+2872*q^5+6073*q^6+20411*q^32+11330*q^7+19151*q^8+29864*q^9+44*q^62+43507*q^
10+59758*q^11+77899*q^12+96925*q^13+115608*q^14+132611*q^15+146634*q^16+10*q^66
+18*q^70+3*q^78+1391*q^43+381*q^49+52922*q^28+156640*q^17+162006*q^18+162532*q^
19+158392*q^20+65120*q^27+10*q^72+2766*q^40+6*q^73+31*q^63+26134*q^31+2*q^75+
9494*q^35+7*q^80+467*q^48+q^105+2*q^83+6*q^81+3*q^82+72*q^59+7382*q^36+2*q^92+
552*q^47]


 The number of permutations avoiding, {[2, 1, 3], [1, 2, 3, 5, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [1, 2, 3, 5, 4]},

     is given by

[1, 1, 3, 7, 22, 60, 186, 542, 1647, 4903, 14769, 44233, 132844, 398442,

    1195608]

The number of ODD permutations avoiding, {[2, 1, 3], [1, 2, 3, 5, 4]},

     is given by

[0, 1, 2, 7, 19, 62, 179, 552, 1634, 4939, 14756, 44341, 132877, 398720,

    1195877]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 62, 179, 542, 1647, 4939, 14756, 44233, 132844, 398720,

    1195877]

 ODD:, [0, 1, 3, 7, 19, 60, 186, 552, 1634, 4903, 14769, 44341, 132877, 398442,

    1195608]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.926829268, 9.409836066,

    13.85479452, 19.28336380, 25.70496800, 33.12365373, 41.54113463,

    50.95812541, 61.37491956, 72.79163583, 85.20832161]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.223930205, 2.775047181,

    3.281873884, 3.747291187, 4.175530272, 4.571281868, 4.939132621,

    5.283210992, 5.607040577, 5.913537155, 6.205078015]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 4.94587, 7.70089, 10.7707, 14.0422,

    17.4351, 20.8966, 24.3950, 27.9123, 31.4389, 34.9699, 38.5030]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4401017490,

    -0.4984432291, -0.5576907446, -0.6084808213, -0.6467393830, -0.6723761604,

    -0.6871935952, -0.6935811525, -0.6937972046, -0.6897337477, -0.6828143366]

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

    2.746346005, 3.130310229, 3.504681361, 3.838432006, 4.113128644,

    4.322484058, 4.469279751, 4.561950977, 4.611270293, 4.627907599,

    4.621151946]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [1, 2, 4, 5, 3]},

    {[1, 3, 2], [3, 1, 2, 4, 5]}, {[3, 1, 2], [3, 5, 4, 2, 1]},

    {[3, 1, 2], [4, 3, 5, 2, 1]}, {[2, 3, 1], [5, 4, 2, 1, 3]},

    {[2, 3, 1], [5, 4, 1, 3, 2]}, {[2, 1, 3], [1, 2, 5, 3, 4]},

    {[1, 3, 2], [2, 3, 1, 4, 5]}}

      the member , {[2, 1, 3], [1, 2, 4, 5, 3]}, has a scheme of depth , 3

                                  here it is:

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

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

    [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 0, 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+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+q^2+2*q^3+3*q^4+5*q^5+7*q^6+9*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+q^2+2*q^3+2*q^4+4*q^5+6
*q^6+6*q^7+9*q^8+13*q^9+18*q^10+25*q^11+31*q^12+37*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+q^2+2*q^3+2*q^4+3*q^5+5*q^6+7*q^7+7*q
^8+9*q^9+13*q^10+15*q^11+23*q^12+29*q^13+34*q^14+48*q^15+63*q^16+78*q^17+95*q^
18+106*q^19+114*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+q^2+2*q^3+2*q^4+3*q^5+4*q^6+6*q^7+8*q^8+9*q^9+11*q^10+13*q^11+17*q^12+
21*q^13+27*q^14+34*q^15+39*q^16+51*q^17+65*q^18+81*q^19+107*q^20+130*q^21+160*q
^22+203*q^23+241*q^24+274*q^25+300*q^26+306*q^27+298*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+q^2+2*q^3+2*q^4+3*q^5+4*q^6+5*q
^7+7*q^8+10*q^9+11*q^10+13*q^11+16*q^12+18*q^13+24*q^14+29*q^15+35*q^16+41*q^17
+48*q^18+57*q^19+70*q^20+88*q^21+104*q^22+131*q^23+160*q^24+190*q^25+241*q^26+
295*q^27+362*q^28+446*q^29+527*q^30+622*q^31+720*q^32+788*q^33+829*q^34+831*q^
35+783*q^36+698*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+q^2+2*q^3+2*q^4+3*q^5+4*q^6+5*q^7+6*q^8+9*q^9+12*q^10+13*q^11+16*q^
12+19*q^13+22*q^14+27*q^15+33*q^16+38*q^17+47*q^18+55*q^19+60*q^20+72*q^21+83*q
^22+97*q^23+119*q^24+140*q^25+163*q^26+193*q^27+231*q^28+273*q^29+333*q^30+403*
q^31+478*q^32+585*q^33+700*q^34+836*q^35+1017*q^36+1210*q^37+1432*q^38+1668*q^
39+1886*q^40+2094*q^41+2244*q^42+2291*q^43+2244*q^44+2086*q^45+1829*q^46+1508*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+q^2
+2*q^3+2*q^4+3*q^5+4*q^6+5*q^7+6*q^8+8*q^9+11*q^10+14*q^11+16*q^12+19*q^13+23*q
^14+27*q^15+32*q^16+37*q^17+45*q^18+52*q^19+63*q^20+73*q^21+80*q^22+92*q^23+106
*q^24+122*q^25+141*q^26+164*q^27+188*q^28+218*q^29+248*q^30+284*q^31+333*q^32+
387*q^33+456*q^34+539*q^35+631*q^36+735*q^37+871*q^38+1031*q^39+1216*q^40+1458*
q^41+1724*q^42+2037*q^43+2429*q^44+2851*q^45+3349*q^46+3920*q^47+4500*q^48+5093
*q^49+5630*q^50+6046*q^51+6312*q^52+6331*q^53+6072*q^54+5571*q^55+4845*q^56+
3976*q^57+3057*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+q^2+2*q^3+2*q^4+3*q^5+4*q^6+5*q^7+6*q^8+8*q^9+10*q^10+13*q^11+17*
q^12+19*q^13+23*q^14+28*q^15+32*q^16+38*q^17+45*q^18+51*q^19+61*q^20+72*q^21+84
*q^22+96*q^23+107*q^24+122*q^25+138*q^26+158*q^27+178*q^28+203*q^29+233*q^30+
262*q^31+300*q^32+335*q^33+371*q^34+425*q^35+487*q^36+553*q^37+637*q^38+730*q^
39+832*q^40+961*q^41+1102*q^42+1270*q^43+1478*q^44+1717*q^45+1998*q^46+2334*q^
47+2717*q^48+3158*q^49+3712*q^50+4346*q^51+5084*q^52+5976*q^53+6959*q^54+8104*q
^55+9414*q^56+10804*q^57+12325*q^58+13869*q^59+15290*q^60+16513*q^61+17353*q^62
+17685*q^63+17430*q^64+16487*q^65+14928*q^66+12900*q^67+10561*q^68+8151*q^69+
5880*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+q^2+2*q^3+2*q^4+3*q^5+4*q^6+5*q^7+6*q^8+8*q^9+10*q^10+12*q^11+16*q^12+20*q^
13+23*q^14+28*q^15+33*q^16+38*q^17+46*q^18+53*q^19+61*q^20+71*q^21+84*q^22+96*q
^23+112*q^24+128*q^25+142*q^26+163*q^27+183*q^28+204*q^29+232*q^30+261*q^31+292
*q^32+333*q^33+373*q^34+416*q^35+463*q^36+513*q^37+570*q^38+639*q^39+720*q^40+
802*q^41+907*q^42+1021*q^43+1141*q^44+1289*q^45+1444*q^46+1626*q^47+1860*q^48+
2118*q^49+2407*q^50+2754*q^51+3141*q^52+3585*q^53+4117*q^54+4722*q^55+5427*q^56
+6271*q^57+7242*q^58+8362*q^59+9688*q^60+11195*q^61+12950*q^62+15040*q^63+17388
*q^64+20085*q^65+23157*q^66+26503*q^67+30213*q^68+34139*q^69+38085*q^70+41954*q
^71+45352*q^72+47931*q^73+49420*q^74+49469*q^75+47936*q^76+44801*q^77+40176*q^
78+34479*q^79+28193*q^80+21827*q^81+15908*q^82+10816*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+q^2+2*q^3+2*q^4+3*q^5+4*q
^6+5*q^7+6*q^8+8*q^9+10*q^10+12*q^11+15*q^12+19*q^13+24*q^14+28*q^15+33*q^16+39
*q^17+46*q^18+54*q^19+63*q^20+73*q^21+84*q^22+97*q^23+113*q^24+129*q^25+149*q^
26+170*q^27+190*q^28+215*q^29+241*q^30+270*q^31+302*q^32+339*q^33+378*q^34+424*
q^35+476*q^36+527*q^37+587*q^38+648*q^39+714*q^40+791*q^41+871*q^42+963*q^43+
1071*q^44+1186*q^45+1316*q^46+1461*q^47+1616*q^48+1787*q^49+1975*q^50+2193*q^51
+2438*q^52+2730*q^53+3060*q^54+3419*q^55+3845*q^56+4308*q^57+4831*q^58+5445*q^
59+6127*q^60+6930*q^61+7874*q^62+8918*q^63+10116*q^64+11513*q^65+13089*q^66+
14913*q^67+17048*q^68+19483*q^69+22297*q^70+25596*q^71+29348*q^72+33655*q^73+
38654*q^74+44287*q^75+50753*q^76+58123*q^77+66240*q^78+75257*q^79+85017*q^80+
95246*q^81+105799*q^82+116031*q^83+125325*q^84+133032*q^85+138155*q^86+139987*q
^87+138016*q^88+131904*q^89+121859*q^90+108425*q^91+92543*q^92+75532*q^93+58663
*q^94+43102*q^95+29765*q^96+19140*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+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+q^13+2*q^12+3*q^11+5*q^10+7*q^9+9*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+q^19+2*q^18+2*q^
17+4*q^16+6*q^15+6*q^14+9*q^13+13*q^12+18*q^11+25*q^10+31*q^9+37*q^8+44*q^7+43*
q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+2*q^25+5*q^22+2*q^24+7*q^21+3*q^23+7*q
+21*q^2+q^26+41*q^3+65*q^4+86*q^5+102*q^6+115*q^7+114*q^8+106*q^9+95*q^10+78*q^
11+63*q^12+48*q^13+34*q^14+29*q^15+23*q^16+q^28+15*q^17+13*q^18+9*q^19+7*q^20+q
^27, 1+13*q^25+27*q^22+17*q^24+34*q^21+21*q^23+8*q+4*q^30+6*q^29+q^34+2*q^33+28
*q^2+11*q^26+63*q^3+112*q^4+167*q^5+219*q^6+2*q^32+268*q^7+298*q^8+306*q^9+300*
q^10+274*q^11+241*q^12+203*q^13+160*q^14+130*q^15+107*q^16+8*q^28+81*q^17+65*q^
18+51*q^19+39*q^20+9*q^27+3*q^31+q^35+q^36, 1+70*q^25+2*q^42+131*q^22+88*q^24+
160*q^21+104*q^23+9*q+q^44+29*q^30+q^45+7*q^37+35*q^29+4*q^39+2*q^41+13*q^34+5*
q^38+16*q^33+36*q^2+57*q^26+92*q^3+182*q^4+301*q^5+434*q^6+18*q^32+574*q^7+698*
q^8+783*q^9+831*q^10+829*q^11+788*q^12+720*q^13+622*q^14+527*q^15+446*q^16+q^43
+41*q^28+362*q^17+295*q^18+241*q^19+190*q^20+48*q^27+3*q^40+24*q^31+11*q^35+10*
q^36, 1+333*q^25+19*q^42+585*q^22+403*q^24+700*q^21+478*q^23+10*q+2*q^51+13*q^
44+q^53+140*q^30+2*q^52+q^55+q^54+12*q^45+47*q^37+163*q^29+9*q^46+33*q^39+22*q^
41+72*q^34+3*q^50+38*q^38+83*q^33+45*q^2+273*q^26+129*q^3+282*q^4+512*q^5+806*q
^6+97*q^32+1149*q^7+1508*q^8+1829*q^9+2086*q^10+2244*q^11+2291*q^12+2244*q^13+
2094*q^14+1886*q^15+1668*q^16+16*q^43+4*q^49+193*q^28+1432*q^17+1210*q^18+1017*
q^19+836*q^20+231*q^27+27*q^40+119*q^31+60*q^35+5*q^48+55*q^36+6*q^47, 1+1458*q
^25+106*q^42+2429*q^22+1724*q^24+2851*q^21+2037*q^23+11*q+27*q^51+80*q^44+19*q^
53+8*q^57+631*q^30+23*q^52+6*q^58+14*q^55+4*q^60+16*q^54+73*q^45+218*q^37+735*q
^29+63*q^46+q^65+164*q^39+3*q^61+122*q^41+333*q^34+32*q^50+q^64+188*q^38+387*q^
33+11*q^56+55*q^2+1216*q^26+175*q^3+420*q^4+831*q^5+1419*q^6+456*q^32+2174*q^7+
3057*q^8+3976*q^9+2*q^62+4845*q^10+5571*q^11+6072*q^12+6331*q^13+6312*q^14+6046
*q^15+5630*q^16+q^66+92*q^43+37*q^49+871*q^28+5093*q^17+4500*q^18+3920*q^19+
3349*q^20+1031*q^27+141*q^40+2*q^63+539*q^31+284*q^35+45*q^48+5*q^59+248*q^36+
52*q^47, 1+5976*q^25+487*q^42+9414*q^22+6959*q^24+10804*q^21+8104*q^23+12*q+158
*q^51+371*q^44+122*q^53+72*q^57+2717*q^30+138*q^52+61*q^58+96*q^55+45*q^60+107*
q^54+335*q^45+5*q^71+2*q^74+961*q^37+13*q^67+3158*q^29+300*q^46+19*q^65+8*q^69+
730*q^39+38*q^61+553*q^41+1478*q^34+178*q^50+23*q^64+832*q^38+1717*q^33+84*q^56
+66*q^2+10*q^68+5084*q^26+231*q^3+605*q^4+1297*q^5+2389*q^6+1998*q^32+3921*q^7+
5880*q^8+8151*q^9+32*q^62+10561*q^10+12900*q^11+14928*q^12+q^76+16487*q^13+
17430*q^14+17685*q^15+17353*q^16+17*q^66+6*q^70+q^77+q^78+425*q^43+203*q^49+
3712*q^28+16513*q^17+15290*q^18+13869*q^19+12325*q^20+4346*q^27+4*q^72+637*q^40
+3*q^73+28*q^63+2334*q^31+2*q^75+1270*q^35+233*q^48+51*q^59+1102*q^36+262*q^47,
1+23157*q^25+2118*q^42+34139*q^22+q^90+26503*q^24+38085*q^21+30213*q^23+13*q+
720*q^51+1626*q^44+570*q^53+373*q^57+11195*q^30+639*q^52+333*q^58+463*q^55+261*
q^60+q^91+513*q^54+1444*q^45+61*q^71+38*q^74+4117*q^37+112*q^67+12950*q^29+1289
*q^46+142*q^65+84*q^69+3141*q^39+232*q^61+2407*q^41+6271*q^34+16*q^79+802*q^50+
163*q^64+3585*q^38+7242*q^33+416*q^56+78*q^2+96*q^68+20085*q^26+298*q^3+847*q^4
+1958*q^5+3872*q^6+8362*q^32+6786*q^7+10816*q^8+15908*q^9+204*q^62+21827*q^10+
28193*q^11+34479*q^12+28*q^76+40176*q^13+44801*q^14+47936*q^15+49469*q^16+128*q
^66+71*q^70+23*q^77+20*q^78+1860*q^43+907*q^49+15040*q^28+49420*q^17+47931*q^18
+45352*q^19+41954*q^20+17388*q^27+53*q^72+2754*q^40+46*q^73+183*q^63+9688*q^31+
33*q^75+5427*q^35+12*q^80+1021*q^48+6*q^83+10*q^81+4*q^85+3*q^86+2*q^87+2*q^88+
q^89+5*q^84+8*q^82+292*q^59+4722*q^36+1141*q^47, 1+85017*q^25+8918*q^42+116031*
q^22+28*q^90+95246*q^24+125325*q^21+105799*q^23+14*q+6*q^97+3060*q^51+6930*q^44
+2438*q^53+1616*q^57+44287*q^30+12*q^94+2730*q^52+1461*q^58+1975*q^55+1186*q^60
+24*q^91+2193*q^54+6127*q^45+378*q^71+270*q^74+8*q^96+5*q^98+17048*q^37+587*q^
67+50753*q^29+5445*q^46+15*q^93+4*q^99+714*q^65+476*q^69+13089*q^39+1071*q^61+
10116*q^41+25596*q^34+149*q^79+3419*q^50+791*q^64+3*q^100+14913*q^38+29348*q^33
+1787*q^56+91*q^2+527*q^68+75257*q^26+377*q^3+1157*q^4+2872*q^5+6073*q^6+33655*
q^32+11330*q^7+19140*q^8+29765*q^9+10*q^95+963*q^62+2*q^101+43102*q^10+58663*q^
11+75532*q^12+215*q^76+92543*q^13+108425*q^14+121859*q^15+131904*q^16+648*q^66+
424*q^70+190*q^77+170*q^78+7874*q^43+2*q^102+3845*q^49+58123*q^28+138016*q^17+
139987*q^18+138155*q^19+133032*q^20+66240*q^27+339*q^72+11513*q^40+302*q^73+q^
103+871*q^63+38654*q^31+241*q^75+22297*q^35+q^104+129*q^80+4308*q^48+q^105+84*q
^83+113*q^81+63*q^85+54*q^86+46*q^87+39*q^88+33*q^89+73*q^84+97*q^82+1316*q^59+
19483*q^36+19*q^92+4831*q^47]


 The number of permutations avoiding, {[2, 1, 3], [1, 2, 4, 5, 3]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [1, 2, 4, 5, 3]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[2, 1, 3], [1, 2, 4, 5, 3]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.902439024, 9.319672131,

    13.65479452, 18.93510055, 25.17829930, 32.39636253, 40.59745978,

    49.78714973, 59.96919325, 71.14612212, 83.31963445]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.271830444, 2.937229634,

    3.622798476, 4.320605801, 5.021401771, 5.716839922, 6.400335536,

    7.067169433, 7.714273242, 8.339914495, 8.943379542]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.16121, 8.62732, 13.1247, 18.6676,

    25.2145, 32.6823, 40.9643, 49.9449, 59.5100, 69.5542, 79.9840]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5034459428,

    -0.6431700048, -0.7870792219, -0.9232279970, -1.047324068, -1.158582042,

    -1.257320772, -1.344208211, -1.419988182, -1.485385911, -1.541169066]

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

    2.821467447, 3.284923600, 3.759151618, 4.239735454, 4.724389681,

    5.209867489, 5.691221434, 6.162695096, 6.618098666, 7.051654372,

    7.458201732]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [5, 4, 3, 1, 2]},

    {[1, 3, 2], [4, 5, 3, 2, 1]}, {[1, 3, 2], [5, 4, 3, 1, 2]},

    {[2, 3, 1], [1, 2, 3, 5, 4]}, {[3, 1, 2], [2, 1, 3, 4, 5]},

    {[3, 1, 2], [1, 2, 3, 5, 4]}, {[2, 1, 3], [4, 5, 3, 2, 1]},

    {[2, 3, 1], [2, 1, 3, 4, 5]}}

      the member , {[1, 3, 2], [4, 5, 3, 2, 1]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

                 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+3*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+15*q^9+15*q^10+15*q^11+11*q^12+6*q^13+3*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+21*q^9+27*q^10+29*q^11+33*q^12+32*q^13+36*q^14+29*q^
15+26*q^16+19*q^17+11*q^18+6*q^19+3*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+25*q^9+33*q^10+39*q^11+48*q^12+54*q^13+62*q^14+65*q^15+72*q^16
+72*q^17+73*q^18+69*q^19+61*q^20+54*q^21+40*q^22+30*q^23+19*q^24+11*q^25+6*q^26
+3*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+27*q^9+37*q^10+
45*q^11+58*q^12+67*q^13+85*q^14+93*q^15+110*q^16+118*q^17+135*q^18+139*q^19+152
*q^20+150*q^21+156*q^22+152*q^23+148*q^24+132*q^25+120*q^26+102*q^27+79*q^28+65
*q^29+44*q^30+30*q^31+19*q^32+11*q^33+6*q^34+3*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+29*q^9+39*q^10+49*q^11+64*q^12+77*q^13+98*q^14+114*
q^15+139*q^16+158*q^17+185*q^18+204*q^19+231*q^20+249*q^21+275*q^22+290*q^23+
306*q^24+314*q^25+323*q^26+328*q^27+321*q^28+316*q^29+292*q^30+279*q^31+249*q^
32+220*q^33+183*q^34+153*q^35+120*q^36+90*q^37+69*q^38+44*q^39+30*q^40+19*q^41+
11*q^42+6*q^43+3*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+29
*q^9+41*q^10+51*q^11+68*q^12+83*q^13+108*q^14+127*q^15+160*q^16+185*q^17+226*q^
18+256*q^19+300*q^20+333*q^21+385*q^22+417*q^23+468*q^24+501*q^25+547*q^26+577*
q^27+612*q^28+633*q^29+660*q^30+672*q^31+678*q^32+680*q^33+671*q^34+646*q^35+
627*q^36+593*q^37+542*q^38+499*q^39+439*q^40+388*q^41+325*q^42+271*q^43+216*q^
44+171*q^45+131*q^46+94*q^47+69*q^48+44*q^49+30*q^50+19*q^51+11*q^52+6*q^53+3*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+29*q^9+41*q^10+53*q^
11+70*q^12+87*q^13+114*q^14+137*q^15+173*q^16+206*q^17+253*q^18+295*q^19+353*q^
20+404*q^21+473*q^22+532*q^23+608*q^24+673*q^25+756*q^26+829*q^27+907*q^28+983*
q^29+1055*q^30+1123*q^31+1189*q^32+1252*q^33+1298*q^34+1346*q^35+1375*q^36+1400
*q^37+1404*q^38+1416*q^39+1392*q^40+1376*q^41+1326*q^42+1274*q^43+1215*q^44+
1135*q^45+1041*q^46+958*q^47+859*q^48+749*q^49+658*q^50+552*q^51+464*q^52+376*q
^53+304*q^54+234*q^55+182*q^56+135*q^57+94*q^58+69*q^59+44*q^60+30*q^61+19*q^62
+11*q^63+6*q^64+3*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+
29*q^9+41*q^10+53*q^11+72*q^12+89*q^13+118*q^14+143*q^15+183*q^16+219*q^17+274*
q^18+322*q^19+392*q^20+455*q^21+545*q^22+622*q^23+727*q^24+818*q^25+941*q^26+
1050*q^27+1183*q^28+1304*q^29+1448*q^30+1578*q^31+1725*q^32+1860*q^33+2003*q^34
+2129*q^35+2266*q^36+2387*q^37+2502*q^38+2612*q^39+2697*q^40+2774*q^41+2835*q^
42+2882*q^43+2910*q^44+2922*q^45+2909*q^46+2879*q^47+2828*q^48+2750*q^49+2660*q
^50+2559*q^51+2422*q^52+2282*q^53+2117*q^54+1953*q^55+1782*q^56+1600*q^57+1417*
q^58+1248*q^59+1078*q^60+912*q^61+771*q^62+628*q^63+515*q^64+409*q^65+322*q^66+
245*q^67+186*q^68+135*q^69+94*q^70+69*q^71+44*q^72+30*q^73+19*q^74+11*q^75+6*q^
76+3*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+29*q^9+41*q^10
+53*q^11+72*q^12+91*q^13+120*q^14+147*q^15+189*q^16+229*q^17+287*q^18+343*q^19+
419*q^20+494*q^21+596*q^22+692*q^23+818*q^24+939*q^25+1090*q^26+1240*q^27+1417*
q^28+1592*q^29+1795*q^30+1999*q^31+2221*q^32+2453*q^33+2695*q^34+2936*q^35+3190
*q^36+3449*q^37+3694*q^38+3968*q^39+4209*q^40+4457*q^41+4693*q^42+4927*q^43+
5127*q^44+5334*q^45+5495*q^46+5650*q^47+5784*q^48+5891*q^49+5949*q^50+6009*q^51
+6027*q^52+5995*q^53+5954*q^54+5873*q^55+5753*q^56+5617*q^57+5443*q^58+5224*q^
59+5006*q^60+4742*q^61+4458*q^62+4170*q^63+3873*q^64+3539*q^65+3229*q^66+2903*q
^67+2590*q^68+2292*q^69+2000*q^70+1719*q^71+1471*q^72+1241*q^73+1025*q^74+847*q
^75+679*q^76+548*q^77+427*q^78+333*q^79+249*q^80+186*q^81+135*q^82+94*q^83+69*q
^84+44*q^85+30*q^86+19*q^87+11*q^88+6*q^89+3*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+29*q^9+41*q^10+53*q^11+72*q^12+91*q^13+122*q^14+149*q
^15+193*q^16+235*q^17+297*q^18+356*q^19+440*q^20+521*q^21+635*q^22+743*q^23+888
*q^24+1028*q^25+1212*q^26+1391*q^27+1611*q^28+1831*q^29+2096*q^30+2358*q^31+
2668*q^32+2979*q^33+3331*q^34+3683*q^35+4078*q^36+4474*q^37+4900*q^38+5337*q^39
+5781*q^40+6243*q^41+6715*q^42+7182*q^43+7662*q^44+8138*q^45+8600*q^46+9063*q^
47+9506*q^48+9926*q^49+10323*q^50+10713*q^51+11053*q^52+11376*q^53+11651*q^54+
11877*q^55+12079*q^56+12231*q^57+12318*q^58+12369*q^59+12377*q^60+12319*q^61+
12222*q^62+12055*q^63+11841*q^64+11590*q^65+11282*q^66+10906*q^67+10521*q^68+
10080*q^69+9586*q^70+9095*q^71+8558*q^72+7991*q^73+7436*q^74+6853*q^75+6268*q^
76+5707*q^77+5150*q^78+4593*q^79+4085*q^80+3585*q^81+3124*q^82+2704*q^83+2306*q
^84+1942*q^85+1634*q^86+1354*q^87+1101*q^88+898*q^89+712*q^90+566*q^91+438*q^92
+337*q^93+249*q^94+186*q^95+135*q^96+94*q^97+69*q^98+44*q^99+30*q^100+19*q^101+
11*q^102+6*q^103+3*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 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+3*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+15*q^6+15*q^5+15*q^4+11*q^3+6*q^2+3*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+21*q^12+27*q^11+29*q^10+33*q^9+32*q^8+36
*q^7+29*q^6+26*q^5+19*q^4+11*q^3+6*q^2+3*q+1, 1+3*q^25+11*q^22+5*q^24+15*q^21+7
*q^23+3*q+6*q^2+2*q^26+11*q^3+19*q^4+30*q^5+40*q^6+54*q^7+61*q^8+69*q^9+73*q^10
+72*q^11+72*q^12+65*q^13+62*q^14+54*q^15+48*q^16+q^28+39*q^17+33*q^18+25*q^19+
20*q^20+q^27, 1+45*q^25+85*q^22+58*q^24+93*q^21+67*q^23+3*q+11*q^30+15*q^29+2*q
^34+3*q^33+6*q^2+37*q^26+11*q^3+19*q^4+30*q^5+44*q^6+5*q^32+65*q^7+79*q^8+102*q
^9+120*q^10+132*q^11+148*q^12+152*q^13+156*q^14+150*q^15+152*q^16+22*q^28+139*q
^17+135*q^18+118*q^19+110*q^20+27*q^27+7*q^31+q^35+q^36, 1+231*q^25+3*q^42+290*
q^22+249*q^24+306*q^21+275*q^23+3*q+q^44+114*q^30+q^45+22*q^37+139*q^29+11*q^39
+5*q^41+49*q^34+15*q^38+64*q^33+6*q^2+204*q^26+11*q^3+19*q^4+30*q^5+44*q^6+77*q
^32+69*q^7+90*q^8+120*q^9+153*q^10+183*q^11+220*q^12+249*q^13+279*q^14+292*q^15
+316*q^16+2*q^43+158*q^28+321*q^17+328*q^18+323*q^19+314*q^20+185*q^27+7*q^40+
98*q^31+39*q^35+29*q^36, 1+660*q^25+83*q^42+680*q^22+672*q^24+671*q^21+678*q^23
+3*q+5*q^51+51*q^44+2*q^53+501*q^30+3*q^52+q^55+q^54+41*q^45+226*q^37+547*q^29+
29*q^46+160*q^39+108*q^41+333*q^34+7*q^50+185*q^38+385*q^33+6*q^2+633*q^26+11*q
^3+19*q^4+30*q^5+44*q^6+417*q^32+69*q^7+94*q^8+131*q^9+171*q^10+216*q^11+271*q^
12+325*q^13+388*q^14+439*q^15+499*q^16+68*q^43+11*q^49+577*q^28+542*q^17+593*q^
18+627*q^19+646*q^20+612*q^27+127*q^40+468*q^31+300*q^35+15*q^48+256*q^36+22*q^
47, 1+1376*q^25+608*q^42+1215*q^22+1326*q^24+1135*q^21+1274*q^23+3*q+137*q^51+
473*q^44+87*q^53+29*q^57+1375*q^30+114*q^52+22*q^58+53*q^55+11*q^60+70*q^54+404
*q^45+983*q^37+1400*q^29+353*q^46+q^65+829*q^39+7*q^61+673*q^41+1189*q^34+173*q
^50+2*q^64+907*q^38+1252*q^33+41*q^56+6*q^2+1392*q^26+11*q^3+19*q^4+30*q^5+44*q
^6+1298*q^32+69*q^7+94*q^8+135*q^9+5*q^62+182*q^10+234*q^11+304*q^12+376*q^13+
464*q^14+552*q^15+658*q^16+q^66+532*q^43+206*q^49+1404*q^28+749*q^17+859*q^18+
958*q^19+1041*q^20+1416*q^27+756*q^40+3*q^63+1346*q^31+1123*q^35+253*q^48+15*q^
59+1055*q^36+295*q^47, 1+2282*q^25+2266*q^42+1782*q^22+2117*q^24+1600*q^21+1953
*q^23+3*q+1050*q^51+2003*q^44+818*q^53+455*q^57+2828*q^30+941*q^52+392*q^58+622
*q^55+274*q^60+727*q^54+1860*q^45+15*q^71+5*q^74+2774*q^37+53*q^67+2750*q^29+
1725*q^46+89*q^65+29*q^69+2612*q^39+219*q^61+2387*q^41+2910*q^34+1183*q^50+118*
q^64+2697*q^38+2922*q^33+545*q^56+6*q^2+41*q^68+2422*q^26+11*q^3+19*q^4+30*q^5+
44*q^6+2909*q^32+69*q^7+94*q^8+135*q^9+183*q^62+186*q^10+245*q^11+322*q^12+2*q^
76+409*q^13+515*q^14+628*q^15+771*q^16+72*q^66+22*q^70+q^77+q^78+2129*q^43+1304
*q^49+2660*q^28+912*q^17+1078*q^18+1248*q^19+1417*q^20+2559*q^27+11*q^72+2502*q
^40+7*q^73+143*q^63+2879*q^31+3*q^75+2882*q^35+1448*q^48+322*q^59+2835*q^36+
1578*q^47, 1+3229*q^25+5891*q^42+2292*q^22+q^90+2903*q^24+2000*q^21+2590*q^23+3
*q+4209*q^51+5650*q^44+3694*q^53+2695*q^57+4742*q^30+3968*q^52+2453*q^58+3190*q
^55+1999*q^60+q^91+3449*q^54+5495*q^45+419*q^71+229*q^74+5954*q^37+818*q^67+
4458*q^29+5334*q^46+1090*q^65+596*q^69+6027*q^39+1795*q^61+5949*q^41+5617*q^34+
72*q^79+4457*q^50+1240*q^64+5995*q^38+5443*q^33+2936*q^56+6*q^2+692*q^68+3539*q
^26+11*q^3+19*q^4+30*q^5+44*q^6+5224*q^32+69*q^7+94*q^8+135*q^9+1592*q^62+186*q
^10+249*q^11+333*q^12+147*q^76+427*q^13+548*q^14+679*q^15+847*q^16+939*q^66+494
*q^70+120*q^77+91*q^78+5784*q^43+4693*q^49+4170*q^28+1025*q^17+1241*q^18+1471*q
^19+1719*q^20+3873*q^27+343*q^72+6009*q^40+287*q^73+1417*q^63+5006*q^31+189*q^
75+5753*q^35+53*q^80+4927*q^48+22*q^83+41*q^81+11*q^85+7*q^86+5*q^87+3*q^88+2*q
^89+15*q^84+29*q^82+2221*q^59+5873*q^36+5127*q^47, 1+4085*q^25+12055*q^42+2704*
q^22+149*q^90+3585*q^24+2306*q^21+3124*q^23+3*q+22*q^97+11651*q^51+12319*q^44+
11053*q^53+9506*q^57+6853*q^30+53*q^94+11376*q^52+9063*q^58+10323*q^55+8138*q^
60+122*q^91+10713*q^54+12377*q^45+3331*q^71+2358*q^74+29*q^96+15*q^98+10521*q^
37+4900*q^67+6268*q^29+12369*q^46+72*q^93+11*q^99+5781*q^65+4078*q^69+11282*q^
39+7662*q^61+11841*q^41+9095*q^34+1212*q^79+11877*q^50+6243*q^64+7*q^100+10906*
q^38+8558*q^33+9926*q^56+6*q^2+4474*q^68+4593*q^26+11*q^3+19*q^4+30*q^5+44*q^6+
7991*q^32+69*q^7+94*q^8+135*q^9+41*q^95+7182*q^62+5*q^101+186*q^10+249*q^11+337
*q^12+1831*q^76+438*q^13+566*q^14+712*q^15+898*q^16+5337*q^66+3683*q^70+1611*q^
77+1391*q^78+12222*q^43+3*q^102+12079*q^49+5707*q^28+1101*q^17+1354*q^18+1634*q
^19+1942*q^20+5150*q^27+2979*q^72+11590*q^40+2668*q^73+2*q^103+6715*q^63+7436*q
^31+2096*q^75+9586*q^35+q^104+1028*q^80+12231*q^48+q^105+635*q^83+888*q^81+440*
q^85+356*q^86+297*q^87+235*q^88+193*q^89+521*q^84+743*q^82+8600*q^59+10080*q^36
+91*q^92+12318*q^47]


 The number of permutations avoiding, {[1, 3, 2], [4, 5, 3, 2, 1]}, is given by

[1, 2, 5, 14, 41, 119, 334, 902, 2351, 5945, 14660, 35408, 84061, 196715,

    454778]

The number of EVEN permutations avoiding, {[1, 3, 2], [4, 5, 3, 2, 1]},

     is given by

 [1, 1, 3, 7, 22, 60, 173, 454, 1191, 2979, 7358, 17715, 42074, 98374, 227451]

The number of ODD permutations avoiding, {[1, 3, 2], [4, 5, 3, 2, 1]},

     is given by

 [0, 1, 2, 7, 19, 59, 161, 448, 1160, 2966, 7302, 17693, 41987, 98341, 227327]

            For the reverse patterns and complement patterns, we get

EVEN:, [

    1, 1, 2, 7, 22, 59, 161, 454, 1191, 2966, 7302, 17715, 42074, 98341, 227327

    ]

 ODD:, [

    0, 1, 3, 7, 19, 60, 173, 448, 1160, 2979, 7358, 17693, 41987, 98374, 227451

    ]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.731707317, 8.647058824,

    12.06886228, 15.99334812, 20.42535091, 25.36921783, 30.82687585,

    36.79820944, 43.28186674, 50.27593727, 57.77839957]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.295535330, 3.030900345,

    3.867342219, 4.817277671, 5.878344468, 7.042494668, 8.300851426,

    9.645438799, 11.06954253, 12.56760016, 14.13498049]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.26948, 9.18636, 14.9563, 23.2062,

    34.5549, 49.5967, 68.9041, 93.0345, 122.535, 157.945, 199.798]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4442359433,

    -0.4295666068, -0.3811358155, -0.3308626606, -0.2897912777, -0.2582251131,

    -0.2336813819, -0.2138141813, -0.1970152545, -0.1822934036, -0.1690764159]

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

    2.664143857, 2.808889903, 2.842659100, 2.828378091, 2.805531535,

    2.787746556, 2.777027136, 2.771974672, 2.770821280, 2.772202873,

    2.775203080]

                                end of this data



For the equivalence class of patterns, {{[2, 3, 1], [1, 5, 3, 2, 4]},

    {[1, 3, 2], [4, 2, 3, 5, 1]}, {[1, 3, 2], [5, 2, 3, 1, 4]},

    {[3, 1, 2], [1, 4, 3, 5, 2]}, {[2, 1, 3], [5, 1, 3, 4, 2]},

    {[2, 3, 1], [4, 1, 3, 2, 5]}, {[2, 1, 3], [2, 5, 3, 4, 1]},

    {[3, 1, 2], [2, 4, 3, 1, 5]}}

      the member , {[1, 3, 2], [5, 2, 3, 1, 4]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

    [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {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+6*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+8*q^6+10*q^7+13
*q^8+14*q^9+13*q^10+15*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+10*q^6+12*q^7+16*q^8+19*q^9+22*q^10+27*q^11+30*q^12+32*q^13+34*q^14+33*q^
15+32*q^16+32*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+10*
q^6+14*q^7+18*q^8+22*q^9+27*q^10+34*q^11+42*q^12+48*q^13+53*q^14+60*q^15+68*q^
16+77*q^17+81*q^18+82*q^19+81*q^20+80*q^21+76*q^22+71*q^23+61*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+10*q^6+14*q^7+20*q^8+24*q^9+30*q^
10+39*q^11+49*q^12+58*q^13+69*q^14+81*q^15+94*q^16+110*q^17+124*q^18+137*q^19+
153*q^20+165*q^21+178*q^22+196*q^23+207*q^24+210*q^25+206*q^26+199*q^27+191*q^
28+182*q^29+166*q^30+143*q^31+107*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+10*q^6+14*q^7+20*q^8+26*q^9+32*q^10+42*q^11+54*q^12+65*q^13+79
*q^14+95*q^15+115*q^16+138*q^17+158*q^18+180*q^19+208*q^20+236*q^21+265*q^22+
302*q^23+335*q^24+363*q^25+392*q^26+422*q^27+453*q^28+487*q^29+510*q^30+529*q^
31+542*q^32+534*q^33+513*q^34+490*q^35+464*q^36+434*q^37+396*q^38+340*q^39+266*
q^40+176*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+10*q^6+
14*q^7+20*q^8+26*q^9+34*q^10+44*q^11+57*q^12+70*q^13+86*q^14+105*q^15+129*q^16+
157*q^17+186*q^18+216*q^19+252*q^20+293*q^21+338*q^22+390*q^23+445*q^24+498*q^
25+557*q^26+619*q^27+685*q^28+759*q^29+835*q^30+911*q^31+986*q^32+1056*q^33+
1114*q^34+1176*q^35+1242*q^36+1305*q^37+1360*q^38+1395*q^39+1399*q^40+1383*q^41
+1344*q^42+1279*q^43+1206*q^44+1133*q^45+1048*q^46+947*q^47+819*q^48+654*q^49+
464*q^50+275*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+10
*q^6+14*q^7+20*q^8+26*q^9+34*q^10+46*q^11+59*q^12+73*q^13+91*q^14+112*q^15+139*
q^16+171*q^17+205*q^18+242*q^19+288*q^20+339*q^21+396*q^22+465*q^23+537*q^24+
611*q^25+696*q^26+791*q^27+896*q^28+1012*q^29+1130*q^30+1259*q^31+1399*q^32+
1537*q^33+1678*q^34+1836*q^35+2000*q^36+2174*q^37+2354*q^38+2530*q^39+2698*q^40
+2860*q^41+3009*q^42+3149*q^43+3286*q^44+3407*q^45+3520*q^46+3623*q^47+3683*q^
48+3690*q^49+3628*q^50+3505*q^51+3359*q^52+3183*q^53+2981*q^54+2772*q^55+2548*q
^56+2286*q^57+1978*q^58+1609*q^59+1189*q^60+768*q^61+412*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+10*q^6+14*q^7+20*q^8+26*q^9+34*q^10+
46*q^11+61*q^12+75*q^13+94*q^14+117*q^15+146*q^16+181*q^17+219*q^18+261*q^19+
314*q^20+373*q^21+442*q^22+525*q^23+613*q^24+705*q^25+813*q^26+935*q^27+1074*q^
28+1230*q^29+1395*q^30+1574*q^31+1776*q^32+1983*q^33+2203*q^34+2454*q^35+2721*q
^36+3006*q^37+3317*q^38+3643*q^39+3975*q^40+4319*q^41+4670*q^42+5037*q^43+5427*
q^44+5823*q^45+6235*q^46+6676*q^47+7099*q^48+7499*q^49+7868*q^50+8201*q^51+8520
*q^52+8829*q^53+9106*q^54+9348*q^55+9554*q^56+9695*q^57+9768*q^58+9749*q^59+
9578*q^60+9266*q^61+8853*q^62+8389*q^63+7905*q^64+7379*q^65+6813*q^66+6216*q^67
+5557*q^68+4804*q^69+3952*q^70+3011*q^71+2058*q^72+1217*q^73+596*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+10*q^6+14*q^7+20*q^8+26*q^9+
34*q^10+46*q^11+61*q^12+77*q^13+96*q^14+120*q^15+151*q^16+188*q^17+229*q^18+275
*q^19+333*q^20+399*q^21+476*q^22+569*q^23+673*q^24+783*q^25+908*q^26+1054*q^27+
1222*q^28+1413*q^29+1621*q^30+1848*q^31+2103*q^32+2378*q^33+2675*q^34+3016*q^35
+3389*q^36+3788*q^37+4223*q^38+4694*q^39+5196*q^40+5734*q^41+6304*q^42+6907*q^
43+7560*q^44+8257*q^45+8986*q^46+9777*q^47+10604*q^48+11442*q^49+12296*q^50+
13172*q^51+14080*q^52+15021*q^53+15980*q^54+16959*q^55+17972*q^56+18990*q^57+
19983*q^58+20955*q^59+21852*q^60+22631*q^61+23334*q^62+23990*q^63+24589*q^64+
25130*q^65+25583*q^66+25904*q^67+26085*q^68+26082*q^69+25842*q^70+25344*q^71+
24561*q^72+23497*q^73+22267*q^74+20970*q^75+19642*q^76+18264*q^77+16795*q^78+
15232*q^79+13562*q^80+11730*q^81+9716*q^82+7564*q^83+5390*q^84+3414*q^85+1859*q
^86+837*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+10*q^6+
14*q^7+20*q^8+26*q^9+34*q^10+46*q^11+61*q^12+77*q^13+98*q^14+122*q^15+154*q^16+
193*q^17+236*q^18+285*q^19+347*q^20+418*q^21+502*q^22+603*q^23+717*q^24+841*q^
25+986*q^26+1151*q^27+1342*q^28+1563*q^29+1808*q^30+2079*q^31+2385*q^32+2716*q^
33+3084*q^34+3506*q^35+3975*q^36+4490*q^37+5057*q^38+5665*q^39+6326*q^40+7052*q
^41+7838*q^42+8689*q^43+9618*q^44+10615*q^45+11688*q^46+12861*q^47+14105*q^48+
15423*q^49+16805*q^50+18250*q^51+19796*q^52+21447*q^53+23180*q^54+25002*q^55+
26931*q^56+28942*q^57+31033*q^58+33200*q^59+35379*q^60+37587*q^61+39834*q^62+
42125*q^63+44477*q^64+46894*q^65+49336*q^66+51784*q^67+54217*q^68+56570*q^69+
58816*q^70+60896*q^71+62757*q^72+64420*q^73+65872*q^74+67126*q^75+68232*q^76+
69157*q^77+69827*q^78+70229*q^79+70276*q^80+69841*q^81+68892*q^82+67343*q^83+
65165*q^84+62468*q^85+59333*q^86+55910*q^87+52406*q^88+48869*q^89+45246*q^90+
41460*q^91+37460*q^92+33232*q^93+28739*q^94+23941*q^95+18920*q^96+13906*q^97+
9272*q^98+5459*q^99+2752*q^100+1146*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+5
*q^6+5*q^5+6*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+8*q^9
+10*q^8+13*q^7+14*q^6+13*q^5+15*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+10*q^15+12*q^14+16*q^13+19*q^12+22*q^11+27*q^10+30*q^9+32*q^8+
34*q^7+33*q^6+32*q^5+32*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+10*q^22+5*q^24+14*q^
21+7*q^23+7*q+21*q^2+2*q^26+41*q^3+61*q^4+71*q^5+76*q^6+80*q^7+81*q^8+82*q^9+81
*q^10+77*q^11+68*q^12+60*q^13+53*q^14+48*q^15+42*q^16+q^28+34*q^17+27*q^18+22*q
^19+18*q^20+q^27, 1+39*q^25+69*q^22+49*q^24+81*q^21+58*q^23+8*q+10*q^30+14*q^29
+2*q^34+3*q^33+28*q^2+30*q^26+63*q^3+107*q^4+143*q^5+166*q^6+5*q^32+182*q^7+191
*q^8+199*q^9+206*q^10+210*q^11+207*q^12+196*q^13+178*q^14+165*q^15+153*q^16+20*
q^28+137*q^17+124*q^18+110*q^19+94*q^20+24*q^27+7*q^31+q^35+q^36, 1+208*q^25+3*
q^42+302*q^22+236*q^24+335*q^21+265*q^23+9*q+q^44+95*q^30+q^45+20*q^37+115*q^29
+10*q^39+5*q^41+42*q^34+14*q^38+54*q^33+36*q^2+180*q^26+92*q^3+176*q^4+266*q^5+
340*q^6+65*q^32+396*q^7+434*q^8+464*q^9+490*q^10+513*q^11+534*q^12+542*q^13+529
*q^14+510*q^15+487*q^16+2*q^43+138*q^28+453*q^17+422*q^18+392*q^19+363*q^20+158
*q^27+7*q^40+79*q^31+32*q^35+26*q^36, 1+835*q^25+70*q^42+1056*q^22+911*q^24+
1114*q^21+986*q^23+10*q+5*q^51+44*q^44+2*q^53+498*q^30+3*q^52+q^55+q^54+34*q^45
+186*q^37+557*q^29+26*q^46+129*q^39+86*q^41+293*q^34+7*q^50+157*q^38+338*q^33+
45*q^2+759*q^26+129*q^3+275*q^4+464*q^5+654*q^6+390*q^32+819*q^7+947*q^8+1048*q
^9+1133*q^10+1206*q^11+1279*q^12+1344*q^13+1383*q^14+1399*q^15+1395*q^16+57*q^
43+10*q^49+619*q^28+1360*q^17+1305*q^18+1242*q^19+1176*q^20+685*q^27+105*q^40+
445*q^31+252*q^35+14*q^48+216*q^36+20*q^47, 1+2860*q^25+537*q^42+3286*q^22+3009
*q^24+3407*q^21+3149*q^23+11*q+112*q^51+396*q^44+73*q^53+26*q^57+2000*q^30+91*q
^52+20*q^58+46*q^55+10*q^60+59*q^54+339*q^45+1012*q^37+2174*q^29+288*q^46+q^65+
791*q^39+7*q^61+611*q^41+1399*q^34+139*q^50+2*q^64+896*q^38+1537*q^33+34*q^56+
55*q^2+2698*q^26+175*q^3+412*q^4+768*q^5+1189*q^6+1678*q^32+1609*q^7+1978*q^8+
2286*q^9+5*q^62+2548*q^10+2772*q^11+2981*q^12+3183*q^13+3359*q^14+3505*q^15+
3628*q^16+q^66+465*q^43+171*q^49+2354*q^28+3690*q^17+3683*q^18+3623*q^19+3520*q
^20+2530*q^27+696*q^40+3*q^63+1836*q^31+1259*q^35+205*q^48+14*q^59+1130*q^36+
242*q^47, 1+8829*q^25+2721*q^42+9554*q^22+9106*q^24+9695*q^21+9348*q^23+12*q+
935*q^51+2203*q^44+705*q^53+373*q^57+7099*q^30+813*q^52+314*q^58+525*q^55+219*q
^60+613*q^54+1983*q^45+14*q^71+5*q^74+4319*q^37+46*q^67+7499*q^29+1776*q^46+75*
q^65+26*q^69+3643*q^39+181*q^61+3006*q^41+5427*q^34+1074*q^50+94*q^64+3975*q^38
+5823*q^33+442*q^56+66*q^2+34*q^68+8520*q^26+231*q^3+596*q^4+1217*q^5+2058*q^6+
6235*q^32+3011*q^7+3952*q^8+4804*q^9+146*q^62+5557*q^10+6216*q^11+6813*q^12+2*q
^76+7379*q^13+7905*q^14+8389*q^15+8853*q^16+61*q^66+20*q^70+q^77+q^78+2454*q^43
+1230*q^49+7868*q^28+9266*q^17+9578*q^18+9749*q^19+9768*q^20+8201*q^27+10*q^72+
3317*q^40+7*q^73+117*q^63+6676*q^31+3*q^75+5037*q^35+1395*q^48+261*q^59+4670*q^
36+1574*q^47, 1+25583*q^25+11442*q^42+26082*q^22+q^90+25904*q^24+25842*q^21+
26085*q^23+13*q+5196*q^51+9777*q^44+4223*q^53+2675*q^57+22631*q^30+4694*q^52+
2378*q^58+3389*q^55+1848*q^60+q^91+3788*q^54+8986*q^45+333*q^71+188*q^74+15980*
q^37+673*q^67+23334*q^29+8257*q^46+908*q^65+476*q^69+14080*q^39+1621*q^61+12296
*q^41+18990*q^34+61*q^79+5734*q^50+1054*q^64+15021*q^38+19983*q^33+3016*q^56+78
*q^2+569*q^68+25130*q^26+298*q^3+837*q^4+1859*q^5+3414*q^6+20955*q^32+5390*q^7+
7564*q^8+9716*q^9+1413*q^62+11730*q^10+13562*q^11+15232*q^12+120*q^76+16795*q^
13+18264*q^14+19642*q^15+20970*q^16+783*q^66+399*q^70+96*q^77+77*q^78+10604*q^
43+6304*q^49+23990*q^28+22267*q^17+23497*q^18+24561*q^19+25344*q^20+24589*q^27+
275*q^72+13172*q^40+229*q^73+1222*q^63+21852*q^31+151*q^75+17972*q^35+46*q^80+
6907*q^48+20*q^83+34*q^81+10*q^85+7*q^86+5*q^87+3*q^88+2*q^89+14*q^84+26*q^82+
2103*q^59+16959*q^36+7560*q^47, 1+70276*q^25+42125*q^42+67343*q^22+122*q^90+
69841*q^24+65165*q^21+68892*q^23+14*q+20*q^97+23180*q^51+37587*q^44+19796*q^53+
14105*q^57+67126*q^30+46*q^94+21447*q^52+12861*q^58+16805*q^55+10615*q^60+98*q^
91+18250*q^54+35379*q^45+3084*q^71+2079*q^74+26*q^96+14*q^98+54217*q^37+5057*q^
67+68232*q^29+33200*q^46+61*q^93+10*q^99+6326*q^65+3975*q^69+49336*q^39+9618*q^
61+44477*q^41+60896*q^34+986*q^79+25002*q^50+7052*q^64+7*q^100+51784*q^38+62757
*q^33+15423*q^56+91*q^2+4490*q^68+70229*q^26+377*q^3+1146*q^4+2752*q^5+5459*q^6
+64420*q^32+9272*q^7+13906*q^8+18920*q^9+34*q^95+8689*q^62+5*q^101+23941*q^10+
28739*q^11+33232*q^12+1563*q^76+37460*q^13+41460*q^14+45246*q^15+48869*q^16+
5665*q^66+3506*q^70+1342*q^77+1151*q^78+39834*q^43+3*q^102+26931*q^49+69157*q^
28+52406*q^17+55910*q^18+59333*q^19+62468*q^20+69827*q^27+2716*q^72+46894*q^40+
2385*q^73+2*q^103+7838*q^63+65872*q^31+1808*q^75+58816*q^35+q^104+841*q^80+
28942*q^48+q^105+502*q^83+717*q^81+347*q^85+285*q^86+236*q^87+193*q^88+154*q^89
+418*q^84+603*q^82+11688*q^59+56570*q^36+77*q^92+31033*q^47]


 The number of permutations avoiding, {[1, 3, 2], [5, 2, 3, 1, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 2, 3, 1, 4]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 2, 3, 1, 4]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.804878049, 8.975409836,

    12.89863014, 17.59872029, 23.09356903, 29.39575290, 36.51386960,

    44.45371102, 53.21915091, 62.81277206, 73.23629460]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.350085750, 3.178986112,

    4.106543164, 5.120025404, 6.208121487, 7.361920999, 8.574853594,

    9.842282418, 11.16103686, 12.52899468, 13.94474654]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.52290, 10.1060, 16.8637, 26.2147,

    38.5408, 54.1979, 73.5281, 96.8705, 124.569, 156.976, 194.456]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4418810426,

    -0.4781319015, -0.5094299837, -0.5379144894, -0.5632391122, -0.5849036937,

    -0.6027413815, -0.6168748963, -0.6276271677, -0.6354076022, -0.6406602782]

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

    2.565707530, 2.669682760, 2.755661216, 2.837151078, 2.914938489,

    2.987275504, 3.052540785, 3.109860439, 3.158898207, 3.199976192,

    3.233638178]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [5, 4, 3, 2, 1]},

    {[1, 3, 2], [5, 4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4, 5]},

    {[2, 3, 1], [1, 2, 3, 4, 5]}}

      the member , {[1, 3, 2], [5, 4, 3, 2, 1]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

                 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, 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+15*q^10+15*q^11+11*q^12+6*q^13, 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+27*q^10+30*q^11+32*q^12+32*q^13+30*q^14+28*q^15+21*q^16+15*q^17
+4*q^18, 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+33*q^10+40*q^
11+48*q^12+53*q^13+55*q^14+60*q^15+58*q^16+60*q^17+53*q^18+52*q^19+45*q^20+36*q
^21+18*q^22+12*q^23+q^24, 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+37*q^10+46*q^11+58*q^12+67*q^13+77*q^14+87*q^15+91*q^16+96*q^17+99*q^18+101*
q^19+103*q^20+100*q^21+96*q^22+90*q^23+88*q^24+75*q^25+60*q^26+37*q^27+30*q^28+
12*q^29+4*q^30, 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+39*q^10
+50*q^11+64*q^12+77*q^13+91*q^14+107*q^15+119*q^16+131*q^17+138*q^18+149*q^19+
153*q^20+159*q^21+160*q^22+168*q^23+162*q^24+168*q^25+159*q^26+154*q^27+149*q^
28+141*q^29+124*q^30+108*q^31+90*q^32+60*q^33+39*q^34+36*q^35+8*q^36+6*q^37, 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+41*q^10+52*q^11+68*q^12+
83*q^13+101*q^14+121*q^15+139*q^16+157*q^17+174*q^18+190*q^19+204*q^20+213*q^21
+222*q^22+233*q^23+242*q^24+246*q^25+255*q^26+248*q^27+264*q^28+249*q^29+258*q^
30+244*q^31+255*q^32+213*q^33+231*q^34+204*q^35+197*q^36+160*q^37+150*q^38+93*q
^39+90*q^40+54*q^41+52*q^42+15*q^43+12*q^44+4*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+41*q^10+54*q^11+70*q^12+87*q^13+107*q^14+131*q^15+
153*q^16+177*q^17+200*q^18+224*q^19+246*q^20+266*q^21+279*q^22+299*q^23+312*q^
24+329*q^25+333*q^26+348*q^27+362*q^28+369*q^29+373*q^30+376*q^31+385*q^32+373*
q^33+383*q^34+380*q^35+371*q^36+364*q^37+359*q^38+321*q^39+339*q^40+312*q^41+
298*q^42+255*q^43+231*q^44+187*q^45+159*q^46+132*q^47+89*q^48+78*q^49+48*q^50+
36*q^51+6*q^52+12*q^53+q^54, 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+41*q^10+54*q^11+72*q^12+89*q^13+111*q^14+137*q^15+163*q^16+191*q^17+220*
q^18+250*q^19+280*q^20+306*q^21+333*q^22+358*q^23+381*q^24+403*q^25+421*q^26+
431*q^27+464*q^28+469*q^29+495*q^30+494*q^31+528*q^32+507*q^33+551*q^34+530*q^
35+555*q^36+526*q^37+577*q^38+512*q^39+574*q^40+517*q^41+552*q^42+491*q^43+540*
q^44+463*q^45+507*q^46+432*q^47+461*q^48+383*q^49+402*q^50+303*q^51+309*q^52+
228*q^53+227*q^54+138*q^55+171*q^56+75*q^57+96*q^58+48*q^59+34*q^60+12*q^61+12*
q^62+4*q^63, 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+41*q^10+54
*q^11+72*q^12+91*q^13+113*q^14+141*q^15+169*q^16+201*q^17+234*q^18+270*q^19+306
*q^20+340*q^21+373*q^22+410*q^23+441*q^24+474*q^25+498*q^26+523*q^27+552*q^28+
576*q^29+599*q^30+620*q^31+646*q^32+651*q^33+692*q^34+704*q^35+713*q^36+718*q^
37+751*q^38+724*q^39+778*q^40+744*q^41+774*q^42+753*q^43+783*q^44+734*q^45+787*
q^46+724*q^47+748*q^48+726*q^49+751*q^50+674*q^51+697*q^52+633*q^53+659*q^54+
594*q^55+603*q^56+519*q^57+513*q^58+432*q^59+392*q^60+325*q^61+288*q^62+274*q^
63+207*q^64+150*q^65+138*q^66+111*q^67+63*q^68+66*q^69+12*q^70+24*q^71+8*q^72+6
*q^73, 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+41*q^10+54*q^11+
72*q^12+91*q^13+115*q^14+143*q^15+173*q^16+207*q^17+244*q^18+284*q^19+326*q^20+
366*q^21+407*q^22+450*q^23+493*q^24+532*q^25+570*q^26+602*q^27+647*q^28+668*q^
29+711*q^30+729*q^31+776*q^32+775*q^33+838*q^34+844*q^35+892*q^36+878*q^37+951*
q^38+904*q^39+1008*q^40+944*q^41+1024*q^42+966*q^43+1059*q^44+968*q^45+1089*q^
46+996*q^47+1081*q^48+989*q^49+1115*q^50+958*q^51+1085*q^52+975*q^53+1075*q^54+
950*q^55+1063*q^56+900*q^57+1023*q^58+870*q^59+944*q^60+811*q^61+938*q^62+769*q
^63+819*q^64+666*q^65+742*q^66+546*q^67+597*q^68+438*q^69+492*q^70+330*q^71+389
*q^72+225*q^73+288*q^74+145*q^75+174*q^76+120*q^77+108*q^78+30*q^79+60*q^80+19*
q^81+12*q^82+12*q^83+4*q^84]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q*(q^9+q^8+2*q^7+3*q^6
+5*q^5+5*q^4+7*q^3+7*q^2+6*q+4), q^2*(q^13+q^12+2*q^11+3*q^10+5*q^9+7*q^8+9*q^7
+11*q^6+14*q^5+16*q^4+15*q^3+15*q^2+11*q+6), q^3*(q^18+q^17+2*q^16+3*q^15+5*q^
14+7*q^13+11*q^12+13*q^11+18*q^10+22*q^9+27*q^8+30*q^7+32*q^6+32*q^5+30*q^4+28*
q^3+21*q^2+15*q+4), q^4*(1+2*q^22+q^24+3*q^21+q^23+12*q+18*q^2+36*q^3+45*q^4+52
*q^5+53*q^6+60*q^7+58*q^8+60*q^9+55*q^10+53*q^11+48*q^12+40*q^13+33*q^14+26*q^
15+20*q^16+15*q^17+11*q^18+7*q^19+5*q^20), q^6*(4+7*q^25+22*q^22+11*q^24+28*q^
21+15*q^23+12*q+q^30+q^29+30*q^2+5*q^26+37*q^3+60*q^4+75*q^5+88*q^6+90*q^7+96*q
^8+100*q^9+103*q^10+101*q^11+99*q^12+96*q^13+91*q^14+87*q^15+77*q^16+2*q^28+67*
q^17+58*q^18+46*q^19+37*q^20+3*q^27), q^8*(6+64*q^25+107*q^22+77*q^24+119*q^21+
91*q^23+8*q+15*q^30+q^37+22*q^29+3*q^34+5*q^33+36*q^2+50*q^26+39*q^3+60*q^4+90*
q^5+108*q^6+7*q^32+124*q^7+141*q^8+149*q^9+154*q^10+159*q^11+168*q^12+162*q^13+
168*q^14+160*q^15+159*q^16+30*q^28+153*q^17+149*q^18+138*q^19+131*q^20+39*q^27+
11*q^31+2*q^35+q^36), q^10*(4+204*q^25+3*q^42+233*q^22+213*q^24+242*q^21+222*q^
23+12*q+q^44+121*q^30+q^45+22*q^37+139*q^29+11*q^39+5*q^41+52*q^34+15*q^38+68*q
^33+15*q^2+190*q^26+52*q^3+54*q^4+90*q^5+93*q^6+83*q^32+150*q^7+160*q^8+197*q^9
+204*q^10+231*q^11+213*q^12+255*q^13+244*q^14+258*q^15+249*q^16+2*q^43+157*q^28
+264*q^17+248*q^18+255*q^19+246*q^20+174*q^27+7*q^40+101*q^31+41*q^35+30*q^36),
q^12*(1+369*q^25+70*q^42+385*q^22+373*q^24+373*q^21+376*q^23+12*q+3*q^51+41*q^
44+q^53+312*q^30+2*q^52+q^54+30*q^45+177*q^37+329*q^29+22*q^46+131*q^39+87*q^41
+246*q^34+5*q^50+153*q^38+266*q^33+6*q^2+362*q^26+36*q^3+48*q^4+78*q^5+89*q^6+
279*q^32+132*q^7+159*q^8+187*q^9+231*q^10+255*q^11+298*q^12+312*q^13+339*q^14+
321*q^15+359*q^16+54*q^43+7*q^49+333*q^28+364*q^17+371*q^18+380*q^19+383*q^20+
348*q^27+107*q^40+299*q^31+224*q^35+11*q^48+200*q^36+15*q^47), q^15*(4+577*q^25
+306*q^42+517*q^22+512*q^24+552*q^21+574*q^23+12*q+72*q^51+250*q^44+41*q^53+11*
q^57+507*q^30+54*q^52+7*q^58+22*q^55+3*q^60+30*q^54+220*q^45+421*q^37+551*q^29+
191*q^46+381*q^39+2*q^61+333*q^41+469*q^34+89*q^50+403*q^38+495*q^33+15*q^56+12
*q^2+526*q^26+34*q^3+48*q^4+96*q^5+75*q^6+494*q^32+171*q^7+138*q^8+227*q^9+q^62
+228*q^10+309*q^11+303*q^12+402*q^13+383*q^14+461*q^15+432*q^16+280*q^43+111*q^
49+530*q^28+507*q^17+463*q^18+540*q^19+491*q^20+555*q^27+358*q^40+q^63+528*q^31
+464*q^35+137*q^48+5*q^59+431*q^36+163*q^47), q^18*(6+748*q^25+620*q^42+674*q^
22+726*q^24+697*q^21+751*q^23+8*q+373*q^51+576*q^44+306*q^53+169*q^57+753*q^30+
340*q^52+141*q^58+234*q^55+91*q^60+270*q^54+552*q^45+2*q^71+713*q^37+11*q^67+
783*q^29+523*q^46+22*q^65+5*q^69+692*q^39+72*q^61+646*q^41+724*q^34+410*q^50+30
*q^64+704*q^38+778*q^33+201*q^56+24*q^2+7*q^68+724*q^26+12*q^3+66*q^4+63*q^5+
111*q^6+744*q^32+138*q^7+150*q^8+207*q^9+54*q^62+274*q^10+288*q^11+325*q^12+392
*q^13+432*q^14+513*q^15+519*q^16+15*q^66+3*q^70+599*q^43+441*q^49+734*q^28+603*
q^17+594*q^18+659*q^19+633*q^20+787*q^27+q^72+651*q^40+q^73+41*q^63+774*q^31+
751*q^35+474*q^48+113*q^59+718*q^36+498*q^47), q^21*(4+870*q^25+1024*q^42+938*q
^22+944*q^24+769*q^21+811*q^23+12*q+775*q^51+1008*q^44+729*q^53+602*q^57+1075*q
^30+776*q^52+570*q^58+668*q^55+493*q^60+711*q^54+904*q^45+91*q^71+41*q^74+996*q
^37+207*q^67+950*q^29+951*q^46+284*q^65+143*q^69+968*q^39+450*q^61+966*q^41+
1115*q^34+7*q^79+838*q^50+326*q^64+1089*q^38+958*q^33+647*q^56+12*q^2+173*q^68+
1023*q^26+19*q^3+60*q^4+30*q^5+108*q^6+1085*q^32+120*q^7+174*q^8+145*q^9+407*q^
62+288*q^10+225*q^11+389*q^12+22*q^76+330*q^13+492*q^14+438*q^15+597*q^16+244*q
^66+115*q^70+15*q^77+11*q^78+944*q^43+844*q^49+1063*q^28+546*q^17+742*q^18+666*
q^19+819*q^20+900*q^27+72*q^72+1059*q^40+54*q^73+366*q^63+975*q^31+30*q^75+989*
q^35+5*q^80+892*q^48+q^83+3*q^81+q^84+2*q^82+532*q^59+1081*q^36+878*q^47)]


 The number of permutations avoiding, {[1, 3, 2], [5, 4, 3, 2, 1]}, is given by

 [1, 2, 5, 14, 41, 116, 302, 715, 1549, 3106, 5831, 10352, 17525, 28484, 44696]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 4, 3, 2, 1]},

     is given by

   [1, 1, 3, 7, 21, 57, 151, 350, 784, 1536, 2996, 5186, 9118, 14442, 23486]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 4, 3, 2, 1]},

     is given by

   [0, 1, 2, 7, 20, 59, 151, 365, 765, 1570, 2835, 5166, 8407, 14042, 21210]

            For the reverse patterns and complement patterns, we get

EVEN:,

    [1, 1, 2, 7, 21, 59, 151, 350, 784, 1570, 2835, 5186, 9118, 14042, 21210]

 ODD:,

    [0, 1, 3, 7, 20, 57, 151, 365, 765, 1536, 2996, 5166, 8407, 14442, 23486]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.707317073, 8.456896552,

    11.49337748, 14.79860140, 18.38540994, 22.26915647, 26.46149889,

    30.97044049, 35.80136947, 40.95794130, 46.44267943]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.255009626, 2.880948842,

    3.592434450, 4.449718874, 5.471146823, 6.656046669, 7.999025691,

    9.494858257, 11.13952382, 12.93013158, 14.86463829]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.08507, 8.29987, 12.9056, 19.8000,

    29.9334, 44.3030, 63.9844, 90.1523, 124.089, 167.188, 220.957]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5273097919,

    -0.5731424794, -0.5089495891, -0.4076945591, -0.3178899248, -0.2512420369,

    -0.2044438859, -0.1717276791, -0.1484707552, -0.1315191713, -0.1188235186]

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

    2.631899840, 2.840370583, 2.859514258, 2.749668401, 2.620578007,

    2.517745701, 2.445479135, 2.396450386, 2.363145288, 2.340161851,

    2.323911549]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [1, 4, 5, 2, 3]},

    {[1, 3, 2], [3, 4, 1, 2, 5]}, {[3, 1, 2], [3, 2, 5, 4, 1]},

    {[2, 3, 1], [5, 2, 1, 4, 3]}}

      the member , {[2, 1, 3], [1, 4, 5, 2, 3]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

                 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+4*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+4*q^4+7*q^5+7*q^6+9*q^7+11*
q^8+14*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+4*q^4+7
*q^5+9*q^6+11*q^7+13*q^8+17*q^9+20*q^10+23*q^11+26*q^12+30*q^13+36*q^14+40*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+4*q^4+7*q^5+9*q^6
+13*q^7+15*q^8+21*q^9+24*q^10+29*q^11+36*q^12+39*q^13+45*q^14+51*q^15+59*q^16+
67*q^17+75*q^18+82*q^19+92*q^20+100*q^21+98*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+4*q^4+7*q^5+9*q^6+13*q^7+17*q^8+23*q^9+28*q^10+
35*q^11+42*q^12+51*q^13+61*q^14+70*q^15+79*q^16+89*q^17+99*q^18+110*q^19+126*q^
20+135*q^21+152*q^22+173*q^23+191*q^24+210*q^25+224*q^26+239*q^27+250*q^28+244*
q^29+214*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+4*
q^4+7*q^5+9*q^6+13*q^7+17*q^8+25*q^9+30*q^10+39*q^11+48*q^12+59*q^13+73*q^14+86
*q^15+103*q^16+118*q^17+138*q^18+152*q^19+173*q^20+193*q^21+215*q^22+236*q^23+
256*q^24+279*q^25+306*q^26+338*q^27+373*q^28+411*q^29+450*q^30+498*q^31+541*q^
32+579*q^33+607*q^34+627*q^35+634*q^36+610*q^37+539*q^38+428*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+4*q^4+7*q^5+9*q^6+13*q^7+17*q
^8+25*q^9+32*q^10+41*q^11+52*q^12+65*q^13+81*q^14+100*q^15+119*q^16+142*q^17+
170*q^18+197*q^19+228*q^20+259*q^21+294*q^22+330*q^23+371*q^24+407*q^25+449*q^
26+490*q^27+531*q^28+572*q^29+623*q^30+674*q^31+728*q^32+793*q^33+854*q^34+939*
q^35+1024*q^36+1120*q^37+1221*q^38+1319*q^39+1420*q^40+1514*q^41+1589*q^42+1634
*q^43+1651*q^44+1628*q^45+1540*q^46+1363*q^47+1101*q^48+799*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+4*q^4+7*q^5+9*q^6+13*q^7+17*
q^8+25*q^9+32*q^10+43*q^11+54*q^12+69*q^13+87*q^14+108*q^15+133*q^16+160*q^17+
194*q^18+229*q^19+276*q^20+318*q^21+371*q^22+424*q^23+486*q^24+549*q^25+618*q^
26+689*q^27+764*q^28+847*q^29+924*q^30+1006*q^31+1088*q^32+1176*q^33+1265*q^34+
1362*q^35+1454*q^36+1560*q^37+1669*q^38+1786*q^39+1921*q^40+2082*q^41+2249*q^42
+2437*q^43+2641*q^44+2853*q^45+3093*q^46+3334*q^47+3571*q^48+3804*q^49+4021*q^
50+4204*q^51+4339*q^52+4389*q^53+4354*q^54+4214*q^55+3929*q^56+3466*q^57+2835*q
^58+2111*q^59+1411*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+4*q^4+7*q^5+9*q^6+13*q^7+17*q^8+25*q^9+32*q^10+43*q^11+56*q^12+71*q^
13+91*q^14+114*q^15+141*q^16+174*q^17+212*q^18+255*q^19+308*q^20+366*q^21+433*q
^22+505*q^23+589*q^24+677*q^25+779*q^26+887*q^27+1003*q^28+1126*q^29+1263*q^30+
1405*q^31+1557*q^32+1708*q^33+1866*q^34+2034*q^35+2205*q^36+2382*q^37+2558*q^38
+2748*q^39+2934*q^40+3132*q^41+3324*q^42+3531*q^43+3749*q^44+3993*q^45+4250*q^
46+4537*q^47+4844*q^48+5189*q^49+5568*q^50+5988*q^51+6470*q^52+6970*q^53+7505*q
^54+8048*q^55+8620*q^56+9199*q^57+9783*q^58+10332*q^59+10841*q^60+11292*q^61+
11621*q^62+11803*q^63+11773*q^64+11503*q^65+10967*q^66+10106*q^67+8876*q^68+
7312*q^69+5558*q^70+3841*q^71+2380*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+4*q^4+7*q^5+9*q^6+13*q^7+17*q^8+25*q^9+32*q^10+43*q^
11+56*q^12+73*q^13+93*q^14+118*q^15+147*q^16+182*q^17+226*q^18+273*q^19+334*q^
20+400*q^21+481*q^22+567*q^23+673*q^24+784*q^25+916*q^26+1059*q^27+1218*q^28+
1392*q^29+1586*q^30+1792*q^31+2016*q^32+2265*q^33+2519*q^34+2800*q^35+3084*q^36
+3392*q^37+3707*q^38+4047*q^39+4385*q^40+4740*q^41+5106*q^42+5486*q^43+5871*q^
44+6263*q^45+6651*q^46+7060*q^47+7484*q^48+7912*q^49+8363*q^50+8832*q^51+9329*q
^52+9857*q^53+10428*q^54+11034*q^55+11716*q^56+12471*q^57+13308*q^58+14221*q^59
+15225*q^60+16294*q^61+17464*q^62+18710*q^63+20032*q^64+21418*q^65+22810*q^66+
24215*q^67+25633*q^68+27022*q^69+28367*q^70+29601*q^71+30655*q^72+31494*q^73+
31988*q^74+32041*q^75+31549*q^76+30441*q^77+28665*q^78+26156*q^79+22874*q^80+
18921*q^81+14601*q^82+10368*q^83+6687*q^84+3862*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+4*q^4+7*q^5+9*q^6+13*q^7+17*q^8+25*q^9
+32*q^10+43*q^11+56*q^12+73*q^13+95*q^14+120*q^15+151*q^16+188*q^17+234*q^18+
287*q^19+352*q^20+426*q^21+515*q^22+617*q^23+735*q^24+868*q^25+1026*q^26+1200*q
^27+1399*q^28+1618*q^29+1867*q^30+2140*q^31+2445*q^32+2776*q^33+3140*q^34+3539*
q^35+3967*q^36+4428*q^37+4924*q^38+5453*q^39+6012*q^40+6609*q^41+7226*q^42+7883
*q^43+8562*q^44+9274*q^45+10008*q^46+10779*q^47+11551*q^48+12356*q^49+13169*q^
50+14008*q^51+14861*q^52+15738*q^53+16620*q^54+17526*q^55+18473*q^56+19429*q^57
+20427*q^58+21447*q^59+22521*q^60+23671*q^61+24903*q^62+26213*q^63+27651*q^64+
29214*q^65+30925*q^66+32784*q^67+34836*q^68+37040*q^69+39494*q^70+42132*q^71+
44956*q^72+47955*q^73+51092*q^74+54360*q^75+57753*q^76+61271*q^77+64807*q^78+
68388*q^79+71868*q^80+75279*q^81+78541*q^82+81534*q^83+84124*q^84+86134*q^85+
87415*q^86+87730*q^87+86842*q^88+84526*q^89+80669*q^90+75188*q^91+68037*q^92+
59258*q^93+49140*q^94+38343*q^95+27816*q^96+18537*q^97+11210*q^98+6062*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+4
*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+4*q^11+7*q^10+7*q^9
+9*q^8+11*q^7+14*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
+4*q^17+7*q^16+9*q^15+11*q^14+13*q^13+17*q^12+20*q^11+23*q^10+26*q^9+30*q^8+36*
q^7+40*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+9*q^22+4*q^24+13*q^21+7*
q^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+86*q^5+98*q^6+100*q^7+92*q^8+82*q^9+75*q^
10+67*q^11+59*q^12+51*q^13+45*q^14+39*q^15+36*q^16+q^28+29*q^17+24*q^18+21*q^19
+15*q^20+q^27, 1+35*q^25+61*q^22+42*q^24+70*q^21+51*q^23+8*q+9*q^30+13*q^29+2*q
^34+3*q^33+28*q^2+28*q^26+63*q^3+112*q^4+167*q^5+214*q^6+4*q^32+244*q^7+250*q^8
+239*q^9+224*q^10+210*q^11+191*q^12+173*q^13+152*q^14+135*q^15+126*q^16+17*q^28
+110*q^17+99*q^18+89*q^19+79*q^20+23*q^27+7*q^31+q^35+q^36, 1+173*q^25+3*q^42+
236*q^22+193*q^24+256*q^21+215*q^23+9*q+q^44+86*q^30+q^45+17*q^37+103*q^29+9*q^
39+4*q^41+39*q^34+13*q^38+48*q^33+36*q^2+152*q^26+92*q^3+182*q^4+301*q^5+428*q^
6+59*q^32+539*q^7+610*q^8+634*q^9+627*q^10+607*q^11+579*q^12+541*q^13+498*q^14+
450*q^15+411*q^16+2*q^43+118*q^28+373*q^17+338*q^18+306*q^19+279*q^20+138*q^27+
7*q^40+73*q^31+30*q^35+25*q^36, 1+623*q^25+65*q^42+793*q^22+674*q^24+854*q^21+
728*q^23+10*q+4*q^51+41*q^44+2*q^53+407*q^30+3*q^52+q^55+q^54+32*q^45+170*q^37+
449*q^29+25*q^46+119*q^39+81*q^41+259*q^34+7*q^50+142*q^38+294*q^33+45*q^2+572*
q^26+129*q^3+282*q^4+512*q^5+799*q^6+330*q^32+1101*q^7+1363*q^8+1540*q^9+1628*q
^10+1651*q^11+1634*q^12+1589*q^13+1514*q^14+1420*q^15+1319*q^16+52*q^43+9*q^49+
490*q^28+1221*q^17+1120*q^18+1024*q^19+939*q^20+531*q^27+100*q^40+371*q^31+228*
q^35+13*q^48+197*q^36+17*q^47, 1+2082*q^25+486*q^42+2641*q^22+2249*q^24+2853*q^
21+2437*q^23+11*q+108*q^51+371*q^44+69*q^53+25*q^57+1454*q^30+87*q^52+17*q^58+
43*q^55+9*q^60+54*q^54+318*q^45+847*q^37+1560*q^29+276*q^46+q^65+689*q^39+7*q^
61+549*q^41+1088*q^34+133*q^50+2*q^64+764*q^38+1176*q^33+32*q^56+55*q^2+1921*q^
26+175*q^3+420*q^4+831*q^5+1411*q^6+1265*q^32+2111*q^7+2835*q^8+3466*q^9+4*q^62
+3929*q^10+4214*q^11+4354*q^12+4389*q^13+4339*q^14+4204*q^15+4021*q^16+q^66+424
*q^43+160*q^49+1669*q^28+3804*q^17+3571*q^18+3334*q^19+3093*q^20+1786*q^27+618*
q^40+3*q^63+1362*q^31+1006*q^35+194*q^48+13*q^59+924*q^36+229*q^47, 1+6970*q^25
+2205*q^42+8620*q^22+7505*q^24+9199*q^21+8048*q^23+12*q+887*q^51+1866*q^44+677*
q^53+366*q^57+4844*q^30+779*q^52+308*q^58+505*q^55+212*q^60+589*q^54+1708*q^45+
13*q^71+4*q^74+3132*q^37+43*q^67+5189*q^29+1557*q^46+71*q^65+25*q^69+2748*q^39+
174*q^61+2382*q^41+3749*q^34+1003*q^50+91*q^64+2934*q^38+3993*q^33+433*q^56+66*
q^2+32*q^68+6470*q^26+231*q^3+605*q^4+1297*q^5+2380*q^6+4250*q^32+3841*q^7+5558
*q^8+7312*q^9+141*q^62+8876*q^10+10106*q^11+10967*q^12+2*q^76+11503*q^13+11773*
q^14+11803*q^15+11621*q^16+56*q^66+17*q^70+q^77+q^78+2034*q^43+1126*q^49+5568*q
^28+11292*q^17+10841*q^18+10332*q^19+9783*q^20+5988*q^27+9*q^72+2558*q^40+7*q^
73+114*q^63+4537*q^31+3*q^75+3531*q^35+1263*q^48+255*q^59+3324*q^36+1405*q^47,
1+22810*q^25+7912*q^42+27022*q^22+q^90+24215*q^24+28367*q^21+25633*q^23+13*q+
4385*q^51+7060*q^44+3707*q^53+2519*q^57+16294*q^30+4047*q^52+2265*q^58+3084*q^
55+1792*q^60+q^91+3392*q^54+6651*q^45+334*q^71+182*q^74+10428*q^37+673*q^67+
17464*q^29+6263*q^46+916*q^65+481*q^69+9329*q^39+1586*q^61+8363*q^41+12471*q^34
+56*q^79+4740*q^50+1059*q^64+9857*q^38+13308*q^33+2800*q^56+78*q^2+567*q^68+
21418*q^26+298*q^3+847*q^4+1958*q^5+3862*q^6+14221*q^32+6687*q^7+10368*q^8+
14601*q^9+1392*q^62+18921*q^10+22874*q^11+26156*q^12+118*q^76+28665*q^13+30441*
q^14+31549*q^15+32041*q^16+784*q^66+400*q^70+93*q^77+73*q^78+7484*q^43+5106*q^
49+18710*q^28+31988*q^17+31494*q^18+30655*q^19+29601*q^20+20032*q^27+273*q^72+
8832*q^40+226*q^73+1218*q^63+15225*q^31+147*q^75+11716*q^35+43*q^80+5486*q^48+
17*q^83+32*q^81+9*q^85+7*q^86+4*q^87+3*q^88+2*q^89+13*q^84+25*q^82+2016*q^59+
11034*q^36+5871*q^47, 1+71868*q^25+26213*q^42+81534*q^22+120*q^90+75279*q^24+
84124*q^21+78541*q^23+14*q+17*q^97+16620*q^51+23671*q^44+14861*q^53+11551*q^57+
54360*q^30+43*q^94+15738*q^52+10779*q^58+13169*q^55+9274*q^60+95*q^91+14008*q^
54+22521*q^45+3140*q^71+2140*q^74+25*q^96+13*q^98+34836*q^37+4924*q^67+57753*q^
29+21447*q^46+56*q^93+9*q^99+6012*q^65+3967*q^69+30925*q^39+8562*q^61+27651*q^
41+42132*q^34+1026*q^79+17526*q^50+6609*q^64+7*q^100+32784*q^38+44956*q^33+
12356*q^56+91*q^2+4428*q^68+68388*q^26+377*q^3+1157*q^4+2872*q^5+6062*q^6+47955
*q^32+11210*q^7+18537*q^8+27816*q^9+32*q^95+7883*q^62+4*q^101+38343*q^10+49140*
q^11+59258*q^12+1618*q^76+68037*q^13+75188*q^14+80669*q^15+84526*q^16+5453*q^66
+3539*q^70+1399*q^77+1200*q^78+24903*q^43+3*q^102+18473*q^49+61271*q^28+86842*q
^17+87730*q^18+87415*q^19+86134*q^20+64807*q^27+2776*q^72+29214*q^40+2445*q^73+
2*q^103+7226*q^63+51092*q^31+1867*q^75+39494*q^35+q^104+868*q^80+19429*q^48+q^
105+515*q^83+735*q^81+352*q^85+287*q^86+234*q^87+188*q^88+151*q^89+426*q^84+617
*q^82+10008*q^59+37040*q^36+73*q^92+20427*q^47]


 The number of permutations avoiding, {[2, 1, 3], [1, 4, 5, 2, 3]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [1, 4, 5, 2, 3]},

     is given by

[1, 1, 3, 7, 21, 60, 182, 547, 1640, 4921, 14762, 44288, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[2, 1, 3], [1, 4, 5, 2, 3]},

     is given by

[0, 1, 2, 7, 20, 62, 183, 547, 1641, 4921, 14763, 44286, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 62, 183, 547, 1640, 4921, 14763, 44288, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 60, 182, 547, 1641, 4921, 14762, 44286, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.853658537, 9.131147541,

    13.21369863, 18.12157221, 23.87412374, 30.48963625, 37.98486029,

    46.37480525, 55.67274698, 65.89034726, 77.03781458]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.332809210, 3.138734263,

    4.048111857, 5.055345505, 6.153403499, 7.334944209, 8.592512407,

    9.918603870, 11.30575549, 12.74665262, 14.23422863]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.44200, 9.85165, 16.3872, 25.5565,

    37.8644, 53.8014, 73.8313, 98.3787, 127.820, 162.477, 202.613]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5032000763,

    -0.5983240179, -0.6707187002, -0.7284811497, -0.7784501496, -0.8247061786,

    -0.8693900526, -0.9135712991, -0.9577115658, -1.001966694, -1.046321209]

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

    2.673308354, 2.856596548, 2.984538741, 3.087494852, 3.181034613,

    3.273506057, 3.369637667, 3.472155416, 3.582541959, 3.701576987,

    3.829530684]

                                end of this data



For the equivalence class of patterns, {{[1, 2, 3], [4, 3, 5, 2, 1]},

    {[3, 2, 1], [3, 1, 2, 4, 5]}, {[3, 2, 1], [1, 2, 5, 3, 4]},

    {[3, 2, 1], [1, 2, 4, 5, 3]}, {[3, 2, 1], [2, 3, 1, 4, 5]},

    {[1, 2, 3], [5, 4, 1, 3, 2]}, {[1, 2, 3], [5, 4, 2, 1, 3]},

    {[1, 2, 3], [3, 5, 4, 2, 1]}}

      the member , {[3, 2, 1], [1, 2, 5, 3, 4]}, has a scheme of depth , 5

                                  here it is:

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

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

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

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

    [[3, 4, 5, 2, 1], {[0, 0, 0, 0, 0, 0]}, {1}],

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

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

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

    [2, 3, 4, 1, 5], {[0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0],

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

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

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

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

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

    [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {}], [[2, 3, 1, 4, 5], {

    [0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 1, 0],

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

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

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

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

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

    [[3, 4, 5, 1, 2],

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

    [2, 4, 5, 1, 3], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 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], [0, 0, 0, 0, 1, 0],

    [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 5, 3], {

    [0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0],

    [0, 1, 0, 0, 0, 0]}, {3}], [[2, 3, 1, 5, 4], {[0, 0, 1, 0, 0, 0],

    [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0],

    [0, 1, 0, 0, 0, 0]}, {3}], [[3, 4, 2, 5, 1], {[0, 0, 0, 0, 0, 0]}, {3}], [

    [3, 4, 1, 5, 2],

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

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

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

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

%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, 1+3*q+5*q^2+4*q^3+q^4, 1+4*q+8*q^2+12*q^3+10*q^4+4*q^5+2*
q^6, 1+5*q+12*q^2+20*q^3+26*q^4+23*q^5+15*q^6+9*q^7+4*q^8+q^9, 1+6*q+17*q^2+32*
q^3+46*q^4+53*q^5+47*q^6+38*q^7+30*q^8+20*q^9+11*q^10+4*q^11+2*q^12, 1+7*q+23*q
^2+49*q^3+78*q^4+99*q^5+104*q^6+93*q^7+79*q^8+69*q^9+57*q^10+42*q^11+29*q^12+17
*q^13+8*q^14+4*q^15+q^16, 1+8*q+30*q^2+72*q^3+127*q^4+177*q^5+203*q^6+201*q^7+
180*q^8+157*q^9+139*q^10+121*q^11+101*q^12+83*q^13+64*q^14+47*q^15+33*q^16+19*q
^17+10*q^18+4*q^19+2*q^20, 1+9*q+38*q^2+102*q^3+199*q^4+304*q^5+380*q^6+404*q^7
+385*q^8+345*q^9+305*q^10+271*q^11+237*q^12+207*q^13+178*q^14+148*q^15+125*q^16
+103*q^17+82*q^18+62*q^19+44*q^20+29*q^21+15*q^22+8*q^23+4*q^24+q^25, 1+10*q+47
*q^2+140*q^3+301*q^4+503*q^5+684*q^6+784*q^7+789*q^8+734*q^9+658*q^10+585*q^11+
519*q^12+457*q^13+402*q^14+351*q^15+304*q^16+263*q^17+225*q^18+193*q^19+165*q^
20+141*q^21+114*q^22+88*q^23+69*q^24+49*q^25+31*q^26+18*q^27+10*q^28+4*q^29+2*q
^30, 1+11*q+57*q^2+187*q^3+441*q^4+804*q^5+1187*q^6+1468*q^7+1573*q^8+1523*q^9+
1396*q^10+1251*q^11+1113*q^12+987*q^13+872*q^14+768*q^15+674*q^16+594*q^17+521*
q^18+453*q^19+395*q^20+346*q^21+306*q^22+264*q^23+228*q^24+198*q^25+166*q^26+
137*q^27+111*q^28+87*q^29+64*q^30+44*q^31+27*q^32+15*q^33+8*q^34+4*q^35+q^36, 1
+12*q+68*q^2+244*q^3+628*q^4+1245*q^5+1991*q^6+2655*q^7+3041*q^8+3096*q^9+2919*
q^10+2651*q^11+2372*q^12+2109*q^13+1870*q^14+1653*q^15+1457*q^16+1285*q^17+1136
*q^18+1003*q^19+883*q^20+778*q^21+689*q^22+608*q^23+535*q^24+478*q^25+425*q^26+
372*q^27+324*q^28+284*q^29+248*q^30+215*q^31+183*q^32+150*q^33+120*q^34+95*q^35
+71*q^36+47*q^37+30*q^38+18*q^39+10*q^40+4*q^41+2*q^42, 1+13*q+80*q^2+312*q^3+
872*q^4+1873*q^5+3236*q^6+4646*q^7+5696*q^8+6137*q^9+6015*q^10+5570*q^11+5027*q
^12+4489*q^13+3988*q^14+3534*q^15+3123*q^16+2757*q^17+2438*q^18+2158*q^19+1909*
q^20+1692*q^21+1504*q^22+1336*q^23+1182*q^24+1051*q^25+944*q^26+843*q^27+751*q^
28+670*q^29+599*q^30+533*q^31+473*q^32+421*q^33+371*q^34+329*q^35+293*q^36+253*
q^37+212*q^38+178*q^39+148*q^40+117*q^41+89*q^42+64*q^43+42*q^44+27*q^45+15*q^
46+8*q^47+4*q^48+q^49, 1+14*q+93*q^2+392*q^3+1184*q^4+2745*q^5+5109*q^6+7882*q^
7+10342*q^8+11833*q^9+12152*q^10+11585*q^11+10597*q^12+9520*q^13+8485*q^14+7531
*q^15+6668*q^16+5893*q^17+5210*q^18+4613*q^19+4086*q^20+3622*q^21+3221*q^22+
2873*q^23+2557*q^24+2274*q^25+2036*q^26+1827*q^27+1635*q^28+1465*q^29+1318*q^30
+1190*q^31+1071*q^32+962*q^33+861*q^34+770*q^35+697*q^36+632*q^37+565*q^38+503*
q^39+451*q^40+403*q^41+354*q^42+312*q^43+271*q^44+231*q^45+194*q^46+158*q^47+
127*q^48+97*q^49+69*q^50+46*q^51+30*q^52+18*q^53+10*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+8*q^4+12*q^
3+10*q^2+4*q+2), q^6*(q^9+5*q^8+12*q^7+20*q^6+26*q^5+23*q^4+15*q^3+9*q^2+4*q+1)
, q^9*(q^12+6*q^11+17*q^10+32*q^9+46*q^8+53*q^7+47*q^6+38*q^5+30*q^4+20*q^3+11*
q^2+4*q+2), q^12*(q^16+7*q^15+23*q^14+49*q^13+78*q^12+99*q^11+104*q^10+93*q^9+
79*q^8+69*q^7+57*q^6+42*q^5+29*q^4+17*q^3+8*q^2+4*q+1), q^16*(q^20+8*q^19+30*q^
18+72*q^17+127*q^16+177*q^15+203*q^14+201*q^13+180*q^12+157*q^11+139*q^10+121*q
^9+101*q^8+83*q^7+64*q^6+47*q^5+33*q^4+19*q^3+10*q^2+4*q+2), q^20*(1+q^25+102*q
^22+9*q^24+199*q^21+38*q^23+4*q+8*q^2+15*q^3+29*q^4+44*q^5+62*q^6+82*q^7+103*q^
8+125*q^9+148*q^10+178*q^11+207*q^12+237*q^13+271*q^14+305*q^15+345*q^16+385*q^
17+404*q^18+380*q^19+304*q^20), q^25*(2+503*q^25+789*q^22+684*q^24+734*q^21+784
*q^23+4*q+q^30+10*q^29+10*q^2+301*q^26+18*q^3+31*q^4+49*q^5+69*q^6+88*q^7+114*q
^8+141*q^9+165*q^10+193*q^11+225*q^12+263*q^13+304*q^14+351*q^15+402*q^16+47*q^
28+457*q^17+519*q^18+585*q^19+658*q^20+140*q^27), q^30*(1+1251*q^25+872*q^22+
1113*q^24+768*q^21+987*q^23+4*q+1187*q^30+1468*q^29+57*q^34+187*q^33+8*q^2+1396
*q^26+15*q^3+27*q^4+44*q^5+64*q^6+441*q^32+87*q^7+111*q^8+137*q^9+166*q^10+198*
q^11+228*q^12+264*q^13+306*q^14+346*q^15+395*q^16+1573*q^28+453*q^17+521*q^18+
594*q^19+674*q^20+1523*q^27+804*q^31+11*q^35+q^36), q^36*(2+1285*q^25+q^42+883*
q^22+1136*q^24+778*q^21+1003*q^23+4*q+2372*q^30+1245*q^37+2109*q^29+244*q^39+12
*q^41+3041*q^34+628*q^38+3096*q^33+10*q^2+1457*q^26+18*q^3+30*q^4+47*q^5+71*q^6
+2919*q^32+95*q^7+120*q^8+150*q^9+183*q^10+215*q^11+248*q^12+284*q^13+324*q^14+
372*q^15+425*q^16+1870*q^28+478*q^17+535*q^18+608*q^19+689*q^20+1653*q^27+68*q^
40+2651*q^31+2655*q^35+1991*q^36), q^42*(1+1182*q^25+4646*q^42+843*q^22+1051*q^
24+751*q^21+944*q^23+4*q+1873*q^44+2158*q^30+872*q^45+5027*q^37+1909*q^29+312*q
^46+6015*q^39+5696*q^41+3534*q^34+5570*q^38+3123*q^33+8*q^2+1336*q^26+15*q^3+27
*q^4+42*q^5+64*q^6+2757*q^32+89*q^7+117*q^8+148*q^9+178*q^10+212*q^11+253*q^12+
293*q^13+329*q^14+371*q^15+421*q^16+3236*q^43+q^49+1692*q^28+473*q^17+533*q^18+
599*q^19+670*q^20+1504*q^27+6137*q^40+2438*q^31+3988*q^35+13*q^48+4489*q^36+80*
q^47), q^49*(2+1190*q^25+8485*q^42+861*q^22+1071*q^24+770*q^21+962*q^23+4*q+
2745*q^51+10597*q^44+392*q^53+2036*q^30+1184*q^52+14*q^55+93*q^54+11585*q^45+
4613*q^37+1827*q^29+12152*q^46+5893*q^39+7531*q^41+3221*q^34+5109*q^50+5210*q^
38+2873*q^33+q^56+10*q^2+1318*q^26+18*q^3+30*q^4+46*q^5+69*q^6+2557*q^32+97*q^7
+127*q^8+158*q^9+194*q^10+231*q^11+271*q^12+312*q^13+354*q^14+403*q^15+451*q^16
+9520*q^43+7882*q^49+1635*q^28+503*q^17+565*q^18+632*q^19+697*q^20+1465*q^27+
6668*q^40+2274*q^31+3622*q^35+10342*q^48+4086*q^36+11833*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [1, 2, 5, 3, 4]}, is given by

[1, 2, 5, 14, 41, 116, 307, 760, 1779, 3986, 8641, 18282, 38005, 78024, 158791]

The number of EVEN permutations avoiding, {[3, 2, 1], [1, 2, 5, 3, 4]},

     is given by

   [1, 1, 3, 7, 21, 58, 154, 380, 890, 1993, 4321, 9141, 19003, 39012, 79396]

The number of ODD permutations avoiding, {[3, 2, 1], [1, 2, 5, 3, 4]},

     is given by

   [0, 1, 2, 7, 20, 58, 153, 380, 889, 1993, 4320, 9141, 19002, 39012, 79395]

            For the reverse patterns and complement patterns, we get

EVEN:,

    [1, 1, 2, 7, 21, 58, 153, 380, 890, 1993, 4320, 9141, 19003, 39012, 79395]

 ODD:,

    [0, 1, 3, 7, 20, 58, 154, 380, 889, 1993, 4321, 9141, 19002, 39012, 79396]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.121951220, 4.327586207,

    5.638436482, 7.005263158, 8.383923553, 9.735825389, 11.02927902,

    12.24105678, 13.35724247, 14.37296217, 15.29104924]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.364982300, 1.784878350,

    2.303422217, 2.917275890, 3.613943098, 4.372943546, 5.168042870,

    5.970328594, 6.751477045, 7.486680131, 8.156793029]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.86318, 3.18579, 5.30575, 8.51050,

    13.0606, 19.1226, 26.7087, 35.6448, 45.5824, 56.0504, 66.5333]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.009381844257,

    0.1284731399, 0.2388928662, 0.3553925142, 0.4825245484, 0.6193695034,

    0.7637950563, 0.9138809086, 1.067934120, 1.224252727, 1.380973122]

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

    2.681797978, 2.687201391, 2.642628850, 2.636306279, 2.702954430,

    2.853690672, 3.092711294, 3.423150566, 3.847931671, 4.369109106,

    4.986739488]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [1, 4, 5, 3, 2]},

    {[1, 3, 2], [4, 3, 1, 2, 5]}, {[1, 3, 2], [3, 4, 2, 1, 5]},

    {[2, 3, 1], [5, 2, 1, 3, 4]}, {[3, 1, 2], [3, 2, 4, 5, 1]},

    {[3, 1, 2], [2, 3, 5, 4, 1]}, {[2, 3, 1], [5, 1, 2, 4, 3]},

    {[2, 1, 3], [1, 5, 4, 2, 3]}}

      the member , {[2, 1, 3], [1, 4, 5, 3, 2]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

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

                 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+
4*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+6*q^5+8*q^6+9*q^7+11*
q^8+14*q^9+14*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+6
*q^5+10*q^6+10*q^7+14*q^8+16*q^9+20*q^10+21*q^11+26*q^12+29*q^13+35*q^14+37*q^
15+37*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+6*q^5+10*
q^6+12*q^7+15*q^8+19*q^9+23*q^10+27*q^11+32*q^12+38*q^13+38*q^14+48*q^15+54*q^
16+61*q^17+70*q^18+81*q^19+86*q^20+94*q^21+90*q^22+82*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+6*q^5+10*q^6+12*q^7+17*q^8+20*q^9+26*q^
10+30*q^11+39*q^12+43*q^13+51*q^14+59*q^15+66*q^16+74*q^17+84*q^18+92*q^19+106*
q^20+117*q^21+132*q^22+154*q^23+176*q^24+192*q^25+213*q^26+226*q^27+230*q^28+
226*q^29+199*q^30+162*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+6*q^5+10*q^6+12*q^7+17*q^8+22*q^9+27*q^10+33*q^11+42*q^12+50*q^13+57*q^
14+71*q^15+79*q^16+92*q^17+103*q^18+117*q^19+129*q^20+148*q^21+160*q^22+176*q^
23+196*q^24+216*q^25+241*q^26+272*q^27+306*q^28+350*q^29+388*q^30+440*q^31+488*
q^32+530*q^33+558*q^34+587*q^35+583*q^36+558*q^37+499*q^38+404*q^39+295*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+6*q^5+10*q^6+12*q^7
+17*q^8+22*q^9+29*q^10+34*q^11+45*q^12+53*q^13+64*q^14+77*q^15+92*q^16+104*q^17
+123*q^18+140*q^19+157*q^20+180*q^21+201*q^22+221*q^23+247*q^24+275*q^25+297*q^
26+326*q^27+357*q^28+386*q^29+432*q^30+473*q^31+517*q^32+576*q^33+652*q^34+723*
q^35+823*q^36+921*q^37+1025*q^38+1135*q^39+1239*q^40+1341*q^41+1430*q^42+1488*q
^43+1509*q^44+1494*q^45+1399*q^46+1248*q^47+1026*q^48+764*q^49+505*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+6*q^5+10*q^6+12*q^7+17*
q^8+22*q^9+29*q^10+36*q^11+46*q^12+56*q^13+67*q^14+84*q^15+98*q^16+117*q^17+136
*q^18+159*q^19+182*q^20+212*q^21+234*q^22+269*q^23+302*q^24+335*q^25+372*q^26+
412*q^27+451*q^28+495*q^29+541*q^30+585*q^31+640*q^32+693*q^33+747*q^34+818*q^
35+888*q^36+961*q^37+1054*q^38+1165*q^39+1284*q^40+1440*q^41+1598*q^42+1781*q^
43+1994*q^44+2232*q^45+2459*q^46+2728*q^47+2969*q^48+3222*q^49+3455*q^50+3672*q
^51+3837*q^52+3940*q^53+3929*q^54+3813*q^55+3558*q^56+3141*q^57+2609*q^58+1985*
q^59+1363*q^60+823*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+6*q^5+10*q^6+12*q^7+17*q^8+22*q^9+29*q^10+36*q^11+48*q^12+57*q^13+70*q^
14+87*q^15+105*q^16+123*q^17+149*q^18+172*q^19+202*q^20+236*q^21+268*q^22+306*q
^23+351*q^24+395*q^25+440*q^26+498*q^27+551*q^28+607*q^29+673*q^30+738*q^31+806
*q^32+881*q^33+952*q^34+1035*q^35+1124*q^36+1208*q^37+1307*q^38+1405*q^39+1514*
q^40+1625*q^41+1760*q^42+1898*q^43+2062*q^44+2239*q^45+2452*q^46+2692*q^47+2968
*q^48+3271*q^49+3640*q^50+4045*q^51+4503*q^52+4998*q^53+5513*q^54+6068*q^55+
6667*q^56+7261*q^57+7860*q^58+8480*q^59+9031*q^60+9562*q^61+9997*q^62+10295*q^
63+10392*q^64+10257*q^65+9805*q^66+9066*q^67+7992*q^68+6646*q^69+5152*q^70+3645
*q^71+2317*q^72+1288*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+6*q^5+10*q^6+12*q^7+17*q^8+22*q^9+29*q^10+36*q^11+48*q^12+59*q^13+71*q
^14+90*q^15+108*q^16+130*q^17+155*q^18+185*q^19+215*q^20+256*q^21+293*q^22+339*
q^23+390*q^24+448*q^25+501*q^26+571*q^27+643*q^28+716*q^29+801*q^30+888*q^31+
977*q^32+1080*q^33+1185*q^34+1288*q^35+1412*q^36+1534*q^37+1656*q^38+1792*q^39+
1939*q^40+2080*q^41+2239*q^42+2406*q^43+2570*q^44+2762*q^45+2950*q^46+3152*q^47
+3391*q^48+3642*q^49+3899*q^50+4209*q^51+4539*q^52+4912*q^53+5345*q^54+5848*q^
55+6389*q^56+7042*q^57+7754*q^58+8566*q^59+9464*q^60+10458*q^61+11508*q^62+
12714*q^63+13955*q^64+15270*q^65+16626*q^66+18044*q^67+19488*q^68+21006*q^69+
22456*q^70+23885*q^71+25193*q^72+26288*q^73+27112*q^74+27550*q^75+27456*q^76+
26755*q^77+25360*q^78+23217*q^79+20421*q^80+17035*q^81+13348*q^82+9688*q^83+
6399*q^84+3782*q^85+1948*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+6*q^5+10*q^6+12*q^7+17*q^8+22*q^9+29*q^10+36*q^11+48*q^12+59*q^13+
73*q^14+91*q^15+111*q^16+133*q^17+162*q^18+191*q^19+228*q^20+269*q^21+313*q^22+
364*q^23+424*q^24+486*q^25+556*q^26+636*q^27+717*q^28+813*q^29+916*q^30+1023*q^
31+1141*q^32+1271*q^33+1402*q^34+1551*q^35+1705*q^36+1865*q^37+2043*q^38+2226*q
^39+2412*q^40+2620*q^41+2839*q^42+3051*q^43+3291*q^44+3534*q^45+3789*q^46+4061*
q^47+4343*q^48+4626*q^49+4951*q^50+5276*q^51+5611*q^52+5987*q^53+6390*q^54+6793
*q^55+7260*q^56+7753*q^57+8280*q^58+8881*q^59+9540*q^60+10269*q^61+11130*q^62+
12049*q^63+13094*q^64+14287*q^65+15627*q^66+17105*q^67+18807*q^68+20672*q^69+
22701*q^70+24966*q^71+27390*q^72+29990*q^73+32778*q^74+35730*q^75+38827*q^76+
42152*q^77+45555*q^78+49070*q^79+52696*q^80+56388*q^81+60063*q^82+63656*q^83+
66899*q^84+69773*q^85+71963*q^86+73316*q^87+73591*q^88+72559*q^89+69945*q^90+
65718*q^91+59801*q^92+52379*q^93+43827*q^94+34642*q^95+25610*q^96+17460*q^97+
10805*q^98+5963*q^99+2861*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+q^2+q^3, 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+4*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+6*q^10+8*q^9
+9*q^8+11*q^7+14*q^6+14*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+6*q^16+10*q^15+10*q^14+14*q^13+16*q^12+20*q^11+21*q^10+26*q^9+29*q^8+35
*q^7+37*q^6+37*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+10*q^22+5*q^24+12*q^21+
6*q^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+82*q^5+90*q^6+94*q^7+86*q^8+81*q^9+70*q^
10+61*q^11+54*q^12+48*q^13+38*q^14+38*q^15+32*q^16+q^28+27*q^17+23*q^18+19*q^19
+15*q^20+q^27, 1+30*q^25+51*q^22+39*q^24+59*q^21+43*q^23+8*q+10*q^30+12*q^29+2*
q^34+3*q^33+28*q^2+26*q^26+63*q^3+112*q^4+162*q^5+199*q^6+5*q^32+226*q^7+230*q^
8+226*q^9+213*q^10+192*q^11+176*q^12+154*q^13+132*q^14+117*q^15+106*q^16+17*q^
28+92*q^17+84*q^18+74*q^19+66*q^20+20*q^27+6*q^31+q^35+q^36, 1+129*q^25+3*q^42+
176*q^22+148*q^24+196*q^21+160*q^23+9*q+q^44+71*q^30+q^45+17*q^37+79*q^29+10*q^
39+5*q^41+33*q^34+12*q^38+42*q^33+36*q^2+117*q^26+92*q^3+182*q^4+295*q^5+404*q^
6+50*q^32+499*q^7+558*q^8+583*q^9+587*q^10+558*q^11+530*q^12+488*q^13+440*q^14+
388*q^15+350*q^16+2*q^43+92*q^28+306*q^17+272*q^18+241*q^19+216*q^20+103*q^27+6
*q^40+57*q^31+27*q^35+22*q^36, 1+432*q^25+53*q^42+576*q^22+473*q^24+652*q^21+
517*q^23+10*q+5*q^51+34*q^44+2*q^53+275*q^30+3*q^52+q^55+q^54+29*q^45+123*q^37+
297*q^29+22*q^46+92*q^39+64*q^41+180*q^34+6*q^50+104*q^38+201*q^33+45*q^2+386*q
^26+129*q^3+282*q^4+505*q^5+764*q^6+221*q^32+1026*q^7+1248*q^8+1399*q^9+1494*q^
10+1509*q^11+1488*q^12+1430*q^13+1341*q^14+1239*q^15+1135*q^16+45*q^43+10*q^49+
326*q^28+1025*q^17+921*q^18+823*q^19+723*q^20+357*q^27+77*q^40+247*q^31+157*q^
35+12*q^48+140*q^36+17*q^47, 1+1440*q^25+302*q^42+1994*q^22+1598*q^24+2232*q^21
+1781*q^23+11*q+84*q^51+234*q^44+56*q^53+22*q^57+888*q^30+67*q^52+17*q^58+36*q^
55+10*q^60+46*q^54+212*q^45+495*q^37+961*q^29+182*q^46+q^65+412*q^39+6*q^61+335
*q^41+640*q^34+98*q^50+2*q^64+451*q^38+693*q^33+29*q^56+55*q^2+1284*q^26+175*q^
3+420*q^4+823*q^5+1363*q^6+747*q^32+1985*q^7+2609*q^8+3141*q^9+5*q^62+3558*q^10
+3813*q^11+3929*q^12+3940*q^13+3837*q^14+3672*q^15+3455*q^16+q^66+269*q^43+117*
q^49+1054*q^28+3222*q^17+2969*q^18+2728*q^19+2459*q^20+1165*q^27+372*q^40+3*q^
63+818*q^31+585*q^35+136*q^48+12*q^59+541*q^36+159*q^47, 1+4998*q^25+1124*q^42+
6667*q^22+5513*q^24+7261*q^21+6068*q^23+12*q+498*q^51+952*q^44+395*q^53+236*q^
57+2968*q^30+440*q^52+202*q^58+306*q^55+149*q^60+351*q^54+881*q^45+12*q^71+5*q^
74+1625*q^37+36*q^67+3271*q^29+806*q^46+57*q^65+22*q^69+1405*q^39+123*q^61+1208
*q^41+2062*q^34+551*q^50+70*q^64+1514*q^38+2239*q^33+268*q^56+66*q^2+29*q^68+
4503*q^26+231*q^3+605*q^4+1288*q^5+2317*q^6+2452*q^32+3645*q^7+5152*q^8+6646*q^
9+105*q^62+7992*q^10+9066*q^11+9805*q^12+2*q^76+10257*q^13+10392*q^14+10295*q^
15+9997*q^16+48*q^66+17*q^70+q^77+q^78+1035*q^43+607*q^49+3640*q^28+9562*q^17+
9031*q^18+8480*q^19+7860*q^20+4045*q^27+10*q^72+1307*q^40+6*q^73+87*q^63+2692*q
^31+3*q^75+1898*q^35+673*q^48+172*q^59+1760*q^36+738*q^47, 1+16626*q^25+3642*q^
42+21006*q^22+q^90+18044*q^24+22456*q^21+19488*q^23+13*q+1939*q^51+3152*q^44+
1656*q^53+1185*q^57+10458*q^30+1792*q^52+1080*q^58+1412*q^55+888*q^60+q^91+1534
*q^54+2950*q^45+215*q^71+130*q^74+5345*q^37+390*q^67+11508*q^29+2762*q^46+501*q
^65+293*q^69+4539*q^39+801*q^61+3899*q^41+7042*q^34+48*q^79+2080*q^50+571*q^64+
4912*q^38+7754*q^33+1288*q^56+78*q^2+339*q^68+15270*q^26+298*q^3+847*q^4+1948*q
^5+3782*q^6+8566*q^32+6399*q^7+9688*q^8+13348*q^9+716*q^62+17035*q^10+20421*q^
11+23217*q^12+90*q^76+25360*q^13+26755*q^14+27456*q^15+27550*q^16+448*q^66+256*
q^70+71*q^77+59*q^78+3391*q^43+2239*q^49+12714*q^28+27112*q^17+26288*q^18+25193
*q^19+23885*q^20+13955*q^27+185*q^72+4209*q^40+155*q^73+643*q^63+9464*q^31+108*
q^75+6389*q^35+36*q^80+2406*q^48+17*q^83+29*q^81+10*q^85+6*q^86+5*q^87+3*q^88+2
*q^89+12*q^84+22*q^82+977*q^59+5848*q^36+2570*q^47, 1+52696*q^25+12049*q^42+
63656*q^22+91*q^90+56388*q^24+66899*q^21+60063*q^23+14*q+17*q^97+6390*q^51+
10269*q^44+5611*q^53+4343*q^57+35730*q^30+36*q^94+5987*q^52+4061*q^58+4951*q^55
+3534*q^60+73*q^91+5276*q^54+9540*q^45+1402*q^71+1023*q^74+22*q^96+12*q^98+
18807*q^37+2043*q^67+38827*q^29+8881*q^46+48*q^93+10*q^99+2412*q^65+1705*q^69+
15627*q^39+3291*q^61+13094*q^41+24966*q^34+556*q^79+6793*q^50+2620*q^64+6*q^100
+17105*q^38+27390*q^33+4626*q^56+91*q^2+1865*q^68+49070*q^26+377*q^3+1157*q^4+
2861*q^5+5963*q^6+29990*q^32+10805*q^7+17460*q^8+25610*q^9+29*q^95+3051*q^62+5*
q^101+34642*q^10+43827*q^11+52379*q^12+813*q^76+59801*q^13+65718*q^14+69945*q^
15+72559*q^16+2226*q^66+1551*q^70+717*q^77+636*q^78+11130*q^43+3*q^102+7260*q^
49+42152*q^28+73591*q^17+73316*q^18+71963*q^19+69773*q^20+45555*q^27+1271*q^72+
14287*q^40+1141*q^73+2*q^103+2839*q^63+32778*q^31+916*q^75+22701*q^35+q^104+486
*q^80+7753*q^48+q^105+313*q^83+424*q^81+228*q^85+191*q^86+162*q^87+133*q^88+111
*q^89+269*q^84+364*q^82+3789*q^59+20672*q^36+59*q^92+8280*q^47]


 The number of permutations avoiding, {[2, 1, 3], [1, 4, 5, 3, 2]}, is given by

[1, 2, 5, 14, 41, 121, 355, 1033, 2986, 8594, 24674, 70757, 202814, 581272,

    1666003]

The number of EVEN permutations avoiding, {[2, 1, 3], [1, 4, 5, 3, 2]},

     is given by

[1, 1, 3, 7, 22, 60, 181, 513, 1498, 4287, 12343, 35353, 101407, 290579, 832970

    ]

The number of ODD permutations avoiding, {[2, 1, 3], [1, 4, 5, 3, 2]},

     is given by

[0, 1, 2, 7, 19, 61, 174, 520, 1488, 4307, 12331, 35404, 101407, 290693, 833033

    ]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 61, 174, 513, 1498, 4307, 12331, 35353, 101407, 290693,

    833033]

 ODD:, [0, 1, 3, 7, 19, 60, 181, 520, 1488, 4287, 12343, 35404, 101407, 290579,

    832970]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.829268293, 9.082644628,

    13.17746479, 18.16456922, 24.08506363, 30.96986269, 38.83987193,

    47.70733638, 57.57804195, 68.45368261, 80.33383313]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.346792715, 3.174242350,

    4.103526382, 5.115512089, 6.185955015, 7.289224216, 8.401352999,

    9.502302839, 10.57715198, 11.61619621, 12.61425784]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.50744, 10.0758, 16.8389, 26.1685,

    38.2660, 53.1328, 70.5827, 90.2938, 111.876, 134.936, 159.120]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4739251896,

    -0.5623704924, -0.6555858067, -0.7525321878, -0.8523915691, -0.9541103254,

    -1.056144209, -1.156561649, -1.253337310, -1.344584807, -1.428762988]

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

    2.598851773, 2.758157854, 2.922677583, 3.114967769, 3.345403474,

    3.619007522, 3.936998123, 4.296765306, 4.692351929, 5.115048948,

    5.554565002]

                                end of this data



For the equivalence class of patterns, {{[2, 3, 1], [1, 3, 2, 4, 5]},

    {[1, 3, 2], [5, 3, 4, 2, 1]}, {[2, 1, 3], [5, 3, 4, 2, 1]},

    {[1, 3, 2], [5, 4, 2, 3, 1]}, {[3, 1, 2], [1, 2, 4, 3, 5]},

    {[3, 1, 2], [1, 3, 2, 4, 5]}, {[2, 1, 3], [5, 4, 2, 3, 1]},

    {[2, 3, 1], [1, 2, 4, 3, 5]}}

      the member , {[1, 3, 2], [5, 3, 4, 2, 1]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

    [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {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+3*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+15*q^9+14*q^10+15*q^11+12*q^12+6*q^13+3*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+21*q^9+26*q^10+28*q^11+32*q^12+31*q^13+34*q^14+30*q^
15+27*q^16+22*q^17+12*q^18+6*q^19+3*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+25*q^9+32*q^10+38*q^11+46*q^12+51*q^13+57*q^14+61*q^15+68*q^16
+70*q^17+70*q^18+69*q^19+64*q^20+57*q^21+49*q^22+36*q^23+22*q^24+12*q^25+6*q^26
+3*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+27*q^9+36*q^10+
44*q^11+56*q^12+63*q^13+78*q^14+86*q^15+99*q^16+108*q^17+121*q^18+128*q^19+139*
q^20+142*q^21+147*q^22+151*q^23+149*q^24+141*q^25+130*q^26+119*q^27+98*q^28+82*
q^29+60*q^30+36*q^31+22*q^32+12*q^33+6*q^34+3*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+29*q^9+38*q^10+48*q^11+62*q^12+73*q^13+90*q^14+105*q
^15+125*q^16+141*q^17+161*q^18+179*q^19+201*q^20+218*q^21+240*q^22+256*q^23+274
*q^24+285*q^25+296*q^26+309*q^27+314*q^28+317*q^29+308*q^30+300*q^31+287*q^32+
260*q^33+232*q^34+202*q^35+164*q^36+129*q^37+95*q^38+60*q^39+36*q^40+22*q^41+12
*q^42+6*q^43+3*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+29*q
^9+40*q^10+50*q^11+66*q^12+79*q^13+100*q^14+117*q^15+144*q^16+165*q^17+195*q^18
+221*q^19+254*q^20+282*q^21+321*q^22+350*q^23+388*q^24+424*q^25+456*q^26+488*q^
27+522*q^28+552*q^29+578*q^30+608*q^31+625*q^32+639*q^33+655*q^34+655*q^35+654*
q^36+651*q^37+622*q^38+595*q^39+557*q^40+507*q^41+451*q^42+389*q^43+326*q^44+
258*q^45+201*q^46+144*q^47+95*q^48+60*q^49+36*q^50+22*q^51+12*q^52+6*q^53+3*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+29*q^9+40*q^10+52*q^
11+68*q^12+83*q^13+106*q^14+127*q^15+156*q^16+184*q^17+219*q^18+253*q^19+297*q^
20+337*q^21+387*q^22+433*q^23+489*q^24+541*q^25+602*q^26+661*q^27+721*q^28+784*
q^29+845*q^30+906*q^31+969*q^32+1024*q^33+1085*q^34+1141*q^35+1191*q^36+1238*q^
37+1272*q^38+1315*q^39+1338*q^40+1359*q^41+1371*q^42+1364*q^43+1352*q^44+1323*q
^45+1280*q^46+1227*q^47+1151*q^48+1057*q^49+963*q^50+854*q^51+733*q^52+618*q^53
+505*q^54+393*q^55+301*q^56+218*q^57+144*q^58+95*q^59+60*q^60+36*q^61+22*q^62+
12*q^63+6*q^64+3*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+29
*q^9+40*q^10+52*q^11+70*q^12+85*q^13+110*q^14+133*q^15+166*q^16+196*q^17+238*q^
18+277*q^19+329*q^20+378*q^21+443*q^22+501*q^23+574*q^24+644*q^25+726*q^26+812*
q^27+903*q^28+994*q^29+1094*q^30+1198*q^31+1304*q^32+1409*q^33+1519*q^34+1629*q
^35+1741*q^36+1859*q^37+1963*q^38+2077*q^39+2181*q^40+2278*q^41+2377*q^42+2470*
q^43+2551*q^44+2625*q^45+2703*q^46+2755*q^47+2795*q^48+2819*q^49+2831*q^50+2833
*q^51+2807*q^52+2763*q^53+2685*q^54+2589*q^55+2478*q^56+2329*q^57+2168*q^58+
1980*q^59+1776*q^60+1569*q^61+1361*q^62+1142*q^63+938*q^64+756*q^65+583*q^66+
442*q^67+320*q^68+218*q^69+144*q^70+95*q^71+60*q^72+36*q^73+22*q^74+12*q^75+6*q
^76+3*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+29*q^9+40*q^
10+52*q^11+70*q^12+87*q^13+112*q^14+137*q^15+172*q^16+206*q^17+250*q^18+296*q^
19+353*q^20+410*q^21+484*q^22+555*q^23+643*q^24+731*q^25+831*q^26+938*q^27+1061
*q^28+1181*q^29+1315*q^30+1460*q^31+1611*q^32+1766*q^33+1940*q^34+2107*q^35+
2285*q^36+2478*q^37+2658*q^38+2859*q^39+3063*q^40+3254*q^41+3459*q^42+3672*q^43
+3866*q^44+4063*q^45+4269*q^46+4451*q^47+4642*q^48+4832*q^49+4987*q^50+5157*q^
51+5315*q^52+5440*q^53+5558*q^54+5674*q^55+5748*q^56+5805*q^57+5861*q^58+5857*q
^59+5834*q^60+5784*q^61+5684*q^62+5560*q^63+5402*q^64+5180*q^65+4932*q^66+4653*
q^67+4323*q^68+3979*q^69+3614*q^70+3217*q^71+2826*q^72+2445*q^73+2068*q^74+1711
*q^75+1385*q^76+1098*q^77+845*q^78+638*q^79+463*q^80+320*q^81+218*q^82+144*q^83
+95*q^84+60*q^85+36*q^86+22*q^87+12*q^88+6*q^89+3*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+29*q^9+40*q^10+52*q^11+70*q^12+87*q^13+114*q^14+
139*q^15+176*q^16+212*q^17+260*q^18+308*q^19+372*q^20+434*q^21+516*q^22+596*q^
23+697*q^24+798*q^25+919*q^26+1045*q^27+1189*q^28+1341*q^29+1509*q^30+1686*q^31
+1884*q^32+2088*q^33+2314*q^34+2548*q^35+2796*q^36+3065*q^37+3334*q^38+3628*q^
39+3926*q^40+4237*q^41+4564*q^42+4898*q^43+5239*q^44+5589*q^45+5951*q^46+6314*q
^47+6680*q^48+7049*q^49+7410*q^50+7786*q^51+8158*q^52+8515*q^53+8881*q^54+9231*
q^55+9572*q^56+9900*q^57+10217*q^58+10511*q^59+10800*q^60+11056*q^61+11305*q^62
+11518*q^63+11699*q^64+11850*q^65+11964*q^66+12040*q^67+12060*q^68+12050*q^69+
11969*q^70+11845*q^71+11661*q^72+11387*q^73+11072*q^74+10680*q^75+10224*q^76+
9715*q^77+9138*q^78+8504*q^79+7842*q^80+7146*q^81+6422*q^82+5714*q^83+4996*q^84
+4299*q^85+3652*q^86+3045*q^87+2480*q^88+1992*q^89+1567*q^90+1198*q^91+906*q^92
+661*q^93+463*q^94+320*q^95+218*q^96+144*q^97+95*q^98+60*q^99+36*q^100+22*q^101
+12*q^102+6*q^103+3*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+q^2+q^3, 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+3*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+15*q^6+14*q^5+15*q^4+12*q^3+6*q^2+3*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+21*q^12+26*q^11+28*q^10+32*q^9+31*q^8+34
*q^7+30*q^6+27*q^5+22*q^4+12*q^3+6*q^2+3*q+1, 1+3*q^25+11*q^22+5*q^24+15*q^21+7
*q^23+3*q+6*q^2+2*q^26+12*q^3+22*q^4+36*q^5+49*q^6+57*q^7+64*q^8+69*q^9+70*q^10
+70*q^11+68*q^12+61*q^13+57*q^14+51*q^15+46*q^16+q^28+38*q^17+32*q^18+25*q^19+
20*q^20+q^27, 1+44*q^25+78*q^22+56*q^24+86*q^21+63*q^23+3*q+11*q^30+15*q^29+2*q
^34+3*q^33+6*q^2+36*q^26+12*q^3+22*q^4+36*q^5+60*q^6+5*q^32+82*q^7+98*q^8+119*q
^9+130*q^10+141*q^11+149*q^12+151*q^13+147*q^14+142*q^15+139*q^16+22*q^28+128*q
^17+121*q^18+108*q^19+99*q^20+27*q^27+7*q^31+q^35+q^36, 1+201*q^25+3*q^42+256*q
^22+218*q^24+274*q^21+240*q^23+3*q+q^44+105*q^30+q^45+22*q^37+125*q^29+11*q^39+
5*q^41+48*q^34+15*q^38+62*q^33+6*q^2+179*q^26+12*q^3+22*q^4+36*q^5+60*q^6+73*q^
32+95*q^7+129*q^8+164*q^9+202*q^10+232*q^11+260*q^12+287*q^13+300*q^14+308*q^15
+317*q^16+2*q^43+141*q^28+314*q^17+309*q^18+296*q^19+285*q^20+161*q^27+7*q^40+
90*q^31+38*q^35+29*q^36, 1+578*q^25+79*q^42+639*q^22+608*q^24+655*q^21+625*q^23
+3*q+5*q^51+50*q^44+2*q^53+424*q^30+3*q^52+q^55+q^54+40*q^45+195*q^37+456*q^29+
29*q^46+144*q^39+100*q^41+282*q^34+7*q^50+165*q^38+321*q^33+6*q^2+552*q^26+12*q
^3+22*q^4+36*q^5+60*q^6+350*q^32+95*q^7+144*q^8+201*q^9+258*q^10+326*q^11+389*q
^12+451*q^13+507*q^14+557*q^15+595*q^16+66*q^43+11*q^49+488*q^28+622*q^17+651*q
^18+654*q^19+655*q^20+522*q^27+117*q^40+388*q^31+254*q^35+15*q^48+221*q^36+22*q
^47, 1+1359*q^25+489*q^42+1352*q^22+1371*q^24+1323*q^21+1364*q^23+3*q+127*q^51+
387*q^44+83*q^53+29*q^57+1191*q^30+106*q^52+22*q^58+52*q^55+11*q^60+68*q^54+337
*q^45+784*q^37+1238*q^29+297*q^46+q^65+661*q^39+7*q^61+541*q^41+969*q^34+156*q^
50+2*q^64+721*q^38+1024*q^33+40*q^56+6*q^2+1338*q^26+12*q^3+22*q^4+36*q^5+60*q^
6+1085*q^32+95*q^7+144*q^8+218*q^9+5*q^62+301*q^10+393*q^11+505*q^12+618*q^13+
733*q^14+854*q^15+963*q^16+q^66+433*q^43+184*q^49+1272*q^28+1057*q^17+1151*q^18
+1227*q^19+1280*q^20+1315*q^27+602*q^40+3*q^63+1141*q^31+906*q^35+219*q^48+15*q
^59+845*q^36+253*q^47, 1+2763*q^25+1741*q^42+2478*q^22+2685*q^24+2329*q^21+2589
*q^23+3*q+812*q^51+1519*q^44+644*q^53+378*q^57+2795*q^30+726*q^52+329*q^58+501*
q^55+238*q^60+574*q^54+1409*q^45+15*q^71+5*q^74+2278*q^37+52*q^67+2819*q^29+
1304*q^46+85*q^65+29*q^69+2077*q^39+196*q^61+1859*q^41+2551*q^34+903*q^50+110*q
^64+2181*q^38+2625*q^33+443*q^56+6*q^2+40*q^68+2807*q^26+12*q^3+22*q^4+36*q^5+
60*q^6+2703*q^32+95*q^7+144*q^8+218*q^9+166*q^62+320*q^10+442*q^11+583*q^12+2*q
^76+756*q^13+938*q^14+1142*q^15+1361*q^16+70*q^66+22*q^70+q^77+q^78+1629*q^43+
994*q^49+2831*q^28+1569*q^17+1776*q^18+1980*q^19+2168*q^20+2833*q^27+11*q^72+
1963*q^40+7*q^73+133*q^63+2755*q^31+3*q^75+2470*q^35+1094*q^48+277*q^59+2377*q^
36+1198*q^47, 1+4932*q^25+4832*q^42+3979*q^22+q^90+4653*q^24+3614*q^21+4323*q^
23+3*q+3063*q^51+4451*q^44+2658*q^53+1940*q^57+5784*q^30+2859*q^52+1766*q^58+
2285*q^55+1460*q^60+q^91+2478*q^54+4269*q^45+353*q^71+206*q^74+5558*q^37+643*q^
67+5684*q^29+4063*q^46+831*q^65+484*q^69+5315*q^39+1315*q^61+4987*q^41+5805*q^
34+70*q^79+3254*q^50+938*q^64+5440*q^38+5861*q^33+2107*q^56+6*q^2+555*q^68+5180
*q^26+12*q^3+22*q^4+36*q^5+60*q^6+5857*q^32+95*q^7+144*q^8+218*q^9+1181*q^62+
320*q^10+463*q^11+638*q^12+137*q^76+845*q^13+1098*q^14+1385*q^15+1711*q^16+731*
q^66+410*q^70+112*q^77+87*q^78+4642*q^43+3459*q^49+5560*q^28+2068*q^17+2445*q^
18+2826*q^19+3217*q^20+5402*q^27+296*q^72+5157*q^40+250*q^73+1061*q^63+5834*q^
31+172*q^75+5748*q^35+52*q^80+3672*q^48+22*q^83+40*q^81+11*q^85+7*q^86+5*q^87+3
*q^88+2*q^89+15*q^84+29*q^82+1611*q^59+5674*q^36+3866*q^47, 1+7842*q^25+11518*q
^42+5714*q^22+139*q^90+7146*q^24+4996*q^21+6422*q^23+3*q+22*q^97+8881*q^51+
11056*q^44+8158*q^53+6680*q^57+10680*q^30+52*q^94+8515*q^52+6314*q^58+7410*q^55
+5589*q^60+114*q^91+7786*q^54+10800*q^45+2314*q^71+1686*q^74+29*q^96+15*q^98+
12060*q^37+3334*q^67+10224*q^29+10511*q^46+70*q^93+11*q^99+3926*q^65+2796*q^69+
11964*q^39+5239*q^61+11699*q^41+11845*q^34+919*q^79+9231*q^50+4237*q^64+7*q^100
+12040*q^38+11661*q^33+7049*q^56+6*q^2+3065*q^68+8504*q^26+12*q^3+22*q^4+36*q^5
+60*q^6+11387*q^32+95*q^7+144*q^8+218*q^9+40*q^95+4898*q^62+5*q^101+320*q^10+
463*q^11+661*q^12+1341*q^76+906*q^13+1198*q^14+1567*q^15+1992*q^16+3628*q^66+
2548*q^70+1189*q^77+1045*q^78+11305*q^43+3*q^102+9572*q^49+9715*q^28+2480*q^17+
3045*q^18+3652*q^19+4299*q^20+9138*q^27+2088*q^72+11850*q^40+1884*q^73+2*q^103+
4564*q^63+11072*q^31+1509*q^75+11969*q^35+q^104+798*q^80+9900*q^48+q^105+516*q^
83+697*q^81+372*q^85+308*q^86+260*q^87+212*q^88+176*q^89+434*q^84+596*q^82+5951
*q^59+12050*q^36+87*q^92+10217*q^47]


 The number of permutations avoiding, {[1, 3, 2], [5, 3, 4, 2, 1]}, is given by

[1, 2, 5, 14, 41, 119, 334, 902, 2351, 5945, 14660, 35408, 84061, 196715,

    454778]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 3, 4, 2, 1]},

     is given by

 [1, 1, 3, 7, 22, 60, 171, 454, 1183, 2979, 7342, 17715, 42048, 98374, 227413]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 3, 4, 2, 1]},

     is given by

 [0, 1, 2, 7, 19, 59, 163, 448, 1168, 2966, 7318, 17693, 42013, 98341, 227365]

            For the reverse patterns and complement patterns, we get

EVEN:, [

    1, 1, 2, 7, 22, 59, 163, 454, 1183, 2966, 7318, 17715, 42048, 98341, 227365

    ]

 ODD:, [

    0, 1, 3, 7, 19, 60, 171, 448, 1168, 2979, 7342, 17693, 42013, 98374, 227413

    ]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.731707317, 8.663865546,

    12.14670659, 16.20399113, 20.86431306, 26.15138772, 32.08240109,

    38.66891663, 45.91828553, 53.83492362, 62.42130226]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.295535330, 3.043873336,

    3.912138096, 4.910559873, 6.032089004, 7.264390039, 8.595528755,

    10.01580556, 11.51799585, 13.09703248, 14.74954422]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.26948, 9.26516, 15.3048, 24.1136,

    36.3861, 52.7714, 73.8831, 100.316, 132.664, 171.532, 217.549]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4442359433,

    -0.4301685913, -0.3954324870, -0.3716639132, -0.3637573904, -0.3670645731,

    -0.3764233317, -0.3882880611, -0.4006015389, -0.4123213401, -0.4229998485]

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

    2.664143857, 2.782666866, 2.785645608, 2.757961452, 2.739955989,

    2.738786779, 2.750465443, 2.769528921, 2.791723551, 2.814266428,

    2.835514424]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [3, 4, 5, 1, 2]},

    {[1, 3, 2], [4, 5, 1, 2, 3]}, {[1, 3, 2], [3, 4, 5, 1, 2]},

    {[3, 1, 2], [3, 2, 1, 5, 4]}, {[2, 1, 3], [4, 5, 1, 2, 3]},

    {[2, 3, 1], [2, 1, 5, 4, 3]}, {[2, 3, 1], [3, 2, 1, 5, 4]},

    {[3, 1, 2], [2, 1, 5, 4, 3]}}

      the member , {[2, 1, 3], [3, 4, 5, 1, 2]}, has a scheme of depth , 5

                                  here it is:

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

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

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

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

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

    [[1, 3, 4, 2], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {}], [[1, 4, 5, 3, 2],

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

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

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

    [[1, 4, 5, 2, 3], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2, 3}],

    [[1, 3, 5, 2, 4], {[0, 0, 0, 0, 0, 0]}, {4}],

    [[1, 3, 4, 2, 5], {[0, 0, 0, 0, 0, 0]}, {4}], [[1, 2, 3], {}, {}],

    [[1, 2, 3, 4], {}, {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+6*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+8*q^6+11*q^7+13
*q^8+14*q^9+13*q^10+14*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+10*q^6+13*q^7+17*q^8+20*q^9+24*q^10+28*q^11+31*q^12+34*q^13+33*q^14+31*q^
15+29*q^16+29*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+10*
q^6+15*q^7+19*q^8+24*q^9+30*q^10+38*q^11+46*q^12+54*q^13+60*q^14+66*q^15+72*q^
16+79*q^17+83*q^18+82*q^19+78*q^20+70*q^21+64*q^22+60*q^23+55*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+10*q^6+15*q^7+21*q^8+26*q^9+34*q^
10+44*q^11+56*q^12+68*q^13+81*q^14+96*q^15+111*q^16+128*q^17+145*q^18+161*q^19+
173*q^20+182*q^21+189*q^22+198*q^23+205*q^24+205*q^25+195*q^26+178*q^27+159*q^
28+140*q^29+128*q^30+117*q^31+97*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+10*q^6+15*q^7+21*q^8+28*q^9+36*q^10+48*q^11+62*q^12+78*q^13+95*
q^14+116*q^15+142*q^16+170*q^17+198*q^18+230*q^19+263*q^20+296*q^21+329*q^22+
365*q^23+402*q^24+433*q^25+459*q^26+479*q^27+492*q^28+498*q^29+503*q^30+507*q^
31+505*q^32+486*q^33+450*q^34+404*q^35+355*q^36+310*q^37+274*q^38+249*q^39+216*
q^40+161*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+10*q^6+
15*q^7+21*q^8+28*q^9+38*q^10+50*q^11+66*q^12+84*q^13+105*q^14+130*q^15+162*q^16
+200*q^17+241*q^18+286*q^19+336*q^20+393*q^21+455*q^22+523*q^23+595*q^24+671*q^
25+749*q^26+825*q^27+897*q^28+970*q^29+1043*q^30+1115*q^31+1181*q^32+1236*q^33+
1275*q^34+1295*q^35+1301*q^36+1295*q^37+1282*q^38+1265*q^39+1241*q^40+1196*q^41
+1122*q^42+1020*q^43+905*q^44+790*q^45+683*q^46+595*q^47+529*q^48+469*q^49+379*
q^50+254*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+10*q^6
+15*q^7+21*q^8+28*q^9+38*q^10+52*q^11+68*q^12+88*q^13+111*q^14+140*q^15+176*q^
16+220*q^17+271*q^18+328*q^19+393*q^20+469*q^21+556*q^22+656*q^23+765*q^24+883*
q^25+1012*q^26+1154*q^27+1304*q^28+1459*q^29+1624*q^30+1800*q^31+1983*q^32+2164
*q^33+2341*q^34+2512*q^35+2675*q^36+2827*q^37+2969*q^38+3103*q^39+3228*q^40+
3331*q^41+3403*q^42+3438*q^43+3435*q^44+3396*q^45+3327*q^46+3244*q^47+3156*q^48
+3059*q^49+2933*q^50+2761*q^51+2539*q^52+2282*q^53+2010*q^54+1746*q^55+1504*q^
56+1296*q^57+1135*q^58+1004*q^59+852*q^60+635*q^61+384*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+10*q^6+15*q^7+21*q^8+28*q^9+38*q^10+52*
q^11+70*q^12+90*q^13+115*q^14+146*q^15+186*q^16+234*q^17+291*q^18+358*q^19+435*
q^20+525*q^21+633*q^22+760*q^23+902*q^24+1060*q^25+1236*q^26+1436*q^27+1659*q^
28+1904*q^29+2172*q^30+2464*q^31+2779*q^32+3115*q^33+3470*q^34+3844*q^35+4228*q
^36+4621*q^37+5030*q^38+5456*q^39+5884*q^40+6303*q^41+6714*q^42+7111*q^43+7481*
q^44+7812*q^45+8108*q^46+8376*q^47+8618*q^48+8826*q^49+8987*q^50+9087*q^51+9117
*q^52+9067*q^53+8938*q^54+8735*q^55+8469*q^56+8162*q^57+7842*q^58+7523*q^59+
7177*q^60+6760*q^61+6251*q^62+5670*q^63+5052*q^64+4433*q^65+3841*q^66+3301*q^67
+2831*q^68+2449*q^69+2150*q^70+1862*q^71+1491*q^72+1021*q^73+560*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+10*q^6+15*q^7+21*q^8+28*q^9+
38*q^10+52*q^11+70*q^12+92*q^13+117*q^14+150*q^15+192*q^16+244*q^17+305*q^18+
378*q^19+465*q^20+567*q^21+689*q^22+836*q^23+1007*q^24+1200*q^25+1417*q^26+1667
*q^27+1953*q^28+2278*q^29+2643*q^30+3051*q^31+3501*q^32+3992*q^33+4531*q^34+
5124*q^35+5765*q^36+6446*q^37+7175*q^38+7959*q^39+8795*q^40+9670*q^41+10579*q^
42+11520*q^43+12484*q^44+13459*q^45+14442*q^46+15436*q^47+16438*q^48+17428*q^49
+18395*q^50+19334*q^51+20227*q^52+21044*q^53+21772*q^54+22412*q^55+22954*q^56+
23393*q^57+23742*q^58+24018*q^59+24204*q^60+24261*q^61+24161*q^62+23898*q^63+
23473*q^64+22893*q^65+22169*q^66+21325*q^67+20393*q^68+19422*q^69+18458*q^70+
17494*q^71+16462*q^72+15285*q^73+13955*q^74+12534*q^75+11103*q^76+9717*q^77+
8415*q^78+7227*q^79+6183*q^80+5311*q^81+4617*q^82+4023*q^83+3359*q^84+2516*q^85
+1583*q^86+792*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+
10*q^6+15*q^7+21*q^8+28*q^9+38*q^10+52*q^11+70*q^12+92*q^13+119*q^14+152*q^15+
196*q^16+250*q^17+315*q^18+392*q^19+485*q^20+597*q^21+731*q^22+892*q^23+1083*q^
24+1304*q^25+1558*q^26+1851*q^27+2188*q^28+2579*q^29+3029*q^30+3541*q^31+4114*q
^32+4753*q^33+5467*q^34+6265*q^35+7153*q^36+8133*q^37+9207*q^38+10380*q^39+
11655*q^40+13037*q^41+14527*q^42+16127*q^43+17832*q^44+19634*q^45+21528*q^46+
23521*q^47+25622*q^48+27820*q^49+30088*q^50+32409*q^51+34784*q^52+37201*q^53+
39634*q^54+42054*q^55+44445*q^56+46800*q^57+49114*q^58+51376*q^59+53563*q^60+
55639*q^61+57567*q^62+59313*q^63+60859*q^64+62185*q^65+63260*q^66+64060*q^67+
64597*q^68+64906*q^69+65005*q^70+64887*q^71+64521*q^72+63864*q^73+62870*q^74+
61522*q^75+59840*q^76+57860*q^77+55617*q^78+53157*q^79+50540*q^80+47850*q^81+
45172*q^82+42544*q^83+39903*q^84+37110*q^85+34071*q^86+30820*q^87+27500*q^88+
24262*q^89+21202*q^90+18364*q^91+15783*q^92+13495*q^93+11545*q^94+9959*q^95+
8658*q^96+7393*q^97+5881*q^98+4103*q^99+2377*q^100+1091*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+q^2+q^3, 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+6*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+8*q^9
+11*q^8+13*q^7+14*q^6+13*q^5+14*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+10*q^15+13*q^14+17*q^13+20*q^12+24*q^11+28*q^10+31*q^9+34*q^8+
33*q^7+31*q^6+29*q^5+29*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+10*q^22+5*q^24+15*q^
21+7*q^23+7*q+21*q^2+2*q^26+41*q^3+55*q^4+60*q^5+64*q^6+70*q^7+78*q^8+82*q^9+83
*q^10+79*q^11+72*q^12+66*q^13+60*q^14+54*q^15+46*q^16+q^28+38*q^17+30*q^18+24*q
^19+19*q^20+q^27, 1+44*q^25+81*q^22+56*q^24+96*q^21+68*q^23+8*q+10*q^30+15*q^29
+2*q^34+3*q^33+28*q^2+34*q^26+63*q^3+97*q^4+117*q^5+128*q^6+5*q^32+140*q^7+159*
q^8+178*q^9+195*q^10+205*q^11+205*q^12+198*q^13+189*q^14+182*q^15+173*q^16+21*q
^28+161*q^17+145*q^18+128*q^19+111*q^20+26*q^27+7*q^31+q^35+q^36, 1+263*q^25+3*
q^42+365*q^22+296*q^24+402*q^21+329*q^23+9*q+q^44+116*q^30+q^45+21*q^37+142*q^
29+10*q^39+5*q^41+48*q^34+15*q^38+62*q^33+36*q^2+230*q^26+92*q^3+161*q^4+216*q^
5+249*q^6+78*q^32+274*q^7+310*q^8+355*q^9+404*q^10+450*q^11+486*q^12+505*q^13+
507*q^14+503*q^15+498*q^16+2*q^43+170*q^28+492*q^17+479*q^18+459*q^19+433*q^20+
198*q^27+7*q^40+95*q^31+36*q^35+28*q^36, 1+1043*q^25+84*q^42+1236*q^22+1115*q^
24+1275*q^21+1181*q^23+10*q+5*q^51+50*q^44+2*q^53+671*q^30+3*q^52+q^55+q^54+38*
q^45+241*q^37+749*q^29+28*q^46+162*q^39+105*q^41+393*q^34+7*q^50+200*q^38+455*q
^33+45*q^2+970*q^26+129*q^3+254*q^4+379*q^5+469*q^6+523*q^32+529*q^7+595*q^8+
683*q^9+790*q^10+905*q^11+1020*q^12+1122*q^13+1196*q^14+1241*q^15+1265*q^16+66*
q^43+10*q^49+825*q^28+1282*q^17+1295*q^18+1301*q^19+1295*q^20+897*q^27+130*q^40
+595*q^31+336*q^35+15*q^48+286*q^36+21*q^47, 1+3331*q^25+765*q^42+3435*q^22+
3403*q^24+3396*q^21+3438*q^23+11*q+140*q^51+556*q^44+88*q^53+28*q^57+2675*q^30+
111*q^52+21*q^58+52*q^55+10*q^60+68*q^54+469*q^45+1459*q^37+2827*q^29+393*q^46+
q^65+1154*q^39+7*q^61+883*q^41+1983*q^34+176*q^50+2*q^64+1304*q^38+2164*q^33+38
*q^56+55*q^2+3228*q^26+175*q^3+384*q^4+635*q^5+852*q^6+2341*q^32+1004*q^7+1135*
q^8+1296*q^9+5*q^62+1504*q^10+1746*q^11+2010*q^12+2282*q^13+2539*q^14+2761*q^15
+2933*q^16+q^66+656*q^43+220*q^49+2969*q^28+3059*q^17+3156*q^18+3244*q^19+3327*
q^20+3103*q^27+1012*q^40+3*q^63+2512*q^31+1800*q^35+271*q^48+15*q^59+1624*q^36+
328*q^47, 1+9067*q^25+4228*q^42+8469*q^22+8938*q^24+8162*q^21+8735*q^23+12*q+
1436*q^51+3470*q^44+1060*q^53+525*q^57+8618*q^30+1236*q^52+435*q^58+760*q^55+
291*q^60+902*q^54+3115*q^45+15*q^71+5*q^74+6303*q^37+52*q^67+8826*q^29+2779*q^
46+90*q^65+28*q^69+5456*q^39+234*q^61+4621*q^41+7481*q^34+1659*q^50+115*q^64+
5884*q^38+7812*q^33+633*q^56+66*q^2+38*q^68+9117*q^26+231*q^3+560*q^4+1021*q^5+
1491*q^6+8108*q^32+1862*q^7+2150*q^8+2449*q^9+186*q^62+2831*q^10+3301*q^11+3841
*q^12+2*q^76+4433*q^13+5052*q^14+5670*q^15+6251*q^16+70*q^66+21*q^70+q^77+q^78+
3844*q^43+1904*q^49+8987*q^28+6760*q^17+7177*q^18+7523*q^19+7842*q^20+9087*q^27
+10*q^72+5030*q^40+7*q^73+146*q^63+8376*q^31+3*q^75+7111*q^35+2172*q^48+358*q^
59+6714*q^36+2464*q^47, 1+22169*q^25+17428*q^42+19422*q^22+q^90+21325*q^24+
18458*q^21+20393*q^23+13*q+8795*q^51+15436*q^44+7175*q^53+4531*q^57+24261*q^30+
7959*q^52+3992*q^58+5765*q^55+3051*q^60+q^91+6446*q^54+14442*q^45+465*q^71+244*
q^74+21772*q^37+1007*q^67+24161*q^29+13459*q^46+1417*q^65+689*q^69+20227*q^39+
2643*q^61+18395*q^41+23393*q^34+70*q^79+9670*q^50+1667*q^64+21044*q^38+23742*q^
33+5124*q^56+78*q^2+836*q^68+22893*q^26+298*q^3+792*q^4+1583*q^5+2516*q^6+24018
*q^32+3359*q^7+4023*q^8+4617*q^9+2278*q^62+5311*q^10+6183*q^11+7227*q^12+150*q^
76+8415*q^13+9717*q^14+11103*q^15+12534*q^16+1200*q^66+567*q^70+117*q^77+92*q^
78+16438*q^43+10579*q^49+23898*q^28+13955*q^17+15285*q^18+16462*q^19+17494*q^20
+23473*q^27+378*q^72+19334*q^40+305*q^73+1953*q^63+24204*q^31+192*q^75+22954*q^
35+52*q^80+11520*q^48+21*q^83+38*q^81+10*q^85+7*q^86+5*q^87+3*q^88+2*q^89+15*q^
84+28*q^82+3501*q^59+22412*q^36+12484*q^47, 1+50540*q^25+59313*q^42+42544*q^22+
152*q^90+47850*q^24+39903*q^21+45172*q^23+14*q+21*q^97+39634*q^51+55639*q^44+
34784*q^53+25622*q^57+61522*q^30+52*q^94+37201*q^52+23521*q^58+30088*q^55+19634
*q^60+119*q^91+32409*q^54+53563*q^45+5467*q^71+3541*q^74+28*q^96+15*q^98+64597*
q^37+9207*q^67+59840*q^29+51376*q^46+70*q^93+10*q^99+11655*q^65+7153*q^69+63260
*q^39+17832*q^61+60859*q^41+64887*q^34+1558*q^79+42054*q^50+13037*q^64+7*q^100+
64060*q^38+64521*q^33+27820*q^56+91*q^2+8133*q^68+53157*q^26+377*q^3+1091*q^4+
2377*q^5+4103*q^6+63864*q^32+5881*q^7+7393*q^8+8658*q^9+38*q^95+16127*q^62+5*q^
101+9959*q^10+11545*q^11+13495*q^12+2579*q^76+15783*q^13+18364*q^14+21202*q^15+
24262*q^16+10380*q^66+6265*q^70+2188*q^77+1851*q^78+57567*q^43+3*q^102+44445*q^
49+57860*q^28+27500*q^17+30820*q^18+34071*q^19+37110*q^20+55617*q^27+4753*q^72+
62185*q^40+4114*q^73+2*q^103+14527*q^63+62870*q^31+3029*q^75+65005*q^35+q^104+
1304*q^80+46800*q^48+q^105+731*q^83+1083*q^81+485*q^85+392*q^86+315*q^87+250*q^
88+196*q^89+597*q^84+892*q^82+21528*q^59+64906*q^36+92*q^92+49114*q^47]


 The number of permutations avoiding, {[2, 1, 3], [3, 4, 5, 1, 2]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [3, 4, 5, 1, 2]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[2, 1, 3], [3, 4, 5, 1, 2]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.804878049, 8.942622951,

    12.76164384, 17.25502742, 22.41877476, 28.25086365, 34.75034716,

    41.91680403, 49.75005363, 58.25002070, 67.41667458]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.350085750, 3.178563394,

    4.106214139, 5.122971114, 6.220685069, 7.393088280, 8.635297295,

    9.943381804, 11.31407232, 12.74457502, 14.23245435]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.52290, 10.1033, 16.8610, 26.2448,

    38.6969, 54.6578, 74.5684, 98.8708, 128.008, 162.424, 202.563]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4418810426,

    -0.4514630475, -0.4392958661, -0.4191228611, -0.3972106194, -0.3760992818,

    -0.3566372058, -0.3390180465, -0.3231596246, -0.3088940589, -0.2960318310]

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

    2.565707530, 2.651712655, 2.700946323, 2.734057250, 2.758930052,

    2.778878390, 2.795513025, 2.809716370, 2.822028828, 2.832831415,

    2.842382937]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [5, 1, 4, 3, 2]},

    {[1, 3, 2], [5, 3, 2, 1, 4]}, {[1, 3, 2], [4, 3, 2, 5, 1]},

    {[2, 3, 1], [4, 1, 2, 3, 5]}, {[3, 1, 2], [2, 3, 4, 1, 5]},

    {[2, 3, 1], [1, 5, 2, 3, 4]}, {[3, 1, 2], [1, 3, 4, 5, 2]},

    {[2, 1, 3], [2, 5, 4, 3, 1]}}

      the member , {[1, 3, 2], [5, 3, 2, 1, 4]}, has a scheme of depth , 4

                                  here it is:

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

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

    [[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}],

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

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

    [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {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+6*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+10*q^7+13
*q^8+14*q^9+13*q^10+15*q^11+12*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+12*q^7+17*q^8+19*q^9+23*q^10+25*q^11+30*q^12+31*q^13+34*q^14+31*q^
15+31*q^16+29*q^17+22*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+14*q^7+19*q^8+23*q^9+28*q^10+33*q^11+41*q^12+47*q^13+51*q^14+59*q^15+66*q^
16+72*q^17+75*q^18+80*q^19+77*q^20+77*q^21+68*q^22+65*q^23+53*q^24+37*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+14*q^7+21*q^8+25*q^9+32*q^
10+38*q^11+49*q^12+56*q^13+68*q^14+76*q^15+92*q^16+103*q^17+117*q^18+129*q^19+
145*q^20+154*q^21+166*q^22+183*q^23+194*q^24+191*q^25+193*q^26+186*q^27+176*q^
28+163*q^29+144*q^30+125*q^31+92*q^32+58*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+14*q^7+21*q^8+27*q^9+34*q^10+42*q^11+54*q^12+64*q^13+77*
q^14+91*q^15+110*q^16+129*q^17+148*q^18+170*q^19+193*q^20+221*q^21+243*q^22+277
*q^23+307*q^24+333*q^25+362*q^26+389*q^27+416*q^28+446*q^29+461*q^30+484*q^31+
491*q^32+485*q^33+458*q^34+444*q^35+411*q^36+383*q^37+336*q^38+287*q^39+226*q^
40+152*q^41+86*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+14*
q^7+21*q^8+27*q^9+36*q^10+44*q^11+58*q^12+69*q^13+85*q^14+100*q^15+125*q^16+145
*q^17+175*q^18+201*q^19+234*q^20+270*q^21+311*q^22+353*q^23+405*q^24+450*q^25+
506*q^26+558*q^27+621*q^28+683*q^29+753*q^30+821*q^31+884*q^32+946*q^33+1004*q^
34+1059*q^35+1121*q^36+1169*q^37+1210*q^38+1230*q^39+1232*q^40+1218*q^41+1184*q
^42+1113*q^43+1044*q^44+980*q^45+890*q^46+796*q^47+670*q^48+540*q^49+389*q^50+
240*q^51+122*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+14*q
^7+21*q^8+27*q^9+36*q^10+46*q^11+60*q^12+73*q^13+90*q^14+108*q^15+134*q^16+160*
q^17+191*q^18+226*q^19+266*q^20+311*q^21+360*q^22+422*q^23+484*q^24+549*q^25+
625*q^26+708*q^27+797*q^28+900*q^29+1003*q^30+1119*q^31+1239*q^32+1361*q^33+
1484*q^34+1628*q^35+1774*q^36+1922*q^37+2073*q^38+2229*q^39+2374*q^40+2527*q^41
+2653*q^42+2763*q^43+2879*q^44+2990*q^45+3066*q^46+3146*q^47+3156*q^48+3158*q^
49+3085*q^50+2997*q^51+2851*q^52+2699*q^53+2489*q^54+2315*q^55+2104*q^56+1867*q
^57+1594*q^58+1284*q^59+967*q^60+642*q^61+364*q^62+167*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+14*q^7+21*q^8+27*q^9+36*q^10+46*q^11+62*
q^12+75*q^13+94*q^14+113*q^15+142*q^16+169*q^17+206*q^18+242*q^19+291*q^20+341*
q^21+402*q^22+471*q^23+553*q^24+629*q^25+727*q^26+830*q^27+951*q^28+1080*q^29+
1227*q^30+1383*q^31+1556*q^32+1732*q^33+1924*q^34+2146*q^35+2383*q^36+2618*q^37
+2889*q^38+3162*q^39+3453*q^40+3757*q^41+4067*q^42+4373*q^43+4716*q^44+5056*q^
45+5408*q^46+5777*q^47+6123*q^48+6452*q^49+6770*q^50+7060*q^51+7321*q^52+7561*q
^53+7751*q^54+7920*q^55+8072*q^56+8148*q^57+8164*q^58+8099*q^59+7918*q^60+7658*
q^61+7299*q^62+6904*q^63+6457*q^64+5993*q^65+5475*q^66+4990*q^67+4407*q^68+3782
*q^69+3084*q^70+2363*q^71+1660*q^72+1021*q^73+533*q^74+222*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+14*q^7+21*q^8+27*q^9+36*q^10+46*q^11+
62*q^12+77*q^13+96*q^14+117*q^15+147*q^16+177*q^17+215*q^18+257*q^19+307*q^20+
366*q^21+432*q^22+511*q^23+603*q^24+698*q^25+807*q^26+933*q^27+1076*q^28+1237*q
^29+1413*q^30+1613*q^31+1825*q^32+2063*q^33+2314*q^34+2606*q^35+2930*q^36+3270*
q^37+3634*q^38+4033*q^39+4464*q^40+4928*q^41+5415*q^42+5926*q^43+6486*q^44+7083
*q^45+7695*q^46+8368*q^47+9066*q^48+9775*q^49+10494*q^50+11252*q^51+12007*q^52+
12800*q^53+13591*q^54+14409*q^55+15235*q^56+16074*q^57+16861*q^58+17634*q^59+
18328*q^60+18951*q^61+19467*q^62+19993*q^63+20395*q^64+20783*q^65+21008*q^66+
21189*q^67+21209*q^68+21163*q^69+20840*q^70+20404*q^71+19665*q^72+18783*q^73+
17735*q^74+16662*q^75+15495*q^76+14339*q^77+13092*q^78+11811*q^79+10471*q^80+
8979*q^81+7409*q^82+5769*q^83+4186*q^84+2747*q^85+1571*q^86+757*q^87+288*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+14*q^7+21*q^8+27*q^9+
36*q^10+46*q^11+62*q^12+77*q^13+98*q^14+119*q^15+151*q^16+182*q^17+223*q^18+266
*q^19+322*q^20+382*q^21+457*q^22+541*q^23+643*q^24+746*q^25+877*q^26+1013*q^27+
1179*q^28+1363*q^29+1573*q^30+1802*q^31+2061*q^32+2340*q^33+2652*q^34+3008*q^35
+3409*q^36+3837*q^37+4317*q^38+4823*q^39+5385*q^40+6003*q^41+6672*q^42+7373*q^
43+8166*q^44+9009*q^45+9907*q^46+10890*q^47+11932*q^48+13034*q^49+14215*q^50+
15437*q^51+16724*q^52+18102*q^53+19554*q^54+21069*q^55+22684*q^56+24344*q^57+
26072*q^58+27844*q^59+29636*q^60+31461*q^61+33312*q^62+35177*q^63+37070*q^64+
39012*q^65+40911*q^66+42826*q^67+44709*q^68+46533*q^69+48220*q^70+49792*q^71+
51126*q^72+52304*q^73+53298*q^74+54121*q^75+54772*q^76+55355*q^77+55613*q^78+
55687*q^79+55429*q^80+54871*q^81+53905*q^82+52542*q^83+50649*q^84+48431*q^85+
45769*q^86+43014*q^87+40128*q^88+37281*q^89+34299*q^90+31322*q^91+28127*q^92+
24882*q^93+21414*q^94+17770*q^95+14063*q^96+10444*q^97+7162*q^98+4401*q^99+2347
*q^100+1047*q^101+366*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+q^2+q^3, 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+6*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
+10*q^8+13*q^7+14*q^6+13*q^5+15*q^4+12*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+12*q^14+17*q^13+19*q^12+23*q^11+25*q^10+30*q^9+31*q^8+
34*q^7+31*q^6+31*q^5+29*q^4+22*q^3+15*q^2+6*q+1, 1+3*q^25+11*q^22+5*q^24+14*q^
21+7*q^23+7*q+21*q^2+2*q^26+37*q^3+53*q^4+65*q^5+68*q^6+77*q^7+77*q^8+80*q^9+75
*q^10+72*q^11+66*q^12+59*q^13+51*q^14+47*q^15+41*q^16+q^28+33*q^17+28*q^18+23*q
^19+19*q^20+q^27, 1+38*q^25+68*q^22+49*q^24+76*q^21+56*q^23+8*q+11*q^30+14*q^29
+2*q^34+3*q^33+28*q^2+32*q^26+58*q^3+92*q^4+125*q^5+144*q^6+5*q^32+163*q^7+176*
q^8+186*q^9+193*q^10+191*q^11+194*q^12+183*q^13+166*q^14+154*q^15+145*q^16+21*q
^28+129*q^17+117*q^18+103*q^19+92*q^20+25*q^27+7*q^31+q^35+q^36, 1+193*q^25+3*q
^42+277*q^22+221*q^24+307*q^21+243*q^23+9*q+q^44+91*q^30+q^45+21*q^37+110*q^29+
11*q^39+5*q^41+42*q^34+14*q^38+54*q^33+36*q^2+170*q^26+86*q^3+152*q^4+226*q^5+
287*q^6+64*q^32+336*q^7+383*q^8+411*q^9+444*q^10+458*q^11+485*q^12+491*q^13+484
*q^14+461*q^15+446*q^16+2*q^43+129*q^28+416*q^17+389*q^18+362*q^19+333*q^20+148
*q^27+7*q^40+77*q^31+34*q^35+27*q^36, 1+753*q^25+69*q^42+946*q^22+821*q^24+1004
*q^21+884*q^23+10*q+5*q^51+44*q^44+2*q^53+450*q^30+3*q^52+q^55+q^54+36*q^45+175
*q^37+506*q^29+27*q^46+125*q^39+85*q^41+270*q^34+7*q^50+145*q^38+311*q^33+45*q^
2+683*q^26+122*q^3+240*q^4+389*q^5+540*q^6+353*q^32+670*q^7+796*q^8+890*q^9+980
*q^10+1044*q^11+1113*q^12+1184*q^13+1218*q^14+1232*q^15+1230*q^16+58*q^43+11*q^
49+558*q^28+1210*q^17+1169*q^18+1121*q^19+1059*q^20+621*q^27+100*q^40+405*q^31+
234*q^35+14*q^48+201*q^36+21*q^47, 1+2527*q^25+484*q^42+2879*q^22+2653*q^24+
2990*q^21+2763*q^23+11*q+108*q^51+360*q^44+73*q^53+27*q^57+1774*q^30+90*q^52+21
*q^58+46*q^55+11*q^60+60*q^54+311*q^45+900*q^37+1922*q^29+266*q^46+q^65+708*q^
39+7*q^61+549*q^41+1239*q^34+134*q^50+2*q^64+797*q^38+1361*q^33+36*q^56+55*q^2+
2374*q^26+167*q^3+364*q^4+642*q^5+967*q^6+1484*q^32+1284*q^7+1594*q^8+1867*q^9+
5*q^62+2104*q^10+2315*q^11+2489*q^12+2699*q^13+2851*q^14+2997*q^15+3085*q^16+q^
66+422*q^43+160*q^49+2073*q^28+3158*q^17+3156*q^18+3146*q^19+3066*q^20+2229*q^
27+625*q^40+3*q^63+1628*q^31+1119*q^35+191*q^48+14*q^59+1003*q^36+226*q^47, 1+
7561*q^25+2383*q^42+8072*q^22+7751*q^24+8148*q^21+7920*q^23+12*q+830*q^51+1924*
q^44+629*q^53+341*q^57+6123*q^30+727*q^52+291*q^58+471*q^55+206*q^60+553*q^54+
1732*q^45+14*q^71+5*q^74+3757*q^37+46*q^67+6452*q^29+1556*q^46+75*q^65+27*q^69+
3162*q^39+169*q^61+2618*q^41+4716*q^34+951*q^50+94*q^64+3453*q^38+5056*q^33+402
*q^56+66*q^2+36*q^68+7321*q^26+222*q^3+533*q^4+1021*q^5+1660*q^6+5408*q^32+2363
*q^7+3084*q^8+3782*q^9+142*q^62+4407*q^10+4990*q^11+5475*q^12+2*q^76+5993*q^13+
6457*q^14+6904*q^15+7299*q^16+62*q^66+21*q^70+q^77+q^78+2146*q^43+1080*q^49+
6770*q^28+7658*q^17+7918*q^18+8099*q^19+8164*q^20+7060*q^27+11*q^72+2889*q^40+7
*q^73+113*q^63+5777*q^31+3*q^75+4373*q^35+1227*q^48+242*q^59+4067*q^36+1383*q^
47, 1+21008*q^25+9775*q^42+21163*q^22+q^90+21189*q^24+20840*q^21+21209*q^23+13*
q+4464*q^51+8368*q^44+3634*q^53+2314*q^57+18951*q^30+4033*q^52+2063*q^58+2930*q
^55+1613*q^60+q^91+3270*q^54+7695*q^45+307*q^71+177*q^74+13591*q^37+603*q^67+
19467*q^29+7083*q^46+807*q^65+432*q^69+12007*q^39+1413*q^61+10494*q^41+16074*q^
34+62*q^79+4928*q^50+933*q^64+12800*q^38+16861*q^33+2606*q^56+78*q^2+511*q^68+
20783*q^26+288*q^3+757*q^4+1571*q^5+2747*q^6+17634*q^32+4186*q^7+5769*q^8+7409*
q^9+1237*q^62+8979*q^10+10471*q^11+11811*q^12+117*q^76+13092*q^13+14339*q^14+
15495*q^15+16662*q^16+698*q^66+366*q^70+96*q^77+77*q^78+9066*q^43+5415*q^49+
19993*q^28+17735*q^17+18783*q^18+19665*q^19+20404*q^20+20395*q^27+257*q^72+
11252*q^40+215*q^73+1076*q^63+18328*q^31+147*q^75+15235*q^35+46*q^80+5926*q^48+
21*q^83+36*q^81+11*q^85+7*q^86+5*q^87+3*q^88+2*q^89+14*q^84+27*q^82+1825*q^59+
14409*q^36+6486*q^47, 1+55429*q^25+35177*q^42+52542*q^22+119*q^90+54871*q^24+
50649*q^21+53905*q^23+14*q+21*q^97+19554*q^51+31461*q^44+16724*q^53+11932*q^57+
54121*q^30+46*q^94+18102*q^52+10890*q^58+14215*q^55+9009*q^60+98*q^91+15437*q^
54+29636*q^45+2652*q^71+1802*q^74+27*q^96+14*q^98+44709*q^37+4317*q^67+54772*q^
29+27844*q^46+62*q^93+11*q^99+5385*q^65+3409*q^69+40911*q^39+8166*q^61+37070*q^
41+49792*q^34+877*q^79+21069*q^50+6003*q^64+7*q^100+42826*q^38+51126*q^33+13034
*q^56+91*q^2+3837*q^68+55687*q^26+366*q^3+1047*q^4+2347*q^5+4401*q^6+52304*q^32
+7162*q^7+10444*q^8+14063*q^9+36*q^95+7373*q^62+5*q^101+17770*q^10+21414*q^11+
24882*q^12+1363*q^76+28127*q^13+31322*q^14+34299*q^15+37281*q^16+4823*q^66+3008
*q^70+1179*q^77+1013*q^78+33312*q^43+3*q^102+22684*q^49+55355*q^28+40128*q^17+
43014*q^18+45769*q^19+48431*q^20+55613*q^27+2340*q^72+39012*q^40+2061*q^73+2*q^
103+6672*q^63+53298*q^31+1573*q^75+48220*q^35+q^104+746*q^80+24344*q^48+q^105+
457*q^83+643*q^81+322*q^85+266*q^86+223*q^87+182*q^88+151*q^89+382*q^84+541*q^
82+9907*q^59+46533*q^36+77*q^92+26072*q^47]


 The number of permutations avoiding, {[1, 3, 2], [5, 3, 2, 1, 4]}, is given by

[1, 2, 5, 14, 41, 121, 356, 1044, 3057, 8948, 26192, 76674, 224465, 657137,

    1923817]

The number of EVEN permutations avoiding, {[1, 3, 2], [5, 3, 2, 1, 4]},

     is given by

[1, 1, 3, 7, 22, 60, 182, 519, 1537, 4462, 13109, 38294, 112228, 328423, 961750

    ]

The number of ODD permutations avoiding, {[1, 3, 2], [5, 3, 2, 1, 4]},

     is given by

[0, 1, 2, 7, 19, 61, 174, 525, 1520, 4486, 13083, 38380, 112237, 328714, 962067

    ]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 61, 174, 519, 1537, 4486, 13083, 38294, 112228, 328714,

    962067]

 ODD:, [0, 1, 3, 7, 19, 60, 182, 525, 1520, 4462, 13109, 38380, 112237, 328423,

    961750]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.780487805, 8.900826446,

    12.76966292, 17.42816092, 22.89630357, 29.17836388, 36.27202963,

    44.17429637, 52.88343840, 62.39889095, 72.72064391]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.342733395, 3.178972646,

    4.130971116, 5.172467273, 6.279711928, 7.440100028, 8.650763270,

    9.913578300, 11.23113417, 12.60493218, 14.03517374]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.48840, 10.1059, 17.0649, 26.7544,

    39.4348, 55.3551, 74.8357, 98.2790, 126.138, 158.884, 196.986]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4180400731,

    -0.4327060039, -0.4636471057, -0.5056558645, -0.5464602443, -0.5785612054,

    -0.6000733079, -0.6124686559, -0.6183130637, -0.6199019162, -0.6188628727]

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

    2.578362051, 2.640684162, 2.694123871, 2.777837077, 2.880772978,

    2.981714143, 3.065987288, 3.128692942, 3.171771107, 3.199924677,

    3.217829599]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [5, 2, 4, 3, 1]},

    {[3, 1, 2], [1, 3, 4, 2, 5]}, {[1, 3, 2], [5, 3, 2, 4, 1]},

    {[2, 3, 1], [1, 4, 2, 3, 5]}}

      the member , {[3, 1, 2], [1, 3, 4, 2, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 1+4*q+5*q^2+7*q^3+7*q^
4+5*q^5+5*q^6+3*q^7+2*q^8+q^9+q^10, 1+5*q+8*q^2+11*q^3+15*q^4+15*q^5+14*q^6+13*
q^7+11*q^8+9*q^9+7*q^10+5*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+12*q^2+17*q^3+25*
q^4+29*q^5+36*q^6+35*q^7+32*q^8+29*q^9+28*q^10+25*q^11+20*q^12+17*q^13+13*q^14+
11*q^15+7*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+17*q^2+26*q^3+40*q^4+49*q^
5+64*q^6+80*q^7+81*q^8+80*q^9+78*q^10+71*q^11+70*q^12+66*q^13+55*q^14+49*q^15+
43*q^16+36*q^17+31*q^18+24*q^19+19*q^20+15*q^21+11*q^22+7*q^23+5*q^24+3*q^25+2*
q^26+q^27+q^28, 1+8*q+23*q^2+39*q^3+62*q^4+82*q^5+105*q^6+141*q^7+167*q^8+186*q
^9+193*q^10+196*q^11+182*q^12+183*q^13+174*q^14+166*q^15+155*q^16+140*q^17+122*
q^18+109*q^19+98*q^20+87*q^21+73*q^22+61*q^23+51*q^24+42*q^25+35*q^26+26*q^27+
21*q^28+15*q^29+11*q^30+7*q^31+5*q^32+3*q^33+2*q^34+q^35+q^36, 1+9*q+30*q^2+57*
q^3+94*q^4+134*q^5+172*q^6+235*q^7+291*q^8+357*q^9+415*q^10+448*q^11+464*q^12+
461*q^13+457*q^14+452*q^15+439*q^16+428*q^17+409*q^18+383*q^19+347*q^20+312*q^
21+291*q^22+264*q^23+240*q^24+213*q^25+184*q^26+160*q^27+138*q^28+120*q^29+103*
q^30+85*q^31+69*q^32+57*q^33+46*q^34+37*q^35+28*q^36+21*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+38*q^2+81*q^3+140*q^4+213*q^5+281*q
^6+386*q^7+493*q^8+613*q^9+764*q^10+901*q^11+1004*q^12+1078*q^13+1106*q^14+1117
*q^15+1143*q^16+1113*q^17+1104*q^18+1095*q^19+1068*q^20+1007*q^21+960*q^22+880*
q^23+827*q^24+770*q^25+703*q^26+652*q^27+607*q^28+539*q^29+483*q^30+426*q^31+
381*q^32+338*q^33+300*q^34+257*q^35+223*q^36+190*q^37+158*q^38+136*q^39+115*q^
40+93*q^41+75*q^42+61*q^43+48*q^44+39*q^45+28*q^46+21*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+47*q^2+112*q^3+205*q^4+330*q^5+455*q^
6+627*q^7+827*q^8+1035*q^9+1312*q^10+1611*q^11+1932*q^12+2206*q^13+2428*q^14+
2564*q^15+2675*q^16+2763*q^17+2765*q^18+2777*q^19+2783*q^20+2760*q^21+2733*q^22
+2652*q^23+2543*q^24+2422*q^25+2297*q^26+2153*q^27+2019*q^28+1898*q^29+1781*q^
30+1676*q^31+1533*q^32+1404*q^33+1274*q^34+1157*q^35+1050*q^36+947*q^37+860*q^
38+769*q^39+679*q^40+599*q^41+518*q^42+457*q^43+398*q^44+345*q^45+297*q^46+251*
q^47+210*q^48+174*q^49+148*q^50+123*q^51+99*q^52+79*q^53+63*q^54+50*q^55+39*q^
56+28*q^57+21*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+57*q^2+151*q^3+295*q^4+500*q^5+726*q^6+1010*q^7+1370*q^8+1739*q^9+2223*q^10+
2765*q^11+3403*q^12+4105*q^13+4768*q^14+5332*q^15+5829*q^16+6187*q^17+6489*q^18
+6681*q^19+6792*q^20+6895*q^21+6945*q^22+6960*q^23+6917*q^24+6829*q^25+6666*q^
26+6465*q^27+6194*q^28+5943*q^29+5616*q^30+5355*q^31+5083*q^32+4828*q^33+4563*q
^34+4272*q^35+3970*q^36+3704*q^37+3410*q^38+3149*q^39+2881*q^40+2652*q^41+2439*
q^42+2240*q^43+2011*q^44+1822*q^45+1626*q^46+1459*q^47+1302*q^48+1159*q^49+1028
*q^50+913*q^51+797*q^52+693*q^53+594*q^54+518*q^55+444*q^56+383*q^57+325*q^58+
271*q^59+226*q^60+186*q^61+156*q^62+129*q^63+103*q^64+81*q^65+65*q^66+50*q^67+
39*q^68+28*q^69+21*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+68*q^2+199*q^3+417*q^4+743*q^5+1138*q^6+1614*q^7+2240*q^8+2904*q^9+3739*
q^10+4712*q^11+5833*q^12+7186*q^13+8661*q^14+10149*q^15+11596*q^16+12921*q^17+
13942*q^18+14914*q^19+15643*q^20+16173*q^21+16691*q^22+17011*q^23+17234*q^24+
17395*q^25+17437*q^26+17353*q^27+17223*q^28+16904*q^29+16389*q^30+15936*q^31+
15336*q^32+14730*q^33+14176*q^34+13543*q^35+12925*q^36+12363*q^37+11725*q^38+
11082*q^39+10474*q^40+9788*q^41+9152*q^42+8520*q^43+7929*q^44+7386*q^45+6886*q^
46+6368*q^47+5868*q^48+5402*q^49+4926*q^50+4484*q^51+4091*q^52+3695*q^53+3354*q
^54+3043*q^55+2746*q^56+2451*q^57+2194*q^58+1934*q^59+1717*q^60+1522*q^61+1331*
q^62+1174*q^63+1033*q^64+891*q^65+770*q^66+656*q^67+562*q^68+482*q^69+411*q^70+
345*q^71+287*q^72+238*q^73+194*q^74+162*q^75+133*q^76+105*q^77+83*q^78+65*q^79+
50*q^80+39*q^81+28*q^82+21*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+80*q^2+257*q^3+579*q^4+1085*q^5+1751*q^6+2555*q^7+3618*q^8+4810
*q^9+6246*q^10+7984*q^11+9944*q^12+12356*q^13+15097*q^14+18178*q^15+21496*q^16+
24910*q^17+28054*q^18+30912*q^19+33490*q^20+35755*q^21+37606*q^22+39333*q^23+
40620*q^24+41742*q^25+42602*q^26+43158*q^27+43578*q^28+43915*q^29+43817*q^30+
43413*q^31+42764*q^32+41882*q^33+40933*q^34+39775*q^35+38507*q^36+37217*q^37+
35930*q^38+34521*q^39+33235*q^40+31877*q^41+30501*q^42+29024*q^43+27499*q^44+
26008*q^45+24568*q^46+23128*q^47+21778*q^48+20455*q^49+19217*q^50+17973*q^51+
16813*q^52+15652*q^53+14564*q^54+13437*q^55+12414*q^56+11403*q^57+10517*q^58+
9664*q^59+8874*q^60+8116*q^61+7414*q^62+6714*q^63+6088*q^64+5488*q^65+4953*q^66
+4439*q^67+3986*q^68+3569*q^69+3204*q^70+2831*q^71+2510*q^72+2200*q^73+1941*q^
74+1696*q^75+1479*q^76+1294*q^77+1128*q^78+969*q^79+830*q^80+700*q^81+600*q^82+
510*q^83+431*q^84+361*q^85+299*q^86+246*q^87+200*q^88+166*q^89+135*q^90+107*q^
91+83*q^92+65*q^93+50*q^94+39*q^95+28*q^96+21*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+q+2*q^2+q^3, 1+q+2*q^2+3*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+5*q^8+7*q^7
+7*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1, q^15+5*q^14+8*q^13+11*q^12+15*q^11+15*q^10+
14*q^9+13*q^8+11*q^7+9*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+12*q^19+17*
q^18+25*q^17+29*q^16+36*q^15+35*q^14+32*q^13+29*q^12+28*q^11+25*q^10+20*q^9+17*
q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+26*q^25+64*q^22+40*q^24+80*q^
21+49*q^23+q+2*q^2+17*q^26+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+19*q^8+24*q^9+31*q^
10+36*q^11+43*q^12+49*q^13+55*q^14+66*q^15+70*q^16+q^28+71*q^17+78*q^18+80*q^19
+81*q^20+7*q^27, 1+196*q^25+174*q^22+182*q^24+166*q^21+183*q^23+q+105*q^30+141*
q^29+23*q^34+39*q^33+2*q^2+193*q^26+3*q^3+5*q^4+7*q^5+11*q^6+62*q^32+15*q^7+21*
q^8+26*q^9+35*q^10+42*q^11+51*q^12+61*q^13+73*q^14+87*q^15+98*q^16+167*q^28+109
*q^17+122*q^18+140*q^19+155*q^20+186*q^27+82*q^31+8*q^35+q^36, 1+347*q^25+57*q^
42+264*q^22+312*q^24+240*q^21+291*q^23+q+9*q^44+452*q^30+q^45+291*q^37+439*q^29
+172*q^39+94*q^41+448*q^34+235*q^38+464*q^33+2*q^2+383*q^26+3*q^3+5*q^4+7*q^5+
11*q^6+461*q^32+15*q^7+21*q^8+28*q^9+37*q^10+46*q^11+57*q^12+69*q^13+85*q^14+
103*q^15+120*q^16+30*q^43+428*q^28+138*q^17+160*q^18+184*q^19+213*q^20+409*q^27
+134*q^40+457*q^31+415*q^35+357*q^36, 1+483*q^25+1078*q^42+338*q^22+426*q^24+
300*q^21+381*q^23+q+140*q^51+901*q^44+38*q^53+770*q^30+81*q^52+q^55+10*q^54+764
*q^45+1104*q^37+703*q^29+613*q^46+1143*q^39+1106*q^41+1007*q^34+213*q^50+1113*q
^38+960*q^33+2*q^2+539*q^26+3*q^3+5*q^4+7*q^5+11*q^6+880*q^32+15*q^7+21*q^8+28*
q^9+39*q^10+48*q^11+61*q^12+75*q^13+93*q^14+115*q^15+136*q^16+1004*q^43+281*q^
49+652*q^28+158*q^17+190*q^18+223*q^19+257*q^20+607*q^27+1117*q^40+827*q^31+
1068*q^35+386*q^48+1095*q^36+493*q^47, 1+599*q^25+2543*q^42+398*q^22+518*q^24+
345*q^21+457*q^23+q+2564*q^51+2733*q^44+2206*q^53+1035*q^57+1050*q^30+2428*q^52
+827*q^58+1611*q^55+455*q^60+1932*q^54+2760*q^45+1898*q^37+947*q^29+2783*q^46+
11*q^65+2153*q^39+330*q^61+2422*q^41+1533*q^34+2675*q^50+47*q^64+2019*q^38+1404
*q^33+1312*q^56+2*q^2+679*q^26+3*q^3+5*q^4+7*q^5+11*q^6+1274*q^32+15*q^7+21*q^8
+28*q^9+205*q^62+39*q^10+50*q^11+63*q^12+79*q^13+99*q^14+123*q^15+148*q^16+q^66
+2652*q^43+2763*q^49+860*q^28+174*q^17+210*q^18+251*q^19+297*q^20+769*q^27+2297
*q^40+112*q^63+1157*q^31+1676*q^35+2765*q^48+627*q^59+1781*q^36+2777*q^47, 1+
693*q^25+3970*q^42+444*q^22+594*q^24+383*q^21+518*q^23+q+6465*q^51+4563*q^44+
6829*q^53+6895*q^57+1302*q^30+6666*q^52+6792*q^58+6960*q^55+6489*q^60+6917*q^54
+4828*q^45+1010*q^71+295*q^74+2652*q^37+2765*q^67+1159*q^29+5083*q^46+4105*q^65
+1739*q^69+3149*q^39+6187*q^61+3704*q^41+2011*q^34+6194*q^50+4768*q^64+2881*q^
38+1822*q^33+6945*q^56+2*q^2+2223*q^68+797*q^26+3*q^3+5*q^4+7*q^5+11*q^6+1626*q
^32+15*q^7+21*q^8+28*q^9+5829*q^62+39*q^10+50*q^11+65*q^12+57*q^76+81*q^13+103*
q^14+129*q^15+156*q^16+3403*q^66+1370*q^70+12*q^77+q^78+4272*q^43+5943*q^49+
1028*q^28+186*q^17+226*q^18+271*q^19+325*q^20+913*q^27+726*q^72+3410*q^40+500*q
^73+5332*q^63+1459*q^31+151*q^75+2240*q^35+5616*q^48+6681*q^59+2439*q^36+5355*q
^47, 1+770*q^25+5402*q^42+482*q^22+13*q^90+656*q^24+411*q^21+562*q^23+q+10474*q
^51+6368*q^44+11725*q^53+14176*q^57+1522*q^30+11082*q^52+14730*q^58+12925*q^55+
15936*q^60+q^91+12363*q^54+6886*q^45+15643*q^71+12921*q^74+3354*q^37+17234*q^67
+1331*q^29+7386*q^46+17437*q^65+16691*q^69+4091*q^39+16389*q^61+4926*q^41+2451*
q^34+5833*q^79+9788*q^50+17353*q^64+3695*q^38+2194*q^33+13543*q^56+2*q^2+17011*
q^68+891*q^26+3*q^3+5*q^4+7*q^5+11*q^6+1934*q^32+15*q^7+21*q^8+28*q^9+16904*q^
62+39*q^10+50*q^11+65*q^12+10149*q^76+83*q^13+105*q^14+133*q^15+162*q^16+17395*
q^66+16173*q^70+8661*q^77+7186*q^78+5868*q^43+9152*q^49+1174*q^28+194*q^17+238*
q^18+287*q^19+345*q^20+1033*q^27+14914*q^72+4484*q^40+13942*q^73+17223*q^63+
1717*q^31+11596*q^75+2746*q^35+4712*q^80+8520*q^48+2240*q^83+3739*q^81+1138*q^
85+743*q^86+417*q^87+199*q^88+68*q^89+1614*q^84+2904*q^82+15336*q^59+3043*q^36+
7929*q^47, 1+830*q^25+6714*q^42+510*q^22+18178*q^90+700*q^24+431*q^21+600*q^23+
q+3618*q^97+14564*q^51+8116*q^44+16813*q^53+21778*q^57+1696*q^30+7984*q^94+
15652*q^52+23128*q^58+19217*q^55+26008*q^60+15097*q^91+17973*q^54+8874*q^45+
40933*q^71+43413*q^74+4810*q^96+2555*q^98+3986*q^37+35930*q^67+1479*q^29+9664*q
^46+9944*q^93+1751*q^99+33235*q^65+38507*q^69+4953*q^39+27499*q^61+6088*q^41+
2831*q^34+42602*q^79+13437*q^50+31877*q^64+1085*q^100+4439*q^38+2510*q^33+20455
*q^56+2*q^2+37217*q^68+969*q^26+3*q^3+5*q^4+7*q^5+11*q^6+2200*q^32+15*q^7+21*q^
8+28*q^9+6246*q^95+29024*q^62+579*q^101+39*q^10+50*q^11+65*q^12+43915*q^76+83*q
^13+107*q^14+135*q^15+166*q^16+34521*q^66+39775*q^70+43578*q^77+43158*q^78+7414
*q^43+257*q^102+12414*q^49+1294*q^28+200*q^17+246*q^18+299*q^19+361*q^20+1128*q
^27+41882*q^72+5488*q^40+42764*q^73+80*q^103+30501*q^63+1941*q^31+43817*q^75+
3204*q^35+14*q^104+41742*q^80+11403*q^48+q^105+37606*q^83+40620*q^81+33490*q^85
+30912*q^86+28054*q^87+24910*q^88+21496*q^89+35755*q^84+39333*q^82+24568*q^59+
3569*q^36+12356*q^92+10517*q^47]


 The number of permutations avoiding, {[3, 1, 2], [1, 3, 4, 2, 5]}, is given by

[1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017, 190885, 533294,

    1484021]

The number of EVEN permutations avoiding, {[3, 1, 2], [1, 3, 4, 2, 5]},

     is given by

[1, 1, 2, 7, 21, 60, 178, 518, 1481, 4244, 12076, 34011, 95393, 266659, 742078]

The number of ODD permutations avoiding, {[3, 1, 2], [1, 3, 4, 2, 5]},

     is given by

[0, 1, 3, 7, 20, 61, 177, 514, 1492, 4252, 12035, 34006, 95492, 266635, 741943]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 21, 61, 177, 518, 1481, 4252, 12035, 34011, 95393, 266635,

    741943]

 ODD:, [0, 1, 2, 7, 20, 60, 178, 514, 1492, 4244, 12076, 34006, 95492, 266659,

    742078]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.400000000, 2.642857143, 4.243902439, 6.190082645,

    8.442253521, 10.96317829, 13.72620249, 16.71398305, 19.91543279,

    23.32331623, 26.93272913, 30.74020522, 34.74319905]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.324634645, 3.128947886,

    4.046060925, 5.061310226, 6.158385149, 7.325150954, 8.553841475,

    9.839783438, 11.18019561, 12.57337630, 14.01822074]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.40393, 9.79031, 16.3706, 25.6169,

    37.9257, 53.6578, 73.1682, 96.8213, 124.997, 158.090, 196.511]

Skewness: , [Float(undefined), 0., 0.2715454176, 0.3874842230, 0.4141164409,

    0.4129452046, 0.4250984145, 0.4514844817, 0.4831884548, 0.5140066288,

    0.5412584897, 0.5643127467, 0.5834458435, 0.5992323781, 0.6122651249]

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

    2.623166044, 2.705347317, 2.744251802, 2.795105976, 2.861671392,

    2.934116709, 3.004173563, 3.067514723, 3.122741890, 3.170029590,

    3.210176098]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [3, 5, 4, 2, 1]},

    {[3, 1, 2], [2, 3, 1, 4, 5]}, {[2, 3, 1], [1, 2, 5, 3, 4]},

    {[3, 1, 2], [1, 2, 4, 5, 3]}, {[1, 3, 2], [5, 4, 2, 1, 3]},

    {[1, 3, 2], [4, 3, 5, 2, 1]}, {[2, 3, 1], [3, 1, 2, 4, 5]},

    {[2, 1, 3], [5, 4, 1, 3, 2]}}

      the member , {[3, 1, 2], [2, 3, 1, 4, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 1+4*q+5*q^2+7*q^3+7*q^
4+5*q^5+5*q^6+3*q^7+2*q^8+q^9+q^10, 1+5*q+8*q^2+11*q^3+14*q^4+15*q^5+15*q^6+13*
q^7+11*q^8+9*q^9+7*q^10+5*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+12*q^2+17*q^3+24*
q^4+27*q^5+33*q^6+35*q^7+34*q^8+31*q^9+28*q^10+26*q^11+21*q^12+17*q^13+13*q^14+
11*q^15+7*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+17*q^2+26*q^3+39*q^4+46*q^
5+58*q^6+71*q^7+75*q^8+78*q^9+80*q^10+77*q^11+73*q^12+67*q^13+59*q^14+54*q^15+
46*q^16+37*q^17+32*q^18+25*q^19+19*q^20+15*q^21+11*q^22+7*q^23+5*q^24+3*q^25+2*
q^26+q^27+q^28, 1+8*q+23*q^2+39*q^3+61*q^4+78*q^5+97*q^6+124*q^7+143*q^8+163*q^
9+175*q^10+186*q^11+185*q^12+186*q^13+182*q^14+175*q^15+164*q^16+150*q^17+137*q
^18+122*q^19+107*q^20+93*q^21+80*q^22+67*q^23+55*q^24+43*q^25+36*q^26+27*q^27+
21*q^28+15*q^29+11*q^30+7*q^31+5*q^32+3*q^33+2*q^34+q^35+q^36, 1+9*q+30*q^2+57*
q^3+93*q^4+129*q^5+161*q^6+211*q^7+251*q^8+297*q^9+343*q^10+384*q^11+412*q^12+
426*q^13+439*q^14+450*q^15+448*q^16+438*q^17+429*q^18+409*q^19+382*q^20+353*q^
21+327*q^22+301*q^23+271*q^24+238*q^25+209*q^26+184*q^27+158*q^28+134*q^29+112*
q^30+93*q^31+76*q^32+61*q^33+47*q^34+38*q^35+29*q^36+21*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+38*q^2+81*q^3+139*q^4+207*q^5+266*q
^6+353*q^7+433*q^8+517*q^9+619*q^10+719*q^11+813*q^12+890*q^13+951*q^14+997*q^
15+1050*q^16+1065*q^17+1086*q^18+1090*q^19+1080*q^20+1052*q^21+1026*q^22+974*q^
23+928*q^24+878*q^25+812*q^26+752*q^27+699*q^28+632*q^29+571*q^30+509*q^31+451*
q^32+401*q^33+349*q^34+299*q^35+258*q^36+221*q^37+183*q^38+153*q^39+125*q^40+
102*q^41+82*q^42+65*q^43+49*q^44+40*q^45+29*q^46+21*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+47*q^2+112*q^3+204*q^4+323*q^5+435*q^6+
582*q^7+739*q^8+891*q^9+1082*q^10+1276*q^11+1498*q^12+1703*q^13+1893*q^14+2056*
q^15+2213*q^16+2347*q^17+2448*q^18+2535*q^19+2602*q^20+2648*q^21+2673*q^22+2657
*q^23+2621*q^24+2572*q^25+2492*q^26+2401*q^27+2304*q^28+2199*q^29+2074*q^30+
1952*q^31+1818*q^32+1691*q^33+1552*q^34+1417*q^35+1295*q^36+1170*q^37+1052*q^38
+941*q^39+832*q^40+734*q^41+639*q^42+555*q^43+481*q^44+412*q^45+349*q^46+293*q^
47+246*q^48+202*q^49+166*q^50+134*q^51+108*q^52+86*q^53+67*q^54+51*q^55+40*q^56
+29*q^57+21*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+
57*q^2+151*q^3+294*q^4+492*q^5+700*q^6+949*q^7+1244*q^8+1524*q^9+1872*q^10+2233
*q^11+2649*q^12+3094*q^13+3542*q^14+3971*q^15+4397*q^16+4782*q^17+5129*q^18+
5449*q^19+5710*q^20+5979*q^21+6190*q^22+6363*q^23+6478*q^24+6551*q^25+6549*q^26
+6533*q^27+6443*q^28+6347*q^29+6183*q^30+6013*q^31+5803*q^32+5600*q^33+5349*q^
34+5075*q^35+4798*q^36+4524*q^37+4222*q^38+3947*q^39+3656*q^40+3380*q^41+3100*q
^42+2833*q^43+2573*q^44+2339*q^45+2098*q^46+1878*q^47+1679*q^48+1485*q^49+1307*
q^50+1150*q^51+1001*q^52+868*q^53+745*q^54+637*q^55+545*q^56+460*q^57+384*q^58+
318*q^59+265*q^60+215*q^61+175*q^62+140*q^63+112*q^64+88*q^65+69*q^66+51*q^67+
40*q^68+29*q^69+21*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+68*q^2+199*q^3+416*q^4+734*q^5+1105*q^6+1532*q^7+2063*q^8+2586*q^9+3210*
q^10+3885*q^11+4628*q^12+5484*q^13+6380*q^14+7316*q^15+8287*q^16+9277*q^17+
10158*q^18+11041*q^19+11835*q^20+12594*q^21+13330*q^22+13978*q^23+14572*q^24+
15112*q^25+15512*q^26+15809*q^27+16065*q^28+16216*q^29+16204*q^30+16152*q^31+
15998*q^32+15789*q^33+15551*q^34+15170*q^35+14769*q^36+14342*q^37+13820*q^38+
13260*q^39+12717*q^40+12095*q^41+11475*q^42+10842*q^43+10204*q^44+9578*q^45+
8974*q^46+8334*q^47+7726*q^48+7147*q^49+6565*q^50+6005*q^51+5495*q^52+4992*q^53
+4524*q^54+4076*q^55+3660*q^56+3276*q^57+2925*q^58+2581*q^59+2279*q^60+2009*q^
61+1749*q^62+1522*q^63+1323*q^64+1137*q^65+976*q^66+828*q^67+699*q^68+593*q^69+
495*q^70+409*q^71+337*q^72+278*q^73+224*q^74+181*q^75+144*q^76+114*q^77+90*q^78
+69*q^79+51*q^80+40*q^81+29*q^82+21*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+80*q^2+257*q^3+578*q^4+1075*q^5+1710*q^6+2446*q^7+3373*
q^8+4346*q^9+5456*q^10+6712*q^11+8048*q^12+9627*q^13+11301*q^14+13128*q^15+
15099*q^16+17229*q^17+19315*q^18+21397*q^19+23394*q^20+25367*q^21+27266*q^22+
29091*q^23+30833*q^24+32521*q^25+34071*q^26+35426*q^27+36696*q^28+37814*q^29+
38710*q^30+39374*q^31+39851*q^32+40095*q^33+40288*q^34+40210*q^35+40005*q^36+
39645*q^37+39159*q^38+38483*q^39+37752*q^40+36809*q^41+35835*q^42+34714*q^43+
33499*q^44+32212*q^45+30939*q^46+29569*q^47+28215*q^48+26789*q^49+25397*q^50+
23968*q^51+22579*q^52+21162*q^53+19827*q^54+18459*q^55+17165*q^56+15898*q^57+
14707*q^58+13538*q^59+12443*q^60+11376*q^61+10378*q^62+9428*q^63+8549*q^64+7709
*q^65+6952*q^66+6231*q^67+5570*q^68+4954*q^69+4399*q^70+3886*q^71+3427*q^72+
2997*q^73+2618*q^74+2279*q^75+1968*q^76+1697*q^77+1461*q^78+1246*q^79+1057*q^80
+890*q^81+747*q^82+628*q^83+520*q^84+428*q^85+350*q^86+287*q^87+230*q^88+185*q^
89+146*q^90+116*q^91+90*q^92+69*q^93+51*q^94+40*q^95+29*q^96+21*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+q+2*q^2+q^3, 1+q+2*q^2+3*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+5*q^8+7*q^7
+7*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1, q^15+5*q^14+8*q^13+11*q^12+14*q^11+15*q^10+
15*q^9+13*q^8+11*q^7+9*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+12*q^19+17*
q^18+24*q^17+27*q^16+33*q^15+35*q^14+34*q^13+31*q^12+28*q^11+26*q^10+21*q^9+17*
q^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+26*q^25+58*q^22+39*q^24+71*q^
21+46*q^23+q+2*q^2+17*q^26+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+19*q^8+25*q^9+32*q^
10+37*q^11+46*q^12+54*q^13+59*q^14+67*q^15+73*q^16+q^28+77*q^17+80*q^18+78*q^19
+75*q^20+7*q^27, 1+186*q^25+182*q^22+185*q^24+175*q^21+186*q^23+q+97*q^30+124*q
^29+23*q^34+39*q^33+2*q^2+175*q^26+3*q^3+5*q^4+7*q^5+11*q^6+61*q^32+15*q^7+21*q
^8+27*q^9+36*q^10+43*q^11+55*q^12+67*q^13+80*q^14+93*q^15+107*q^16+143*q^28+122
*q^17+137*q^18+150*q^19+164*q^20+163*q^27+78*q^31+8*q^35+q^36, 1+382*q^25+57*q^
42+301*q^22+353*q^24+271*q^21+327*q^23+q+9*q^44+450*q^30+q^45+251*q^37+448*q^29
+161*q^39+93*q^41+384*q^34+211*q^38+412*q^33+2*q^2+409*q^26+3*q^3+5*q^4+7*q^5+
11*q^6+426*q^32+15*q^7+21*q^8+29*q^9+38*q^10+47*q^11+61*q^12+76*q^13+93*q^14+
112*q^15+134*q^16+30*q^43+438*q^28+158*q^17+184*q^18+209*q^19+238*q^20+429*q^27
+129*q^40+439*q^31+343*q^35+297*q^36, 1+571*q^25+890*q^42+401*q^22+509*q^24+349
*q^21+451*q^23+q+139*q^51+719*q^44+38*q^53+878*q^30+81*q^52+q^55+10*q^54+619*q^
45+1086*q^37+812*q^29+517*q^46+1050*q^39+951*q^41+1052*q^34+207*q^50+1065*q^38+
1026*q^33+2*q^2+632*q^26+3*q^3+5*q^4+7*q^5+11*q^6+974*q^32+15*q^7+21*q^8+29*q^9
+40*q^10+49*q^11+65*q^12+82*q^13+102*q^14+125*q^15+153*q^16+813*q^43+266*q^49+
752*q^28+183*q^17+221*q^18+258*q^19+299*q^20+699*q^27+997*q^40+928*q^31+1080*q^
35+353*q^48+1090*q^36+433*q^47, 1+734*q^25+2621*q^42+481*q^22+639*q^24+412*q^21
+555*q^23+q+2056*q^51+2673*q^44+1703*q^53+891*q^57+1295*q^30+1893*q^52+739*q^58
+1276*q^55+435*q^60+1498*q^54+2648*q^45+2199*q^37+1170*q^29+2602*q^46+11*q^65+
2401*q^39+323*q^61+2572*q^41+1818*q^34+2213*q^50+47*q^64+2304*q^38+1691*q^33+
1082*q^56+2*q^2+832*q^26+3*q^3+5*q^4+7*q^5+11*q^6+1552*q^32+15*q^7+21*q^8+29*q^
9+204*q^62+40*q^10+51*q^11+67*q^12+86*q^13+108*q^14+134*q^15+166*q^16+q^66+2657
*q^43+2347*q^49+1052*q^28+202*q^17+246*q^18+293*q^19+349*q^20+941*q^27+2492*q^
40+112*q^63+1417*q^31+1952*q^35+2448*q^48+582*q^59+2074*q^36+2535*q^47, 1+868*q
^25+4798*q^42+545*q^22+745*q^24+460*q^21+637*q^23+q+6533*q^51+5349*q^44+6551*q^
53+5979*q^57+1679*q^30+6549*q^52+5710*q^58+6363*q^55+5129*q^60+6478*q^54+5600*q
^45+949*q^71+294*q^74+3380*q^37+2233*q^67+1485*q^29+5803*q^46+3094*q^65+1524*q^
69+3947*q^39+4782*q^61+4524*q^41+2573*q^34+6443*q^50+3542*q^64+3656*q^38+2339*q
^33+6190*q^56+2*q^2+1872*q^68+1001*q^26+3*q^3+5*q^4+7*q^5+11*q^6+2098*q^32+15*q
^7+21*q^8+29*q^9+4397*q^62+40*q^10+51*q^11+69*q^12+57*q^76+88*q^13+112*q^14+140
*q^15+175*q^16+2649*q^66+1244*q^70+12*q^77+q^78+5075*q^43+6347*q^49+1307*q^28+
215*q^17+265*q^18+318*q^19+384*q^20+1150*q^27+700*q^72+4222*q^40+492*q^73+3971*
q^63+1878*q^31+151*q^75+2833*q^35+6183*q^48+5449*q^59+3100*q^36+6013*q^47, 1+
976*q^25+7147*q^42+593*q^22+13*q^90+828*q^24+495*q^21+699*q^23+q+12717*q^51+
8334*q^44+13820*q^53+15551*q^57+2009*q^30+13260*q^52+15789*q^58+14769*q^55+
16152*q^60+q^91+14342*q^54+8974*q^45+11835*q^71+9277*q^74+4524*q^37+14572*q^67+
1749*q^29+9578*q^46+15512*q^65+13330*q^69+5495*q^39+16204*q^61+6565*q^41+3276*q
^34+4628*q^79+12095*q^50+15809*q^64+4992*q^38+2925*q^33+15170*q^56+2*q^2+13978*
q^68+1137*q^26+3*q^3+5*q^4+7*q^5+11*q^6+2581*q^32+15*q^7+21*q^8+29*q^9+16216*q^
62+40*q^10+51*q^11+69*q^12+7316*q^76+90*q^13+114*q^14+144*q^15+181*q^16+15112*q
^66+12594*q^70+6380*q^77+5484*q^78+7726*q^43+11475*q^49+1522*q^28+224*q^17+278*
q^18+337*q^19+409*q^20+1323*q^27+11041*q^72+6005*q^40+10158*q^73+16065*q^63+
2279*q^31+8287*q^75+3660*q^35+3885*q^80+10842*q^48+2063*q^83+3210*q^81+1105*q^
85+734*q^86+416*q^87+199*q^88+68*q^89+1532*q^84+2586*q^82+15998*q^59+4076*q^36+
10204*q^47, 1+1057*q^25+9428*q^42+628*q^22+13128*q^90+890*q^24+520*q^21+747*q^
23+q+3373*q^97+19827*q^51+11376*q^44+22579*q^53+28215*q^57+2279*q^30+6712*q^94+
21162*q^52+29569*q^58+25397*q^55+32212*q^60+11301*q^91+23968*q^54+12443*q^45+
40288*q^71+39374*q^74+4346*q^96+2446*q^98+5570*q^37+39159*q^67+1968*q^29+13538*
q^46+8048*q^93+1710*q^99+37752*q^65+40005*q^69+6952*q^39+33499*q^61+8549*q^41+
3886*q^34+34071*q^79+18459*q^50+36809*q^64+1075*q^100+6231*q^38+3427*q^33+26789
*q^56+2*q^2+39645*q^68+1246*q^26+3*q^3+5*q^4+7*q^5+11*q^6+2997*q^32+15*q^7+21*q
^8+29*q^9+5456*q^95+34714*q^62+578*q^101+40*q^10+51*q^11+69*q^12+37814*q^76+90*
q^13+116*q^14+146*q^15+185*q^16+38483*q^66+40210*q^70+36696*q^77+35426*q^78+
10378*q^43+257*q^102+17165*q^49+1697*q^28+230*q^17+287*q^18+350*q^19+428*q^20+
1461*q^27+40095*q^72+7709*q^40+39851*q^73+80*q^103+35835*q^63+2618*q^31+38710*q
^75+4399*q^35+14*q^104+32521*q^80+15898*q^48+q^105+27266*q^83+30833*q^81+23394*
q^85+21397*q^86+19315*q^87+17229*q^88+15099*q^89+25367*q^84+29091*q^82+30939*q^
59+4954*q^36+9627*q^92+14707*q^47]


 The number of permutations avoiding, {[3, 1, 2], [2, 3, 1, 4, 5]}, is given by

[1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496, 24111, 68017, 190885, 533294,

    1484021]

The number of EVEN permutations avoiding, {[3, 1, 2], [2, 3, 1, 4, 5]},

     is given by

[1, 1, 2, 7, 21, 60, 177, 518, 1486, 4244, 12064, 34011, 95399, 266659, 742160]

The number of ODD permutations avoiding, {[3, 1, 2], [2, 3, 1, 4, 5]},

     is given by

[0, 1, 3, 7, 20, 61, 178, 514, 1487, 4252, 12047, 34006, 95486, 266635, 741861]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 21, 61, 178, 518, 1486, 4252, 12047, 34011, 95399, 266635,

    741861]

 ODD:, [0, 1, 2, 7, 20, 60, 177, 514, 1487, 4244, 12064, 34006, 95486, 266659,

    742160]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.400000000, 2.642857143, 4.243902439, 6.206611570,

    8.512676056, 11.14147287, 14.07769929, 17.31144068, 20.83650616,

    24.64898481, 28.74631323, 33.12673122, 37.78897401]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.324634645, 3.122611101,

    4.031108943, 5.043737478, 6.150731717, 7.343501618, 8.615471508,

    9.961702779, 11.37836630, 12.86234746, 14.41099856]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.40393, 9.75070, 16.2498, 25.4393,

    37.8315, 53.9270, 74.2263, 99.2355, 129.467, 165.440, 207.677]

Skewness: , [Float(undefined), 0., 0.2715454176, 0.3874842230, 0.4141164409,

    0.4024317951, 0.3892518453, 0.3830583231, 0.3817175800, 0.3822147706,

    0.3827555256, 0.3825848963, 0.3815109095, 0.3795856037, 0.3769504321]

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

    2.623166044, 2.716674988, 2.750053019, 2.771224404, 2.793080951,

    2.815814429, 2.837494555, 2.856894950, 2.873615937, 2.887700526,

    2.899484454]

                                end of this data



For the equivalence class of patterns, {{[1, 3, 2], [3, 2, 4, 1, 5]},

    {[3, 1, 2], [3, 4, 2, 5, 1]}, {[1, 3, 2], [4, 2, 1, 3, 5]},

    {[2, 1, 3], [1, 3, 5, 4, 2]}, {[2, 1, 3], [1, 5, 2, 4, 3]},

    {[2, 3, 1], [5, 1, 4, 2, 3]}, {[2, 3, 1], [5, 3, 1, 2, 4]},

    {[3, 1, 2], [2, 4, 5, 3, 1]}}

      the member , {[1, 3, 2], [3, 2, 4, 1, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

                 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+4*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+4*q^4+6*q^5+7*q^6+9*q^7+12*
q^8+14*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+4*q^4+6
*q^5+8*q^6+9*q^7+13*q^8+17*q^9+18*q^10+23*q^11+27*q^12+33*q^13+38*q^14+40*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+4*q^4+6*q^5+8*q^6+
10*q^7+13*q^8+18*q^9+20*q^10+26*q^11+32*q^12+36*q^13+43*q^14+52*q^15+59*q^16+68
*q^17+80*q^18+90*q^19+100*q^20+103*q^21+98*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+4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+18*q^9+21*q^10+28
*q^11+34*q^12+41*q^13+50*q^14+57*q^15+65*q^16+83*q^17+90*q^18+107*q^19+120*q^20
+136*q^21+157*q^22+182*q^23+201*q^24+226*q^25+248*q^26+262*q^27+265*q^28+248*q^
29+214*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+4*q^
4+6*q^5+8*q^6+10*q^7+14*q^8+19*q^9+21*q^10+29*q^11+36*q^12+43*q^13+54*q^14+64*q
^15+72*q^16+91*q^17+105*q^18+119*q^19+141*q^20+163*q^21+181*q^22+207*q^23+232*q
^24+264*q^25+306*q^26+336*q^27+378*q^28+425*q^29+476*q^30+533*q^31+587*q^32+632
*q^33+675*q^34+694*q^35+683*q^36+634*q^37+544*q^38+428*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+4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+19
*q^9+22*q^10+29*q^11+37*q^12+45*q^13+56*q^14+68*q^15+78*q^16+98*q^17+115*q^18+
134*q^19+154*q^20+185*q^21+207*q^22+241*q^23+278*q^24+313*q^25+348*q^26+401*q^
27+437*q^28+500*q^29+554*q^30+617*q^31+687*q^32+770*q^33+847*q^34+957*q^35+1061
*q^36+1168*q^37+1292*q^38+1419*q^39+1546*q^40+1670*q^41+1765*q^42+1827*q^43+
1842*q^44+1777*q^45+1629*q^46+1398*q^47+1107*q^48+799*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+4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+
19*q^9+22*q^10+30*q^11+37*q^12+46*q^13+58*q^14+70*q^15+82*q^16+104*q^17+121*q^
18+144*q^19+171*q^20+198*q^21+228*q^22+270*q^23+311*q^24+357*q^25+408*q^26+458*
q^27+519*q^28+595*q^29+652*q^30+732*q^31+830*q^32+915*q^33+1015*q^34+1125*q^35+
1224*q^36+1353*q^37+1509*q^38+1644*q^39+1834*q^40+2022*q^41+2218*q^42+2451*q^43
+2697*q^44+2961*q^45+3268*q^46+3555*q^47+3850*q^48+4160*q^49+4448*q^50+4699*q^
51+4884*q^52+4953*q^53+4890*q^54+4653*q^55+4218*q^56+3612*q^57+2883*q^58+2118*q
^59+1411*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+
4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+19*q^9+22*q^10+30*q^11+38*q^12+46*q^13+59*q^14+
72*q^15+84*q^16+108*q^17+127*q^18+150*q^19+180*q^20+215*q^21+243*q^22+291*q^23+
339*q^24+391*q^25+453*q^26+518*q^27+579*q^28+681*q^29+755*q^30+855*q^31+966*q^
32+1083*q^33+1199*q^34+1348*q^35+1486*q^36+1651*q^37+1824*q^38+2005*q^39+2193*q
^40+2430*q^41+2629*q^42+2893*q^43+3172*q^44+3458*q^45+3757*q^46+4122*q^47+4478*
q^48+4924*q^49+5407*q^50+5902*q^51+6468*q^52+7063*q^53+7688*q^54+8389*q^55+9110
*q^56+9858*q^57+10627*q^58+11344*q^59+12042*q^60+12674*q^61+13155*q^62+13431*q^
63+13413*q^64+13022*q^65+12227*q^66+10996*q^67+9386*q^68+7535*q^69+5621*q^70+
3849*q^71+2380*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+4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+19*q^9+22*q^10+30*q^11+38*q^12+47*q^13+
59*q^14+73*q^15+86*q^16+110*q^17+131*q^18+156*q^19+186*q^20+224*q^21+259*q^22+
306*q^23+362*q^24+419*q^25+486*q^26+564*q^27+638*q^28+743*q^29+846*q^30+956*q^
31+1091*q^32+1238*q^33+1383*q^34+1547*q^35+1733*q^36+1941*q^37+2164*q^38+2402*q
^39+2640*q^40+2927*q^41+3241*q^42+3532*q^43+3880*q^44+4247*q^45+4614*q^46+5045*
q^47+5500*q^48+5966*q^49+6501*q^50+7039*q^51+7604*q^52+8262*q^53+8962*q^54+9663
*q^55+10538*q^56+11432*q^57+12396*q^58+13468*q^59+14634*q^60+15904*q^61+17343*q
^62+18810*q^63+20374*q^64+22084*q^65+23828*q^66+25662*q^67+27567*q^68+29448*q^
69+31311*q^70+33055*q^71+34571*q^72+35829*q^73+36647*q^74+36840*q^75+36265*q^76
+34759*q^77+32257*q^78+28799*q^79+24525*q^80+19760*q^81+14924*q^82+10448*q^83+
6696*q^84+3862*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+4*q^4+6*q^5+8*q^6+10*q^7+14*q^8+19*q^9+22*q^10+30*q^11+38*q^12+47*q^13+
60*q^14+73*q^15+87*q^16+112*q^17+133*q^18+160*q^19+192*q^20+230*q^21+268*q^22+
322*q^23+376*q^24+442*q^25+516*q^26+597*q^27+683*q^28+803*q^29+907*q^30+1047*q^
31+1199*q^32+1363*q^33+1534*q^34+1746*q^35+1946*q^36+2200*q^37+2475*q^38+2759*q
^39+3068*q^40+3435*q^41+3778*q^42+4207*q^43+4637*q^44+5102*q^45+5600*q^46+6172*
q^47+6715*q^48+7350*q^49+8030*q^50+8737*q^51+9487*q^52+10318*q^53+11118*q^54+
12062*q^55+13041*q^56+14026*q^57+15134*q^58+16341*q^59+17544*q^60+18979*q^61+
20418*q^62+21940*q^63+23658*q^64+25477*q^65+27380*q^66+29598*q^67+31933*q^68+
34456*q^69+37347*q^70+40317*q^71+43508*q^72+47045*q^73+50768*q^74+54754*q^75+
59053*q^76+63410*q^77+68019*q^78+72777*q^79+77540*q^80+82351*q^81+87056*q^82+
91429*q^83+95365*q^84+98552*q^85+100737*q^86+101654*q^87+100861*q^88+98019*q^89
+92901*q^90+85406*q^91+75736*q^92+64356*q^93+52003*q^94+39650*q^95+28265*q^96+
18636*q^97+11220*q^98+6062*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+4
*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+4*q^11+6*q^10+7*q^9
+9*q^8+12*q^7+14*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
+4*q^17+6*q^16+8*q^15+9*q^14+13*q^13+17*q^12+18*q^11+23*q^10+27*q^9+33*q^8+38*q
^7+40*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+8*q^22+4*q^24+10*q^21+6*q
^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+86*q^5+98*q^6+103*q^7+100*q^8+90*q^9+80*q^
10+68*q^11+59*q^12+52*q^13+43*q^14+36*q^15+32*q^16+q^28+26*q^17+20*q^18+18*q^19
+13*q^20+q^27, 1+28*q^25+50*q^22+34*q^24+57*q^21+41*q^23+8*q+8*q^30+10*q^29+2*q
^34+3*q^33+28*q^2+21*q^26+63*q^3+112*q^4+167*q^5+214*q^6+4*q^32+248*q^7+265*q^8
+262*q^9+248*q^10+226*q^11+201*q^12+182*q^13+157*q^14+136*q^15+120*q^16+14*q^28
+107*q^17+90*q^18+83*q^19+65*q^20+18*q^27+6*q^31+q^35+q^36, 1+141*q^25+3*q^42+
207*q^22+163*q^24+232*q^21+181*q^23+9*q+q^44+64*q^30+q^45+14*q^37+72*q^29+8*q^
39+4*q^41+29*q^34+10*q^38+36*q^33+36*q^2+119*q^26+92*q^3+182*q^4+301*q^5+428*q^
6+43*q^32+544*q^7+634*q^8+683*q^9+694*q^10+675*q^11+632*q^12+587*q^13+533*q^14+
476*q^15+425*q^16+2*q^43+91*q^28+378*q^17+336*q^18+306*q^19+264*q^20+105*q^27+6
*q^40+54*q^31+21*q^35+19*q^36, 1+554*q^25+45*q^42+770*q^22+617*q^24+847*q^21+
687*q^23+10*q+4*q^51+29*q^44+2*q^53+313*q^30+3*q^52+q^55+q^54+22*q^45+115*q^37+
348*q^29+19*q^46+78*q^39+56*q^41+185*q^34+6*q^50+98*q^38+207*q^33+45*q^2+500*q^
26+129*q^3+282*q^4+512*q^5+799*q^6+241*q^32+1107*q^7+1398*q^8+1629*q^9+1777*q^
10+1842*q^11+1827*q^12+1765*q^13+1670*q^14+1546*q^15+1419*q^16+37*q^43+8*q^49+
401*q^28+1292*q^17+1168*q^18+1061*q^19+957*q^20+437*q^27+68*q^40+278*q^31+154*q
^35+10*q^48+134*q^36+14*q^47, 1+2022*q^25+311*q^42+2697*q^22+2218*q^24+2961*q^
21+2451*q^23+11*q+70*q^51+228*q^44+46*q^53+19*q^57+1224*q^30+58*q^52+14*q^58+30
*q^55+8*q^60+37*q^54+198*q^45+595*q^37+1353*q^29+171*q^46+q^65+458*q^39+6*q^61+
357*q^41+830*q^34+82*q^50+2*q^64+519*q^38+915*q^33+22*q^56+55*q^2+1834*q^26+175
*q^3+420*q^4+831*q^5+1411*q^6+1015*q^32+2118*q^7+2883*q^8+3612*q^9+4*q^62+4218*
q^10+4653*q^11+4890*q^12+4953*q^13+4884*q^14+4699*q^15+4448*q^16+q^66+270*q^43+
104*q^49+1509*q^28+4160*q^17+3850*q^18+3555*q^19+3268*q^20+1644*q^27+408*q^40+3
*q^63+1125*q^31+732*q^35+121*q^48+10*q^59+652*q^36+144*q^47, 1+7063*q^25+1486*q
^42+9110*q^22+7688*q^24+9858*q^21+8389*q^23+12*q+518*q^51+1199*q^44+391*q^53+
215*q^57+4478*q^30+453*q^52+180*q^58+291*q^55+127*q^60+339*q^54+1083*q^45+10*q^
71+4*q^74+2430*q^37+30*q^67+4924*q^29+966*q^46+46*q^65+19*q^69+2005*q^39+108*q^
61+1651*q^41+3172*q^34+579*q^50+59*q^64+2193*q^38+3458*q^33+243*q^56+66*q^2+22*
q^68+6468*q^26+231*q^3+605*q^4+1297*q^5+2380*q^6+3757*q^32+3849*q^7+5621*q^8+
7535*q^9+84*q^62+9386*q^10+10996*q^11+12227*q^12+2*q^76+13022*q^13+13413*q^14+
13431*q^15+13155*q^16+38*q^66+14*q^70+q^77+q^78+1348*q^43+681*q^49+5407*q^28+
12674*q^17+12042*q^18+11344*q^19+10627*q^20+5902*q^27+8*q^72+1824*q^40+6*q^73+
72*q^63+4122*q^31+3*q^75+2893*q^35+755*q^48+150*q^59+2629*q^36+855*q^47, 1+
23828*q^25+5966*q^42+29448*q^22+q^90+25662*q^24+31311*q^21+27567*q^23+13*q+2640
*q^51+5045*q^44+2164*q^53+1383*q^57+15904*q^30+2402*q^52+1238*q^58+1733*q^55+
956*q^60+q^91+1941*q^54+4614*q^45+186*q^71+110*q^74+8962*q^37+362*q^67+17343*q^
29+4247*q^46+486*q^65+259*q^69+7604*q^39+846*q^61+6501*q^41+11432*q^34+38*q^79+
2927*q^50+564*q^64+8262*q^38+12396*q^33+1547*q^56+78*q^2+306*q^68+22084*q^26+
298*q^3+847*q^4+1958*q^5+3862*q^6+13468*q^32+6696*q^7+10448*q^8+14924*q^9+743*q
^62+19760*q^10+24525*q^11+28799*q^12+73*q^76+32257*q^13+34759*q^14+36265*q^15+
36840*q^16+419*q^66+224*q^70+59*q^77+47*q^78+5500*q^43+3241*q^49+18810*q^28+
36647*q^17+35829*q^18+34571*q^19+33055*q^20+20374*q^27+156*q^72+7039*q^40+131*q
^73+638*q^63+14634*q^31+86*q^75+10538*q^35+30*q^80+3532*q^48+14*q^83+22*q^81+8*
q^85+6*q^86+4*q^87+3*q^88+2*q^89+10*q^84+19*q^82+1091*q^59+9663*q^36+3880*q^47,
1+77540*q^25+21940*q^42+91429*q^22+73*q^90+82351*q^24+95365*q^21+87056*q^23+14*
q+14*q^97+11118*q^51+18979*q^44+9487*q^53+6715*q^57+54754*q^30+30*q^94+10318*q^
52+6172*q^58+8030*q^55+5102*q^60+60*q^91+8737*q^54+17544*q^45+1534*q^71+1047*q^
74+19*q^96+10*q^98+31933*q^37+2475*q^67+59053*q^29+16341*q^46+38*q^93+8*q^99+
3068*q^65+1946*q^69+27380*q^39+4637*q^61+23658*q^41+40317*q^34+516*q^79+12062*q
^50+3435*q^64+6*q^100+29598*q^38+43508*q^33+7350*q^56+91*q^2+2200*q^68+72777*q^
26+377*q^3+1157*q^4+2872*q^5+6062*q^6+47045*q^32+11220*q^7+18636*q^8+28265*q^9+
22*q^95+4207*q^62+4*q^101+39650*q^10+52003*q^11+64356*q^12+803*q^76+75736*q^13+
85406*q^14+92901*q^15+98019*q^16+2759*q^66+1746*q^70+683*q^77+597*q^78+20418*q^
43+3*q^102+13041*q^49+63410*q^28+100861*q^17+101654*q^18+100737*q^19+98552*q^20
+68019*q^27+1363*q^72+25477*q^40+1199*q^73+2*q^103+3778*q^63+50768*q^31+907*q^
75+37347*q^35+q^104+442*q^80+14026*q^48+q^105+268*q^83+376*q^81+192*q^85+160*q^
86+133*q^87+112*q^88+87*q^89+230*q^84+322*q^82+5600*q^59+34456*q^36+47*q^92+
15134*q^47]


 The number of permutations avoiding, {[1, 3, 2], [3, 2, 4, 1, 5]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 3, 2], [3, 2, 4, 1, 5]},

     is given by

[1, 1, 3, 7, 21, 61, 182, 547, 1635, 4922, 14742, 44293, 132808, 398600,

    1195648]

The number of ODD permutations avoiding, {[1, 3, 2], [3, 2, 4, 1, 5]},

     is given by

[0, 1, 2, 7, 20, 61, 183, 547, 1646, 4920, 14783, 44281, 132913, 398562,

    1195837]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 183, 547, 1635, 4920, 14783, 44293, 132808, 398562,

    1195837]

 ODD:, [0, 1, 3, 7, 20, 61, 182, 547, 1646, 4922, 14742, 44281, 132913, 398600,

    1195648]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.853658537, 9.155737705,

    13.30684932, 18.33546618, 24.26150564, 31.09997968, 38.86289585,

    47.56024341, 57.20052612, 67.79107383, 79.33824507]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.332809210, 3.117955646,

    3.973760813, 4.889630477, 5.857123095, 6.869376810, 7.920491870,

    9.005150556, 10.11843772, 11.25577424, 12.41289979]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.44200, 9.72165, 15.7908, 23.9085,

    34.3059, 47.1883, 62.7342, 81.0927, 102.383, 126.692, 154.080]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5032000763,

    -0.6153536035, -0.7150147888, -0.8011076530, -0.8758850297, -0.9420859246,

    -1.001950640, -1.057103853, -1.108655933, -1.157379946, -1.203746295]

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

    2.673308354, 2.927421422, 3.163164267, 3.381656114, 3.584925424,

    3.776577598, 3.960407511, 4.139764659, 4.316943037, 4.493658729,

    4.670850074]

                                end of this data



For the equivalence class of patterns, {{[1, 3, 2], [1, 2, 3, 4, 5]},

    {[2, 3, 1], [5, 4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4, 5]},

    {[3, 1, 2], [5, 4, 3, 2, 1]}}

      the member , {[1, 3, 2], [1, 2, 3, 4, 5]}, has a scheme of depth , 2

                                  here it is:

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

    [[1, 2], {[0, 1, 0], [0, 0, 3]}, {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, 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, q^3+3*q^4+5*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, q^6+4*q^7+8*q^8+15*q^9+22*q^10+29*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,
q^10+5*q^11+12*q^12+24*q^13+40*q^14+57*q^15+78*q^16+94*q^17+109*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, q^15+6*
q^16+17*q^17+37*q^18+68*q^19+106*q^20+154*q^21+203*q^22+250*q^23+293*q^24+318*q
^25+333*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, q^21+7*q^22+23*q^23+55*q^24+110*q^25+187*q^26+289*q^27+410*q^
28+538*q^29+671*q^30+783*q^31+873*q^32+925*q^33+929*q^34+895*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, q^28+8*q^
29+30*q^30+79*q^31+171*q^32+315*q^33+519*q^34+785*q^35+1096*q^36+1445*q^37+1795
*q^38+2119*q^39+2391*q^40+2566*q^41+2645*q^42+2600*q^43+2444*q^44+2196*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, q^36+9*q^37+38*q^38+110*q^39+257*q^40+510*q^41+896*q^42+1438*q^43+2129
*q^44+2959*q^45+3884*q^46+4836*q^47+5758*q^48+6544*q^49+7140*q^50+7475*q^51+
7506*q^52+7248*q^53+6699*q^54+5935*q^55+5019*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, q^45+10*q^46+47*q^
47+149*q^48+375*q^49+798*q^50+1493*q^51+2536*q^52+3968*q^53+5803*q^54+8010*q^55
+10482*q^56+13101*q^57+15663*q^58+17973*q^59+19841*q^60+21063*q^61+21558*q^62+
21239*q^63+20148*q^64+18390*q^65+16103*q^66+13516*q^67+10820*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,
q^55+11*q^56+57*q^57+197*q^58+533*q^59+1212*q^60+2410*q^61+4325*q^62+7132*q^63+
10956*q^64+15854*q^65+21737*q^66+28424*q^67+35577*q^68+42741*q^69+49440*q^70+
55098*q^71+59276*q^72+61580*q^73+61779*q^74+59882*q^75+55998*q^76+50504*q^77+
43843*q^78+36546*q^79+29177*q^80+22195*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, q^66+12*q^67+68*q^
68+255*q^69+740*q^70+1793*q^71+3781*q^72+7158*q^73+12413*q^74+19996*q^75+30270*
q^76+43372*q^77+59190*q^78+77297*q^79+96886*q^80+116936*q^81+136119*q^82+153078
*q^83+166517*q^84+175271*q^85+178657*q^86+176294*q^87+168386*q^88+155586*q^89+
138882*q^90+119631*q^91+99202*q^92+78984*q^93+60163*q^94+43606*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 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^12+3*q^11+5*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^15+4*q^14+8*q^13+15*q^12+22*q^11+29*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, q^18+5*q^17+12*q^16
+24*q^15+40*q^14+57*q^13+78*q^12+94*q^11+109*q^10+118*q^9+118*q^8+115*q^7+102*q
^6+86*q^5+65*q^4+41*q^3+21*q^2+7*q+1, q^21+6*q^20+17*q^19+37*q^18+68*q^17+106*q
^16+154*q^15+203*q^14+250*q^13+293*q^12+318*q^11+333*q^10+326*q^9+303*q^8+268*q
^7+219*q^6+167*q^5+112*q^4+63*q^3+28*q^2+8*q+1, 1+23*q^22+q^24+55*q^21+7*q^23+9
*q+36*q^2+92*q^3+182*q^4+301*q^5+434*q^6+574*q^7+704*q^8+813*q^9+895*q^10+929*q
^11+925*q^12+873*q^13+783*q^14+671*q^15+538*q^16+410*q^17+289*q^18+187*q^19+110
*q^20, 1+30*q^25+315*q^22+79*q^24+519*q^21+171*q^23+10*q+45*q^2+8*q^26+129*q^3+
282*q^4+512*q^5+806*q^6+1149*q^7+1515*q^8+1871*q^9+2196*q^10+2444*q^11+2600*q^
12+2645*q^13+2566*q^14+2391*q^15+2119*q^16+1795*q^17+1445*q^18+1096*q^19+785*q^
20+q^27, 1+510*q^25+2129*q^22+896*q^24+2959*q^21+1438*q^23+11*q+q^30+9*q^29+55*
q^2+257*q^26+175*q^3+420*q^4+831*q^5+1419*q^6+2174*q^7+3065*q^8+4032*q^9+5019*q
^10+5935*q^11+6699*q^12+7248*q^13+7506*q^14+7475*q^15+7140*q^16+38*q^28+6544*q^
17+5758*q^18+4836*q^19+3884*q^20+110*q^27, 1+3968*q^25+10482*q^22+5803*q^24+
13101*q^21+8010*q^23+12*q+149*q^30+375*q^29+q^33+66*q^2+2536*q^26+231*q^3+605*q
^4+1297*q^5+2389*q^6+10*q^32+3921*q^7+5889*q^8+8223*q^9+10820*q^10+13516*q^11+
16103*q^12+18390*q^13+20148*q^14+21239*q^15+21558*q^16+798*q^28+21063*q^17+
19841*q^18+17973*q^19+15663*q^20+1493*q^27+47*q^31, 1+21737*q^25+42741*q^22+
28424*q^24+49440*q^21+35577*q^23+13*q+2410*q^30+4325*q^29+57*q^34+197*q^33+78*q
^2+15854*q^26+298*q^3+847*q^4+1958*q^5+3872*q^6+533*q^32+6786*q^7+10826*q^8+
15998*q^9+22195*q^10+29177*q^11+36546*q^12+43843*q^13+50504*q^14+55998*q^15+
59882*q^16+7132*q^28+61779*q^17+61580*q^18+59276*q^19+55098*q^20+10956*q^27+
1212*q^31+11*q^35+q^36, 1+96886*q^25+153078*q^22+116936*q^24+166517*q^21+136119
*q^23+14*q+19996*q^30+68*q^37+30270*q^29+q^39+1793*q^34+12*q^38+3781*q^33+91*q^
2+77297*q^26+377*q^3+1157*q^4+2872*q^5+6073*q^6+7158*q^32+11330*q^7+19151*q^8+
29875*q^9+43606*q^10+60163*q^11+78984*q^12+99202*q^13+119631*q^14+138882*q^15+
155586*q^16+43372*q^28+168386*q^17+176294*q^18+178657*q^19+175271*q^20+59190*q^
27+12413*q^31+740*q^35+255*q^36]


 The number of permutations avoiding, {[1, 3, 2], [1, 2, 3, 4, 5]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[1, 3, 2], [1, 2, 3, 4, 5]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[1, 3, 2], [1, 2, 3, 4, 5]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.951219512, 9.483606557,

    13.99452055, 19.49817185, 25.99939043, 33.49979679, 41.99993226,

    51.49997742, 61.99999247, 73.49999749, 85.99999916]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.163457254, 2.612183329,

    2.998624816, 3.340994655, 3.651205427, 3.936894249, 4.203131772,

    4.453447651, 4.690410142, 4.915958381, 5.131600719]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 4.68055, 6.82350, 8.99175, 11.1622,

    13.3313, 15.4991, 17.6663, 19.8332, 21.9999, 24.1666, 26.3333]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.2984641944,

    -0.2145562449, -0.1562852133, -0.1175696904, -0.09151867201, -0.07346654949,

    -0.06051546079, -0.05092038064, -0.04360275106, -0.03787650345,

    -0.03330079360]

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

    2.361460045, 2.376710441, 2.422550418, 2.475570616, 2.525630258,

    2.569796859, 2.607833418, 2.640408356, 2.668412688, 2.692592263,

    2.713634494]

                                end of this data



For the equivalence class of patterns, {{[1, 3, 2], [4, 3, 2, 1, 5]},

    {[2, 3, 1], [5, 1, 2, 3, 4]}, {[3, 1, 2], [2, 3, 4, 5, 1]},

    {[2, 1, 3], [1, 5, 4, 3, 2]}}

      the member , {[1, 3, 2], [4, 3, 2, 1, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

    [[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}],

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

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

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

                 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+6*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+8*q^6+9*q^7+11*
q^8+12*q^9+15*q^10+15*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+10*q^6+11*q^7+13*q^8+14*q^9+18*q^10+21*q^11+22*q^12+26*q^13+33*q^14+34*q^
15+36*q^16+32*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+10*
q^6+13*q^7+15*q^8+16*q^9+20*q^10+21*q^11+27*q^12+29*q^13+37*q^14+35*q^15+43*q^
16+51*q^17+64*q^18+70*q^19+80*q^20+85*q^21+84*q^22+77*q^23+61*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+10*q^6+13*q^7+17*q^8+18*q^9+22*q^
10+23*q^11+27*q^12+31*q^13+38*q^14+40*q^15+49*q^16+48*q^17+61*q^18+63*q^19+76*q
^20+85*q^21+111*q^22+117*q^23+146*q^24+169*q^25+190*q^26+198*q^27+210*q^28+203*
q^29+185*q^30+151*q^31+107*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+10*q^6+13*q^7+17*q^8+20*q^9+24*q^10+25*q^11+29*q^12+31*q^13+40*q^14+
38*q^15+52*q^16+53*q^17+63*q^18+64*q^19+76*q^20+76*q^21+97*q^22+108*q^23+119*q^
24+132*q^25+156*q^26+188*q^27+218*q^28+258*q^29+311*q^30+361*q^31+399*q^32+447*
q^33+489*q^34+510*q^35+513*q^36+501*q^37+449*q^38+374*q^39+276*q^40+176*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+10*q^6+13*q^7+17*q^8+20*
q^9+26*q^10+27*q^11+31*q^12+33*q^13+40*q^14+40*q^15+50*q^16+53*q^17+66*q^18+65*
q^19+80*q^20+79*q^21+95*q^22+93*q^23+121*q^24+111*q^25+153*q^26+152*q^27+183*q^
28+186*q^29+225*q^30+243*q^31+309*q^32+350*q^33+420*q^34+475*q^35+583*q^36+675*
q^37+783*q^38+880*q^39+1004*q^40+1086*q^41+1174*q^42+1249*q^43+1301*q^44+1289*q
^45+1228*q^46+1111*q^47+926*q^48+707*q^49+476*q^50+275*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+10*q^6+13*q^7+17*q^8+20*q^9+26*q^10+29*
q^11+33*q^12+35*q^13+42*q^14+40*q^15+52*q^16+51*q^17+66*q^18+65*q^19+79*q^20+82
*q^21+101*q^22+92*q^23+113*q^24+116*q^25+133*q^26+143*q^27+164*q^28+179*q^29+
202*q^30+218*q^31+235*q^32+272*q^33+301*q^34+337*q^35+376*q^36+453*q^37+510*q^
38+599*q^39+690*q^40+811*q^41+957*q^42+1154*q^43+1328*q^44+1529*q^45+1743*q^46+
1987*q^47+2213*q^48+2447*q^49+2677*q^50+2912*q^51+3083*q^52+3232*q^53+3298*q^54
+3248*q^55+3059*q^56+2748*q^57+2317*q^58+1798*q^59+1265*q^60+782*q^61+412*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+10*q^6+13*q^7+17*q^8
+20*q^9+26*q^10+29*q^11+35*q^12+37*q^13+44*q^14+42*q^15+52*q^16+53*q^17+64*q^18
+65*q^19+79*q^20+78*q^21+102*q^22+97*q^23+115*q^24+109*q^25+143*q^26+126*q^27+
162*q^28+154*q^29+199*q^30+187*q^31+236*q^32+233*q^33+289*q^34+264*q^35+337*q^
36+328*q^37+406*q^38+433*q^39+505*q^40+535*q^41+631*q^42+679*q^43+806*q^44+912*
q^45+1087*q^46+1218*q^47+1474*q^48+1711*q^49+2017*q^50+2309*q^51+2698*q^52+3073
*q^53+3554*q^54+4009*q^55+4494*q^56+4977*q^57+5540*q^58+6089*q^59+6658*q^60+
7208*q^61+7726*q^62+8133*q^63+8354*q^64+8399*q^65+8191*q^66+7671*q^67+6857*q^68
+5805*q^69+4581*q^70+3316*q^71+2161*q^72+1233*q^73+596*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+10*q^6+13*q^7+17*q^8+20*q^9+26*q^10+29*
q^11+35*q^12+39*q^13+46*q^14+44*q^15+54*q^16+53*q^17+66*q^18+63*q^19+79*q^20+78
*q^21+98*q^22+95*q^23+118*q^24+110*q^25+139*q^26+137*q^27+150*q^28+153*q^29+181
*q^30+184*q^31+211*q^32+224*q^33+246*q^34+271*q^35+299*q^36+313*q^37+337*q^38+
389*q^39+394*q^40+444*q^41+493*q^42+560*q^43+596*q^44+680*q^45+748*q^46+838*q^
47+891*q^48+1039*q^49+1144*q^50+1354*q^51+1536*q^52+1766*q^53+2025*q^54+2367*q^
55+2702*q^56+3196*q^57+3691*q^58+4287*q^59+4902*q^60+5673*q^61+6439*q^62+7299*q
^63+8176*q^64+9186*q^65+10248*q^66+11446*q^67+12640*q^68+13915*q^69+15290*q^70+
16732*q^71+18132*q^72+19412*q^73+20514*q^74+21284*q^75+21637*q^76+21463*q^77+
20722*q^78+19303*q^79+17220*q^80+14606*q^81+11665*q^82+8643*q^83+5851*q^84+3548
*q^85+1877*q^86+837*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+10*q^6+13*q^7+17*q^8+20*q^9+26*q^10+29*q^11+35*q^12+39*q^13+48*q^14+46*q^
15+56*q^16+55*q^17+66*q^18+65*q^19+77*q^20+78*q^21+98*q^22+91*q^23+116*q^24+110
*q^25+138*q^26+132*q^27+164*q^28+142*q^29+185*q^30+167*q^31+213*q^32+199*q^33+
249*q^34+220*q^35+302*q^36+274*q^37+343*q^38+320*q^39+401*q^40+374*q^41+454*q^
42+436*q^43+534*q^44+532*q^45+631*q^46+643*q^47+773*q^48+778*q^49+905*q^50+931*
q^51+1090*q^52+1162*q^53+1318*q^54+1422*q^55+1660*q^56+1822*q^57+2097*q^58+2323
*q^59+2721*q^60+3012*q^61+3498*q^62+3983*q^63+4664*q^64+5384*q^65+6253*q^66+
7103*q^67+8195*q^68+9286*q^69+10622*q^70+12021*q^71+13622*q^72+15248*q^73+17138
*q^74+19134*q^75+21336*q^76+23649*q^77+26230*q^78+29034*q^79+32214*q^80+35504*q
^81+38926*q^82+42403*q^83+45953*q^84+49192*q^85+52054*q^86+54235*q^87+55655*q^
88+55968*q^89+55011*q^90+52610*q^91+48746*q^92+43410*q^93+36913*q^94+29726*q^95
+22434*q^96+15648*q^97+9935*q^98+5628*q^99+2772*q^100+1146*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+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, q^10+q^9+2*q^8+3*q^7+5
*q^6+5*q^5+6*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+8*q^9
+9*q^8+11*q^7+12*q^6+15*q^5+15*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+10*q^15+11*q^14+13*q^13+14*q^12+18*q^11+21*q^10+22*q^9+26*q^8+33
*q^7+34*q^6+36*q^5+32*q^4+25*q^3+15*q^2+6*q+1, 1+3*q^25+10*q^22+5*q^24+13*q^21+
7*q^23+7*q+21*q^2+2*q^26+41*q^3+61*q^4+77*q^5+84*q^6+85*q^7+80*q^8+70*q^9+64*q^
10+51*q^11+43*q^12+35*q^13+37*q^14+29*q^15+27*q^16+q^28+21*q^17+20*q^18+16*q^19
+15*q^20+q^27, 1+23*q^25+38*q^22+27*q^24+40*q^21+31*q^23+8*q+10*q^30+13*q^29+2*
q^34+3*q^33+28*q^2+22*q^26+63*q^3+107*q^4+151*q^5+185*q^6+5*q^32+203*q^7+210*q^
8+198*q^9+190*q^10+169*q^11+146*q^12+117*q^13+111*q^14+85*q^15+76*q^16+17*q^28+
63*q^17+61*q^18+48*q^19+49*q^20+18*q^27+7*q^31+q^35+q^36, 1+76*q^25+3*q^42+108*
q^22+76*q^24+119*q^21+97*q^23+9*q+q^44+38*q^30+q^45+17*q^37+52*q^29+10*q^39+5*q
^41+25*q^34+13*q^38+29*q^33+36*q^2+64*q^26+92*q^3+176*q^4+276*q^5+374*q^6+31*q^
32+449*q^7+501*q^8+513*q^9+510*q^10+489*q^11+447*q^12+399*q^13+361*q^14+311*q^
15+258*q^16+2*q^43+53*q^28+218*q^17+188*q^18+156*q^19+132*q^20+63*q^27+7*q^40+
40*q^31+24*q^35+20*q^36, 1+225*q^25+33*q^42+350*q^22+243*q^24+420*q^21+309*q^23
+10*q+5*q^51+27*q^44+2*q^53+111*q^30+3*q^52+q^55+q^54+26*q^45+66*q^37+153*q^29+
20*q^46+50*q^39+40*q^41+79*q^34+7*q^50+53*q^38+95*q^33+45*q^2+186*q^26+129*q^3+
275*q^4+476*q^5+707*q^6+93*q^32+926*q^7+1111*q^8+1228*q^9+1289*q^10+1301*q^11+
1249*q^12+1174*q^13+1086*q^14+1004*q^15+880*q^16+31*q^43+10*q^49+152*q^28+783*q
^17+675*q^18+583*q^19+475*q^20+183*q^27+40*q^40+121*q^31+80*q^35+13*q^48+65*q^
36+17*q^47, 1+811*q^25+113*q^42+1328*q^22+957*q^24+1529*q^21+1154*q^23+11*q+40*
q^51+101*q^44+35*q^53+20*q^57+376*q^30+42*q^52+17*q^58+29*q^55+10*q^60+33*q^54+
82*q^45+179*q^37+453*q^29+79*q^46+q^65+143*q^39+7*q^61+116*q^41+235*q^34+52*q^
50+2*q^64+164*q^38+272*q^33+26*q^56+55*q^2+690*q^26+175*q^3+412*q^4+782*q^5+
1265*q^6+301*q^32+1798*q^7+2317*q^8+2748*q^9+5*q^62+3059*q^10+3248*q^11+3298*q^
12+3232*q^13+3083*q^14+2912*q^15+2677*q^16+q^66+92*q^43+51*q^49+510*q^28+2447*q
^17+2213*q^18+1987*q^19+1743*q^20+599*q^27+133*q^40+3*q^63+337*q^31+218*q^35+66
*q^48+13*q^59+202*q^36+65*q^47, 1+3073*q^25+337*q^42+4494*q^22+3554*q^24+4977*q
^21+4009*q^23+12*q+126*q^51+289*q^44+109*q^53+78*q^57+1474*q^30+143*q^52+79*q^
58+97*q^55+64*q^60+115*q^54+233*q^45+13*q^71+5*q^74+535*q^37+29*q^67+1711*q^29+
236*q^46+37*q^65+20*q^69+433*q^39+53*q^61+328*q^41+806*q^34+162*q^50+44*q^64+
505*q^38+912*q^33+102*q^56+66*q^2+26*q^68+2698*q^26+231*q^3+596*q^4+1233*q^5+
2161*q^6+1087*q^32+3316*q^7+4581*q^8+5805*q^9+52*q^62+6857*q^10+7671*q^11+8191*
q^12+2*q^76+8399*q^13+8354*q^14+8133*q^15+7726*q^16+35*q^66+17*q^70+q^77+q^78+
264*q^43+154*q^49+2017*q^28+7208*q^17+6658*q^18+6089*q^19+5540*q^20+2309*q^27+
10*q^72+406*q^40+7*q^73+42*q^63+1218*q^31+3*q^75+679*q^35+199*q^48+65*q^59+631*
q^36+187*q^47, 1+10248*q^25+1039*q^42+13915*q^22+q^90+11446*q^24+15290*q^21+
12640*q^23+13*q+394*q^51+838*q^44+337*q^53+246*q^57+5673*q^30+389*q^52+224*q^58
+299*q^55+184*q^60+q^91+313*q^54+748*q^45+79*q^71+53*q^74+2025*q^37+118*q^67+
6439*q^29+680*q^46+139*q^65+98*q^69+1536*q^39+181*q^61+1144*q^41+3196*q^34+35*q
^79+444*q^50+137*q^64+1766*q^38+3691*q^33+271*q^56+78*q^2+95*q^68+9186*q^26+298
*q^3+837*q^4+1877*q^5+3548*q^6+4287*q^32+5851*q^7+8643*q^8+11665*q^9+153*q^62+
14606*q^10+17220*q^11+19303*q^12+44*q^76+20722*q^13+21463*q^14+21637*q^15+21284
*q^16+110*q^66+78*q^70+46*q^77+39*q^78+891*q^43+493*q^49+7299*q^28+20514*q^17+
19412*q^18+18132*q^19+16732*q^20+8176*q^27+63*q^72+1354*q^40+66*q^73+150*q^63+
4902*q^31+54*q^75+2702*q^35+29*q^80+560*q^48+17*q^83+26*q^81+10*q^85+7*q^86+5*q
^87+3*q^88+2*q^89+13*q^84+20*q^82+211*q^59+2367*q^36+596*q^47, 1+32214*q^25+
3983*q^42+42403*q^22+46*q^90+35504*q^24+45953*q^21+38926*q^23+14*q+17*q^97+1318
*q^51+3012*q^44+1090*q^53+773*q^57+19134*q^30+29*q^94+1162*q^52+643*q^58+905*q^
55+532*q^60+48*q^91+931*q^54+2721*q^45+249*q^71+167*q^74+20*q^96+13*q^98+8195*q
^37+343*q^67+21336*q^29+2323*q^46+35*q^93+10*q^99+401*q^65+302*q^69+6253*q^39+
534*q^61+4664*q^41+12021*q^34+138*q^79+1422*q^50+374*q^64+7*q^100+7103*q^38+
13622*q^33+778*q^56+91*q^2+274*q^68+29034*q^26+377*q^3+1146*q^4+2772*q^5+5628*q
^6+15248*q^32+9935*q^7+15648*q^8+22434*q^9+26*q^95+436*q^62+5*q^101+29726*q^10+
36913*q^11+43410*q^12+142*q^76+48746*q^13+52610*q^14+55011*q^15+55968*q^16+320*
q^66+220*q^70+164*q^77+132*q^78+3498*q^43+3*q^102+1660*q^49+23649*q^28+55655*q^
17+54235*q^18+52054*q^19+49192*q^20+26230*q^27+199*q^72+5384*q^40+213*q^73+2*q^
103+454*q^63+17138*q^31+185*q^75+10622*q^35+q^104+110*q^80+1822*q^48+q^105+98*q
^83+116*q^81+77*q^85+65*q^86+66*q^87+55*q^88+56*q^89+78*q^84+91*q^82+631*q^59+
9286*q^36+39*q^92+2097*q^47]


 The number of permutations avoiding, {[1, 3, 2], [4, 3, 2, 1, 5]}, is given by

[1, 2, 5, 14, 41, 119, 336, 927, 2527, 6870, 18717, 51155, 140120, 384147,

    1053147]

The number of EVEN permutations avoiding, {[1, 3, 2], [4, 3, 2, 1, 5]},

     is given by

 [1, 1, 3, 7, 21, 61, 171, 471, 1286, 3447, 9448, 25566, 70321, 191964, 527297]

The number of ODD permutations avoiding, {[1, 3, 2], [4, 3, 2, 1, 5]},

     is given by

 [0, 1, 2, 7, 20, 58, 165, 456, 1241, 3423, 9269, 25589, 69799, 192183, 525850]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 58, 165, 471, 1286, 3423, 9269, 25566, 70321, 192183,

    525850]

 ODD:, [0, 1, 3, 7, 20, 61, 171, 456, 1241, 3447, 9448, 25589, 69799, 191964,

    527297]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.804878049, 9.025210084,

    13.14583333, 18.28694714, 24.52196280, 31.86331878, 40.27712774,

    49.71547258, 60.14354125, 71.54850877, 83.93387248]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.350085750, 3.213579864,

    4.201705354, 5.241901553, 6.239531195, 7.115134005, 7.834014066,

    8.411468617, 8.893102255, 9.327380144, 9.747311232]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.52290, 10.3271, 17.6543, 27.4775,

    38.9317, 50.6251, 61.3718, 70.7528, 79.0873, 87.0000, 95.0101]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.4418810426,

    -0.5180587787, -0.6481844998, -0.8228319761, -1.023249238, -1.227733963,

    -1.411402765, -1.550055189, -1.628705460, -1.648662615, -1.626305121]

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

    2.565707530, 2.650545386, 2.824281553, 3.179667079, 3.754361256,

    4.547191303, 5.498529644, 6.477021065, 7.310000725, 7.859743691,

    8.090346127]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [4, 1, 2, 5, 3]},

    {[3, 2, 1], [2, 5, 1, 3, 4]}, {[3, 2, 1], [3, 1, 4, 5, 2]},

    {[3, 2, 1], [2, 3, 5, 1, 4]}, {[1, 2, 3], [2, 5, 4, 1, 3]},

    {[1, 2, 3], [3, 5, 2, 1, 4]}, {[1, 2, 3], [4, 3, 1, 5, 2]},

    {[1, 2, 3], [4, 1, 5, 3, 2]}}

      the member , {[3, 2, 1], [4, 1, 2, 5, 3]}, has a scheme of depth , 5

                                  here it is:

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

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

    [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {2}], [[3, 1, 5, 2, 4], {

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

    [[4, 1, 5, 2, 3],

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

    [[2, 1, 3], {[1, 0, 0, 0]}, {}], [[3, 1, 4, 2, 5],

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

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

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

    [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 0, 0]}, {3}],

    [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 0, 0]}, {3}], [[2, 1, 4, 3], %1, {1, 2}],

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

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

%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, 1+3*q+5*q^2+4*q^3+q^4, 1+4*q+9*q^2+12*q^3+9*q^4+4*q^5+2*q
^6, 1+5*q+14*q^2+25*q^3+29*q^4+22*q^5+13*q^6+8*q^7+4*q^8+q^9, 1+6*q+20*q^2+44*q
^3+67*q^4+72*q^5+56*q^6+39*q^7+26*q^8+17*q^9+10*q^10+4*q^11+2*q^12, 1+7*q+27*q^
2+70*q^3+130*q^4+178*q^5+182*q^6+149*q^7+110*q^8+80*q^9+56*q^10+39*q^11+26*q^12
+15*q^13+8*q^14+4*q^15+q^16, 1+8*q+35*q^2+104*q^3+226*q^4+372*q^5+471*q^6+473*q
^7+399*q^8+313*q^9+232*q^10+173*q^11+127*q^12+95*q^13+68*q^14+46*q^15+31*q^16+
18*q^17+10*q^18+4*q^19+2*q^20, 1+9*q+44*q^2+147*q^3+364*q^4+695*q^5+1043*q^6+
1255*q^7+1247*q^8+1087*q^9+877*q^10+679*q^11+511*q^12+391*q^13+301*q^14+233*q^
15+179*q^16+133*q^17+97*q^18+68*q^19+45*q^20+28*q^21+15*q^22+8*q^23+4*q^24+q^25
, 1+10*q+54*q^2+200*q^3+554*q^4+1198*q^5+2068*q^6+2899*q^7+3363*q^8+3340*q^9+
2971*q^10+2472*q^11+1958*q^12+1523*q^13+1170*q^14+919*q^15+725*q^16+583*q^17+
464*q^18+365*q^19+285*q^20+218*q^21+161*q^22+114*q^23+80*q^24+51*q^25+32*q^26+
18*q^27+10*q^28+4*q^29+2*q^30, 1+11*q+65*q^2+264*q^3+807*q^4+1943*q^5+3777*q^6+
6029*q^7+8021*q^8+9084*q^9+9028*q^10+8179*q^11+6948*q^12+5660*q^13+4485*q^14+
3532*q^15+2773*q^16+2213*q^17+1789*q^18+1460*q^19+1196*q^20+978*q^21+796*q^22+
638*q^23+506*q^24+395*q^25+299*q^26+221*q^27+159*q^28+111*q^29+74*q^30+47*q^31+
28*q^32+15*q^33+8*q^34+4*q^35+q^36, 1+12*q+77*q^2+340*q^3+1135*q^4+3004*q^5+
6474*q^6+11561*q^7+17340*q^8+22179*q^9+24693*q^10+24607*q^11+22594*q^12+19575*q
^13+16274*q^14+13210*q^15+10566*q^16+8442*q^17+6746*q^18+5468*q^19+4466*q^20+
3699*q^21+3081*q^22+2576*q^23+2147*q^24+1790*q^25+1479*q^26+1208*q^27+976*q^28+
777*q^29+610*q^30+467*q^31+352*q^32+257*q^33+184*q^34+126*q^35+85*q^36+52*q^37+
32*q^38+18*q^39+10*q^40+4*q^41+2*q^42, 1+13*q+90*q^2+429*q^3+1551*q^4+4468*q^5+
10549*q^6+20788*q^7+34648*q^8+49459*q^9+61372*q^10+67525*q^11+67464*q^12+62673*
q^13+55163*q^14+46757*q^15+38656*q^16+31545*q^17+25540*q^18+20698*q^19+16796*q^
20+13748*q^21+11353*q^22+9473*q^23+7964*q^24+6735*q^25+5727*q^26+4867*q^27+4122
*q^28+3480*q^29+2923*q^30+2434*q^31+2007*q^32+1637*q^33+1317*q^34+1048*q^35+821
*q^36+628*q^37+471*q^38+347*q^39+249*q^40+173*q^41+117*q^42+76*q^43+47*q^44+28*
q^45+15*q^46+8*q^47+4*q^48+q^49, 1+14*q+104*q^2+532*q^3+2069*q^4+6436*q^5+16492
*q^6+35479*q^7+64944*q^8+102306*q^9+140343*q^10+170117*q^11+185558*q^12+185901*
q^13+174379*q^14+155662*q^15+134074*q^16+112794*q^17+93515*q^18+76987*q^19+
63136*q^20+51820*q^21+42575*q^22+35209*q^23+29289*q^24+24567*q^25+20761*q^26+
17706*q^27+15163*q^28+13041*q^29+11229*q^30+9674*q^31+8309*q^32+7116*q^33+6060*
q^34+5133*q^35+4322*q^36+3606*q^37+2978*q^38+2432*q^39+1969*q^40+1569*q^41+1233
*q^42+952*q^43+725*q^44+538*q^45+395*q^46+280*q^47+196*q^48+131*q^49+86*q^50+52
*q^51+32*q^52+18*q^53+10*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+9*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+25*q^6+29*q^5+22*q^4+13*q^3+8*q^2+4*q+1),
q^9*(q^12+6*q^11+20*q^10+44*q^9+67*q^8+72*q^7+56*q^6+39*q^5+26*q^4+17*q^3+10*q^
2+4*q+2), q^12*(q^16+7*q^15+27*q^14+70*q^13+130*q^12+178*q^11+182*q^10+149*q^9+
110*q^8+80*q^7+56*q^6+39*q^5+26*q^4+15*q^3+8*q^2+4*q+1), q^16*(q^20+8*q^19+35*q
^18+104*q^17+226*q^16+372*q^15+471*q^14+473*q^13+399*q^12+313*q^11+232*q^10+173
*q^9+127*q^8+95*q^7+68*q^6+46*q^5+31*q^4+18*q^3+10*q^2+4*q+2), q^20*(1+q^25+147
*q^22+9*q^24+364*q^21+44*q^23+4*q+8*q^2+15*q^3+28*q^4+45*q^5+68*q^6+97*q^7+133*
q^8+179*q^9+233*q^10+301*q^11+391*q^12+511*q^13+679*q^14+877*q^15+1087*q^16+
1247*q^17+1255*q^18+1043*q^19+695*q^20), q^25*(2+1198*q^25+3363*q^22+2068*q^24+
3340*q^21+2899*q^23+4*q+q^30+10*q^29+10*q^2+554*q^26+18*q^3+32*q^4+51*q^5+80*q^
6+114*q^7+161*q^8+218*q^9+285*q^10+365*q^11+464*q^12+583*q^13+725*q^14+919*q^15
+1170*q^16+54*q^28+1523*q^17+1958*q^18+2472*q^19+2971*q^20+200*q^27), q^30*(1+
8179*q^25+4485*q^22+6948*q^24+3532*q^21+5660*q^23+4*q+3777*q^30+6029*q^29+65*q^
34+264*q^33+8*q^2+9028*q^26+15*q^3+28*q^4+47*q^5+74*q^6+807*q^32+111*q^7+159*q^
8+221*q^9+299*q^10+395*q^11+506*q^12+638*q^13+796*q^14+978*q^15+1196*q^16+8021*
q^28+1460*q^17+1789*q^18+2213*q^19+2773*q^20+9084*q^27+1943*q^31+11*q^35+q^36),
q^36*(2+8442*q^25+q^42+4466*q^22+6746*q^24+3699*q^21+5468*q^23+4*q+22594*q^30+
3004*q^37+19575*q^29+340*q^39+12*q^41+17340*q^34+1135*q^38+22179*q^33+10*q^2+
10566*q^26+18*q^3+32*q^4+52*q^5+85*q^6+24693*q^32+126*q^7+184*q^8+257*q^9+352*q
^10+467*q^11+610*q^12+777*q^13+976*q^14+1208*q^15+1479*q^16+16274*q^28+1790*q^
17+2147*q^18+2576*q^19+3081*q^20+13210*q^27+77*q^40+24607*q^31+11561*q^35+6474*
q^36), q^42*(1+7964*q^25+20788*q^42+4867*q^22+6735*q^24+4122*q^21+5727*q^23+4*q
+4468*q^44+20698*q^30+1551*q^45+67464*q^37+16796*q^29+429*q^46+61372*q^39+34648
*q^41+46757*q^34+67525*q^38+38656*q^33+8*q^2+9473*q^26+15*q^3+28*q^4+47*q^5+76*
q^6+31545*q^32+117*q^7+173*q^8+249*q^9+347*q^10+471*q^11+628*q^12+821*q^13+1048
*q^14+1317*q^15+1637*q^16+10549*q^43+q^49+13748*q^28+2007*q^17+2434*q^18+2923*q
^19+3480*q^20+11353*q^27+49459*q^40+25540*q^31+55163*q^35+13*q^48+62673*q^36+90
*q^47), q^49*(2+9674*q^25+174379*q^42+6060*q^22+8309*q^24+5133*q^21+7116*q^23+4
*q+6436*q^51+185558*q^44+532*q^53+20761*q^30+2069*q^52+14*q^55+104*q^54+170117*
q^45+76987*q^37+17706*q^29+140343*q^46+112794*q^39+155662*q^41+42575*q^34+16492
*q^50+93515*q^38+35209*q^33+q^56+10*q^2+11229*q^26+18*q^3+32*q^4+52*q^5+86*q^6+
29289*q^32+131*q^7+196*q^8+280*q^9+395*q^10+538*q^11+725*q^12+952*q^13+1233*q^
14+1569*q^15+1969*q^16+185901*q^43+35479*q^49+15163*q^28+2432*q^17+2978*q^18+
3606*q^19+4322*q^20+13041*q^27+134074*q^40+24567*q^31+51820*q^35+64944*q^48+
63136*q^36+102306*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [4, 1, 2, 5, 3]}, is given by

[1, 2, 5, 14, 41, 122, 364, 1083, 3208, 9462, 27812, 81545, 238696, 698005,

    2040025]

The number of EVEN permutations avoiding, {[3, 2, 1], [4, 1, 2, 5, 3]},

     is given by

[1, 1, 3, 7, 21, 61, 182, 541, 1602, 4728, 13898, 40761, 119324, 348967,

    1019949]

The number of ODD permutations avoiding, {[3, 2, 1], [4, 1, 2, 5, 3]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 542, 1606, 4734, 13914, 40784, 119372, 349038,

    1020076]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 541, 1602, 4734, 13914, 40761, 119324, 349038,

    1020076]

 ODD:, [0, 1, 3, 7, 20, 61, 182, 542, 1606, 4728, 13898, 40784, 119372, 348967,

    1019949]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.073170732, 4.172131148,

    5.340659341, 6.555863343, 7.799875312, 9.059395477, 10.32503955,

    11.59067999, 12.85279184, 14.10980151, 15.36146959]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.368464404, 1.749438424,

    2.168555402, 2.618100619, 3.089960412, 3.575773655, 4.067280964,

    4.556846531, 5.037961018, 5.505581310, 5.956250791]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.87269, 3.06053, 4.70263, 6.85445,

    9.54786, 12.7862, 16.5428, 20.7649, 25.3811, 30.3114, 35.4769]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.09640199491,

    0.2963241274, 0.4650354398, 0.6137814432, 0.7494716565, 0.8758583791,

    0.9947852962, 1.106955705, 1.212411611, 1.310772001, 1.401467322]

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

    2.667632108, 2.832832525, 3.028284916, 3.261294914, 3.531200550,

    3.837163467, 4.178295397, 4.552588524, 4.956564459, 5.384863380,

    5.830215452]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [4, 1, 2, 3, 5]},

    {[3, 2, 1], [2, 3, 4, 1, 5]}, {[3, 2, 1], [1, 5, 2, 3, 4]},

    {[1, 2, 3], [2, 5, 4, 3, 1]}, {[1, 2, 3], [4, 3, 2, 5, 1]},

    {[3, 2, 1], [1, 3, 4, 5, 2]}, {[1, 2, 3], [5, 3, 2, 1, 4]},

    {[1, 2, 3], [5, 1, 4, 3, 2]}}

      the member , {[3, 2, 1], [2, 3, 4, 1, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

                 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+11*q^3+10*q^4+4*q^5+2*
q^6, 1+5*q+14*q^2+23*q^3+27*q^4+22*q^5+14*q^6+8*q^7+4*q^8+q^9, 1+6*q+20*q^2+41*
q^3+60*q^4+63*q^5+54*q^6+35*q^7+26*q^8+15*q^9+10*q^10+3*q^11+2*q^12, 1+7*q+27*q
^2+66*q^3+116*q^4+150*q^5+154*q^6+127*q^7+94*q^8+64*q^9+47*q^10+30*q^11+20*q^12
+11*q^13+6*q^14+3*q^15+q^16, 1+8*q+35*q^2+99*q^3+203*q^4+312*q^5+379*q^6+371*q^
7+314*q^8+232*q^9+172*q^10+117*q^11+87*q^12+56*q^13+42*q^14+24*q^15+19*q^16+10*
q^17+7*q^18+2*q^19+2*q^20, 1+9*q+44*q^2+141*q^3+330*q^4+587*q^5+828*q^6+946*q^7
+914*q^8+765*q^9+592*q^10+429*q^11+314*q^12+216*q^13+160*q^14+108*q^15+80*q^16+
54*q^17+41*q^18+28*q^19+20*q^20+13*q^21+8*q^22+4*q^23+3*q^24+q^25, 1+10*q+54*q^
2+193*q^3+507*q^4+1023*q^5+1646*q^6+2158*q^7+2376*q^8+2246*q^9+1902*q^10+1473*q
^11+1105*q^12+789*q^13+576*q^14+400*q^15+298*q^16+206*q^17+158*q^18+109*q^19+87
*q^20+61*q^21+51*q^22+32*q^23+27*q^24+17*q^25+14*q^26+7*q^27+6*q^28+2*q^29+2*q^
30, 1+11*q+65*q^2+256*q^3+745*q^4+1679*q^5+3035*q^6+4497*q^7+5593*q^8+5952*q^9+
5578*q^10+4709*q^11+3716*q^12+2777*q^13+2043*q^14+1454*q^15+1061*q^16+748*q^17+
560*q^18+400*q^19+307*q^20+222*q^21+177*q^22+132*q^23+105*q^24+78*q^25+64*q^26+
46*q^27+37*q^28+27*q^29+22*q^30+15*q^31+11*q^32+6*q^33+4*q^34+3*q^35+q^36, 1+12
*q+77*q^2+331*q^3+1056*q^4+2626*q^5+5266*q^6+8704*q^7+12102*q^8+14396*q^9+14965
*q^10+13863*q^11+11759*q^12+9301*q^13+7065*q^14+5171*q^15+3772*q^16+2690*q^17+
1973*q^18+1414*q^19+1068*q^20+778*q^21+605*q^22+450*q^23+360*q^24+270*q^25+224*
q^26+169*q^27+141*q^28+104*q^29+90*q^30+66*q^31+59*q^32+42*q^33+36*q^34+25*q^35
+23*q^36+14*q^37+11*q^38+6*q^39+6*q^40+2*q^41+2*q^42, 1+13*q+90*q^2+419*q^3+
1453*q^4+3948*q^5+8692*q^6+15851*q^7+24393*q^8+32167*q^9+36966*q^10+37631*q^11+
34635*q^12+29347*q^13+23417*q^14+17823*q^15+13233*q^16+9593*q^17+6984*q^18+5020
*q^19+3711*q^20+2705*q^21+2058*q^22+1534*q^23+1201*q^24+910*q^25+729*q^26+565*q
^27+463*q^28+359*q^29+298*q^30+234*q^31+197*q^32+154*q^33+131*q^34+101*q^35+89*
q^36+71*q^37+59*q^38+44*q^39+38*q^40+29*q^41+24*q^42+18*q^43+13*q^44+9*q^45+6*q
^46+4*q^47+3*q^48+q^49, 1+14*q+104*q^2+521*q^3+1950*q^4+5743*q^5+13762*q^6+
27435*q^7+46310*q^8+67145*q^9+84822*q^10+94662*q^11+94836*q^12+86648*q^13+73538
*q^14+58829*q^15+45174*q^16+33586*q^17+24642*q^18+17828*q^19+13029*q^20+9467*q^
21+7065*q^22+5234*q^23+4023*q^24+3048*q^25+2399*q^26+1851*q^27+1495*q^28+1170*q
^29+959*q^30+757*q^31+635*q^32+504*q^33+428*q^34+336*q^35+289*q^36+232*q^37+204
*q^38+160*q^39+139*q^40+111*q^41+99*q^42+76*q^43+69*q^44+51*q^45+46*q^46+34*q^
47+31*q^48+23*q^49+20*q^50+11*q^51+10*q^52+6*q^53+6*q^54+2*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+11*q^
3+10*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+23*q^6+27*q^5+22*q^4+14*q^3+8*q^2+4*q+1)
, q^9*(q^12+6*q^11+20*q^10+41*q^9+60*q^8+63*q^7+54*q^6+35*q^5+26*q^4+15*q^3+10*
q^2+3*q+2), q^12*(q^16+7*q^15+27*q^14+66*q^13+116*q^12+150*q^11+154*q^10+127*q^
9+94*q^8+64*q^7+47*q^6+30*q^5+20*q^4+11*q^3+6*q^2+3*q+1), q^16*(q^20+8*q^19+35*
q^18+99*q^17+203*q^16+312*q^15+379*q^14+371*q^13+314*q^12+232*q^11+172*q^10+117
*q^9+87*q^8+56*q^7+42*q^6+24*q^5+19*q^4+10*q^3+7*q^2+2*q+2), q^20*(1+q^25+141*q
^22+9*q^24+330*q^21+44*q^23+3*q+4*q^2+8*q^3+13*q^4+20*q^5+28*q^6+41*q^7+54*q^8+
80*q^9+108*q^10+160*q^11+216*q^12+314*q^13+429*q^14+592*q^15+765*q^16+914*q^17+
946*q^18+828*q^19+587*q^20), q^25*(2+1023*q^25+2376*q^22+1646*q^24+2246*q^21+
2158*q^23+2*q+q^30+10*q^29+6*q^2+507*q^26+7*q^3+14*q^4+17*q^5+27*q^6+32*q^7+51*
q^8+61*q^9+87*q^10+109*q^11+158*q^12+206*q^13+298*q^14+400*q^15+576*q^16+54*q^
28+789*q^17+1105*q^18+1473*q^19+1902*q^20+193*q^27), q^30*(1+4709*q^25+2043*q^
22+3716*q^24+1454*q^21+2777*q^23+3*q+3035*q^30+4497*q^29+65*q^34+256*q^33+4*q^2
+5578*q^26+6*q^3+11*q^4+15*q^5+22*q^6+745*q^32+27*q^7+37*q^8+46*q^9+64*q^10+78*
q^11+105*q^12+132*q^13+177*q^14+222*q^15+307*q^16+5593*q^28+400*q^17+560*q^18+
748*q^19+1061*q^20+5952*q^27+1679*q^31+11*q^35+q^36), q^36*(2+2690*q^25+q^42+
1068*q^22+1973*q^24+778*q^21+1414*q^23+2*q+11759*q^30+2626*q^37+9301*q^29+331*q
^39+12*q^41+12102*q^34+1056*q^38+14396*q^33+6*q^2+3772*q^26+6*q^3+11*q^4+14*q^5
+23*q^6+14965*q^32+25*q^7+36*q^8+42*q^9+59*q^10+66*q^11+90*q^12+104*q^13+141*q^
14+169*q^15+224*q^16+7065*q^28+270*q^17+360*q^18+450*q^19+605*q^20+5171*q^27+77
*q^40+13863*q^31+8704*q^35+5266*q^36), q^42*(1+1201*q^25+15851*q^42+565*q^22+
910*q^24+463*q^21+729*q^23+3*q+3948*q^44+5020*q^30+1453*q^45+34635*q^37+3711*q^
29+419*q^46+36966*q^39+24393*q^41+17823*q^34+37631*q^38+13233*q^33+4*q^2+1534*q
^26+6*q^3+9*q^4+13*q^5+18*q^6+9593*q^32+24*q^7+29*q^8+38*q^9+44*q^10+59*q^11+71
*q^12+89*q^13+101*q^14+131*q^15+154*q^16+8692*q^43+q^49+2705*q^28+197*q^17+234*
q^18+298*q^19+359*q^20+2058*q^27+32167*q^40+6984*q^31+23417*q^35+13*q^48+29347*
q^36+90*q^47), q^49*(2+757*q^25+73538*q^42+428*q^22+635*q^24+336*q^21+504*q^23+
2*q+5743*q^51+94836*q^44+521*q^53+2399*q^30+1950*q^52+14*q^55+104*q^54+94662*q^
45+17828*q^37+1851*q^29+84822*q^46+33586*q^39+58829*q^41+7065*q^34+13762*q^50+
24642*q^38+5234*q^33+q^56+6*q^2+959*q^26+6*q^3+10*q^4+11*q^5+20*q^6+4023*q^32+
23*q^7+31*q^8+34*q^9+46*q^10+51*q^11+69*q^12+76*q^13+99*q^14+111*q^15+139*q^16+
86648*q^43+27435*q^49+1495*q^28+160*q^17+204*q^18+232*q^19+289*q^20+1170*q^27+
45174*q^40+3048*q^31+9467*q^35+46310*q^48+13029*q^36+67145*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [2, 3, 4, 1, 5]}, is given by

[1, 2, 5, 14, 41, 119, 336, 924, 2492, 6636, 17536, 46137, 121095, 317434,

    831571]

The number of EVEN permutations avoiding, {[3, 2, 1], [2, 3, 4, 1, 5]},

     is given by

 [1, 1, 3, 7, 22, 60, 173, 466, 1261, 3335, 8810, 23125, 60661, 158884, 416087]

The number of ODD permutations avoiding, {[3, 2, 1], [2, 3, 4, 1, 5]},

     is given by

 [0, 1, 2, 7, 19, 59, 163, 458, 1231, 3301, 8726, 23012, 60434, 158550, 415484]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 59, 163, 466, 1261, 3301, 8726, 23125, 60661, 158550,

    415484]

 ODD:, [0, 1, 3, 7, 19, 60, 173, 458, 1231, 3335, 8810, 23012, 60434, 158884,

    416087]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.097560976, 4.210084034,

    5.336309524, 6.430735931, 7.473113965, 8.460367691, 9.399178832,

    10.30019290, 11.17420207, 12.03021100, 12.87487899]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.375834612, 1.772212924,

    2.189665271, 2.600068075, 2.981429860, 3.318591045, 3.604410349,

    3.839245214, 4.028797415, 4.181443166, 4.305996334]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.89292, 3.14074, 4.79463, 6.76035,

    8.88892, 11.0130, 12.9918, 14.7398, 16.2312, 17.4845, 18.5416]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.04913712930,

    0.2420760298, 0.4265965959, 0.6090730884, 0.7895422344, 0.9619518858,

    1.118467485, 1.251938090, 1.357242928, 1.432078602, 1.477140820]

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

    2.610582899, 2.740614458, 2.961505474, 3.317467468, 3.824261667,

    4.471795956, 5.227770470, 6.041883939, 6.852029988, 7.594900929,

    8.217726963]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [5, 1, 2, 3, 4]},

    {[1, 2, 3], [1, 5, 4, 3, 2]}, {[1, 2, 3], [4, 3, 2, 1, 5]},

    {[3, 2, 1], [2, 3, 4, 5, 1]}}

      the member , {[3, 2, 1], [5, 1, 2, 3, 4]}, has a scheme of depth , 2

                                  here it is:

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

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

                 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+9*q^4+4*q^5+2*q
^6, 1+5*q+14*q^2+25*q^3+29*q^4+23*q^5+14*q^6+7*q^7+3*q^8+q^9, 1+6*q+20*q^2+44*q
^3+67*q^4+74*q^5+62*q^6+42*q^7+26*q^8+14*q^9+6*q^10+2*q^11+q^12, 1+7*q+27*q^2+
70*q^3+130*q^4+181*q^5+195*q^6+169*q^7+127*q^8+86*q^9+51*q^10+28*q^11+14*q^12+5
*q^13+2*q^14+q^15, 1+8*q+35*q^2+104*q^3+226*q^4+376*q^5+493*q^6+524*q^7+472*q^8
+376*q^9+270*q^10+178*q^11+108*q^12+58*q^13+30*q^14+14*q^15+5*q^16+2*q^17+q^18,
1+9*q+44*q^2+147*q^3+364*q^4+700*q^5+1076*q^6+1354*q^7+1437*q^8+1329*q^9+1102*q
^10+837*q^11+586*q^12+378*q^13+229*q^14+129*q^15+65*q^16+32*q^17+15*q^18+5*q^19
+2*q^20+q^21, 1+10*q+54*q^2+200*q^3+554*q^4+1204*q^5+2114*q^6+3066*q^7+3758*q^8
+3990*q^9+3764*q^10+3228*q^11+2550*q^12+1868*q^13+1282*q^14+824*q^15+494*q^16+
282*q^17+151*q^18+72*q^19+35*q^20+16*q^21+5*q^22+2*q^23+q^24, 1+11*q+65*q^2+264
*q^3+807*q^4+1950*q^5+3838*q^6+6287*q^7+8740*q^8+10516*q^9+11181*q^10+10725*q^
11+9433*q^12+7690*q^13+5862*q^14+4197*q^15+2829*q^16+1809*q^17+1093*q^18+620*q^
19+339*q^20+174*q^21+80*q^22+38*q^23+17*q^24+5*q^25+2*q^26+q^27, 1+12*q+77*q^2+
340*q^3+1135*q^4+3012*q^5+6552*q^6+11936*q^7+18537*q^8+24956*q^9+29628*q^10+
31570*q^11+30680*q^12+27542*q^13+23063*q^14+18132*q^15+13448*q^16+9456*q^17+
6311*q^18+4000*q^19+2425*q^20+1396*q^21+758*q^22+400*q^23+199*q^24+88*q^25+41*q
^26+18*q^27+5*q^28+2*q^29+q^30, 1+13*q+90*q^2+429*q^3+1551*q^4+4477*q^5+10646*q
^6+21309*q^7+36516*q^8+54382*q^9+71422*q^10+83964*q^11+89647*q^12+88059*q^13+
80427*q^14+68840*q^15+55551*q^16+42474*q^17+30866*q^18+21371*q^19+14145*q^20+
8943*q^21+5400*q^22+3137*q^23+1736*q^24+908*q^25+466*q^26+225*q^27+96*q^28+44*q
^29+19*q^30+5*q^31+2*q^32+q^33, 1+14*q+104*q^2+532*q^3+2069*q^4+6446*q^5+16610*
q^6+36178*q^7+67719*q^8+110468*q^9+159098*q^10+204922*q^11+239084*q^12+255742*q
^13+253502*q^14+234886*q^15+204833*q^16+169042*q^17+132565*q^18+99112*q^19+
70853*q^20+48498*q^21+31828*q^22+20078*q^23+12149*q^24+7048*q^25+3952*q^26+2114
*q^27+1071*q^28+536*q^29+252*q^30+104*q^31+47*q^32+20*q^33+5*q^34+2*q^35+q^36]
     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+9*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+25*q^6+29*q^5+23*q^4+14*q^3+7*q^2+3*q+1),
q^9*(q^12+6*q^11+20*q^10+44*q^9+67*q^8+74*q^7+62*q^6+42*q^5+26*q^4+14*q^3+6*q^2
+2*q+1), q^13*(q^15+7*q^14+27*q^13+70*q^12+130*q^11+181*q^10+195*q^9+169*q^8+
127*q^7+86*q^6+51*q^5+28*q^4+14*q^3+5*q^2+2*q+1), q^18*(q^18+8*q^17+35*q^16+104
*q^15+226*q^14+376*q^13+493*q^12+524*q^11+472*q^10+376*q^9+270*q^8+178*q^7+108*
q^6+58*q^5+30*q^4+14*q^3+5*q^2+2*q+1), q^24*(q^21+9*q^20+44*q^19+147*q^18+364*q
^17+700*q^16+1076*q^15+1354*q^14+1437*q^13+1329*q^12+1102*q^11+837*q^10+586*q^9
+378*q^8+229*q^7+129*q^6+65*q^5+32*q^4+15*q^3+5*q^2+2*q+1), q^31*(1+54*q^22+q^
24+200*q^21+10*q^23+2*q+5*q^2+16*q^3+35*q^4+72*q^5+151*q^6+282*q^7+494*q^8+824*
q^9+1282*q^10+1868*q^11+2550*q^12+3228*q^13+3764*q^14+3990*q^15+3758*q^16+3066*
q^17+2114*q^18+1204*q^19+554*q^20), q^39*(1+65*q^25+1950*q^22+264*q^24+3838*q^
21+807*q^23+2*q+5*q^2+11*q^26+17*q^3+38*q^4+80*q^5+174*q^6+339*q^7+620*q^8+1093
*q^9+1809*q^10+2829*q^11+4197*q^12+5862*q^13+7690*q^14+9433*q^15+10725*q^16+
11181*q^17+10516*q^18+8740*q^19+6287*q^20+q^27), q^48*(1+3012*q^25+18537*q^22+
6552*q^24+24956*q^21+11936*q^23+2*q+q^30+12*q^29+5*q^2+1135*q^26+18*q^3+41*q^4+
88*q^5+199*q^6+400*q^7+758*q^8+1396*q^9+2425*q^10+4000*q^11+6311*q^12+9456*q^13
+13448*q^14+18132*q^15+23063*q^16+77*q^28+27542*q^17+30680*q^18+31570*q^19+
29628*q^20+340*q^27), q^58*(1+36516*q^25+83964*q^22+54382*q^24+89647*q^21+71422
*q^23+2*q+429*q^30+1551*q^29+q^33+5*q^2+21309*q^26+19*q^3+44*q^4+96*q^5+225*q^6
+13*q^32+466*q^7+908*q^8+1736*q^9+3137*q^10+5400*q^11+8943*q^12+14145*q^13+
21371*q^14+30866*q^15+42474*q^16+4477*q^28+55551*q^17+68840*q^18+80427*q^19+
88059*q^20+10646*q^27+90*q^31), q^69*(1+204922*q^25+253502*q^22+239084*q^24+
234886*q^21+255742*q^23+2*q+16610*q^30+36178*q^29+104*q^34+532*q^33+5*q^2+
159098*q^26+20*q^3+47*q^4+104*q^5+252*q^6+2069*q^32+536*q^7+1071*q^8+2114*q^9+
3952*q^10+7048*q^11+12149*q^12+20078*q^13+31828*q^14+48498*q^15+70853*q^16+
67719*q^28+99112*q^17+132565*q^18+169042*q^19+204833*q^20+110468*q^27+6446*q^31
+14*q^35+q^36)]


 The number of permutations avoiding, {[3, 2, 1], [5, 1, 2, 3, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[3, 2, 1], [5, 1, 2, 3, 4]},

     is given by

[1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581,

    1195743]

The number of ODD permutations avoiding, {[3, 2, 1], [5, 1, 2, 3, 4]},

     is given by

[0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581,

    1195742]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 61, 182, 547, 1641, 4921, 14762, 44287, 132861, 398581,

    1195742]

 ODD:, [0, 1, 3, 7, 20, 61, 183, 547, 1640, 4921, 14763, 44287, 132860, 398581,

    1195743]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.073170732, 4.139344262,

    5.232876712, 6.337294333, 7.445900640, 8.556086161, 9.666858594,

    10.77784677, 11.88891356, 13.00000878, 14.11111422]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.368464404, 1.704937736,

    2.015831505, 2.297470667, 2.553221806, 2.787566753, 3.004446162,

    3.206981476, 3.397574095, 3.578069789, 3.749902877]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.87269, 2.90681, 4.06358, 5.27837,

    6.51894, 7.77053, 9.02670, 10.2847, 11.5435, 12.8026, 14.0618]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.09640199491,

    0.2587083709, 0.3385291301, 0.3729477885, 0.3848465162, 0.3856354209,

    0.3809479695, 0.3736281411, 0.3651591996, 0.3563171707, 0.3475108654]

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

    2.667632108, 2.899032055, 3.035024760, 3.100628524, 3.129616642,

    3.140517239, 3.142475138, 3.140040937, 3.135478419, 3.130021024,

    3.124304951]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [3, 5, 1, 2, 4]},

    {[3, 2, 1], [4, 1, 5, 2, 3]}, {[3, 2, 1], [2, 4, 5, 1, 3]},

    {[3, 2, 1], [3, 4, 1, 5, 2]}, {[1, 2, 3], [3, 2, 5, 1, 4]},

    {[1, 2, 3], [2, 5, 1, 4, 3]}, {[1, 2, 3], [4, 2, 1, 5, 3]},

    {[1, 2, 3], [3, 1, 5, 4, 2]}}

      the member , {[3, 2, 1], [3, 5, 1, 2, 4]}, has a scheme of depth , 2

                                  here it is:

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

    [[1, 2], {[2, 1, 0]}, {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+3*q^5+2*
q^6, 1+5*q+14*q^2+25*q^3+31*q^4+24*q^5+12*q^6+6*q^7+3*q^8+q^9, 1+6*q+20*q^2+44*
q^3+70*q^4+79*q^5+64*q^6+36*q^7+23*q^8+11*q^9+7*q^10+2*q^11+2*q^12, 1+7*q+27*q^
2+70*q^3+134*q^4+192*q^5+209*q^6+170*q^7+112*q^8+72*q^9+44*q^10+24*q^11+15*q^12
+9*q^13+4*q^14+3*q^15+q^16, 1+8*q+35*q^2+104*q^3+231*q^4+395*q^5+530*q^6+557*q^
7+468*q^8+331*q^9+231*q^10+144*q^11+94*q^12+55*q^13+39*q^14+22*q^15+17*q^16+9*q
^17+6*q^18+2*q^19+2*q^20, 1+9*q+44*q^2+147*q^3+370*q^4+729*q^5+1150*q^6+1462*q^
7+1513*q^8+1297*q^9+982*q^10+705*q^11+480*q^12+315*q^13+210*q^14+139*q^15+94*q^
16+67*q^17+46*q^18+31*q^19+21*q^20+15*q^21+7*q^22+4*q^23+3*q^24+q^25, 1+10*q+54
*q^2+200*q^3+561*q^4+1245*q^5+2242*q^6+3314*q^7+4057*q^8+4151*q^9+3645*q^10+
2873*q^11+2155*q^12+1520*q^13+1061*q^14+718*q^15+508*q^16+342*q^17+256*q^18+176
*q^19+131*q^20+95*q^21+73*q^22+45*q^23+36*q^24+23*q^25+15*q^26+8*q^27+6*q^28+2*
q^29+2*q^30, 1+11*q+65*q^2+264*q^3+815*q^4+2005*q^5+4040*q^6+6766*q^7+9511*q^8+
11306*q^9+11518*q^10+10296*q^11+8409*q^12+6489*q^13+4781*q^14+3426*q^15+2440*q^
16+1730*q^17+1242*q^18+904*q^19+658*q^20+488*q^21+369*q^22+279*q^23+206*q^24+
160*q^25+121*q^26+87*q^27+62*q^28+45*q^29+31*q^30+21*q^31+13*q^32+7*q^33+4*q^34
+3*q^35+q^36, 1+12*q+77*q^2+340*q^3+1144*q^4+3083*q^5+6851*q^6+12767*q^7+20172*
q^8+27230*q^9+31683*q^10+32202*q^11+29272*q^12+24529*q^13+19479*q^14+14775*q^15
+10961*q^16+7987*q^17+5857*q^18+4257*q^19+3183*q^20+2336*q^21+1776*q^22+1337*q^
23+1035*q^24+776*q^25+624*q^26+481*q^27+377*q^28+286*q^29+229*q^30+169*q^31+133
*q^32+93*q^33+69*q^34+48*q^35+37*q^36+21*q^37+14*q^38+8*q^39+6*q^40+2*q^41+2*q^
42, 1+13*q+90*q^2+429*q^3+1561*q^4+4566*q^5+11068*q^6+22647*q^7+39592*q^8+59627
*q^9+77936*q^10+89199*q^11+90663*q^12+83536*q^13+71562*q^14+58137*q^15+45351*q^
16+34464*q^17+25846*q^18+19268*q^19+14403*q^20+10822*q^21+8173*q^22+6221*q^23+
4785*q^24+3689*q^25+2850*q^26+2259*q^27+1785*q^28+1408*q^29+1128*q^30+908*q^31+
725*q^32+581*q^33+454*q^34+354*q^35+283*q^36+220*q^37+161*q^38+120*q^39+90*q^40
+64*q^41+45*q^42+31*q^43+19*q^44+13*q^45+7*q^46+4*q^47+3*q^48+q^49, 1+14*q+104*
q^2+532*q^3+2080*q^4+6555*q^5+17184*q^6+38216*q^7+73045*q^8+121072*q^9+175267*q
^10+223163*q^11+252274*q^12+256621*q^13+239284*q^14+208668*q^15+173089*q^16+
138078*q^17+107458*q^18+82265*q^19+62670*q^20+47562*q^21+36378*q^22+27763*q^23+
21486*q^24+16619*q^25+12994*q^26+10130*q^27+8081*q^28+6379*q^29+5117*q^30+4096*
q^31+3355*q^32+2713*q^33+2233*q^34+1794*q^35+1480*q^36+1205*q^37+990*q^38+779*q
^39+632*q^40+500*q^41+400*q^42+305*q^43+241*q^44+175*q^45+135*q^46+95*q^47+72*q
^48+49*q^49+35*q^50+20*q^51+14*q^52+8*q^53+6*q^54+2*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+3*q+2), q^6*(q^9+5*q^8+14*q^7+25*q^6+31*q^5+24*q^4+12*q^3+6*q^2+3*q+1)
, q^9*(q^12+6*q^11+20*q^10+44*q^9+70*q^8+79*q^7+64*q^6+36*q^5+23*q^4+11*q^3+7*q
^2+2*q+2), q^12*(q^16+7*q^15+27*q^14+70*q^13+134*q^12+192*q^11+209*q^10+170*q^9
+112*q^8+72*q^7+44*q^6+24*q^5+15*q^4+9*q^3+4*q^2+3*q+1), q^16*(q^20+8*q^19+35*q
^18+104*q^17+231*q^16+395*q^15+530*q^14+557*q^13+468*q^12+331*q^11+231*q^10+144
*q^9+94*q^8+55*q^7+39*q^6+22*q^5+17*q^4+9*q^3+6*q^2+2*q+2), q^20*(1+q^25+147*q^
22+9*q^24+370*q^21+44*q^23+3*q+4*q^2+7*q^3+15*q^4+21*q^5+31*q^6+46*q^7+67*q^8+
94*q^9+139*q^10+210*q^11+315*q^12+480*q^13+705*q^14+982*q^15+1297*q^16+1513*q^
17+1462*q^18+1150*q^19+729*q^20), q^25*(2+1245*q^25+4057*q^22+2242*q^24+4151*q^
21+3314*q^23+2*q+q^30+10*q^29+6*q^2+561*q^26+8*q^3+15*q^4+23*q^5+36*q^6+45*q^7+
73*q^8+95*q^9+131*q^10+176*q^11+256*q^12+342*q^13+508*q^14+718*q^15+1061*q^16+
54*q^28+1520*q^17+2155*q^18+2873*q^19+3645*q^20+200*q^27), q^30*(1+10296*q^25+
4781*q^22+8409*q^24+3426*q^21+6489*q^23+3*q+4040*q^30+6766*q^29+65*q^34+264*q^
33+4*q^2+11518*q^26+7*q^3+13*q^4+21*q^5+31*q^6+815*q^32+45*q^7+62*q^8+87*q^9+
121*q^10+160*q^11+206*q^12+279*q^13+369*q^14+488*q^15+658*q^16+9511*q^28+904*q^
17+1242*q^18+1730*q^19+2440*q^20+11306*q^27+2005*q^31+11*q^35+q^36), q^36*(2+
7987*q^25+q^42+3183*q^22+5857*q^24+2336*q^21+4257*q^23+2*q+29272*q^30+3083*q^37
+24529*q^29+340*q^39+12*q^41+20172*q^34+1144*q^38+27230*q^33+6*q^2+10961*q^26+8
*q^3+14*q^4+21*q^5+37*q^6+31683*q^32+48*q^7+69*q^8+93*q^9+133*q^10+169*q^11+229
*q^12+286*q^13+377*q^14+481*q^15+624*q^16+19479*q^28+776*q^17+1035*q^18+1337*q^
19+1776*q^20+14775*q^27+77*q^40+32202*q^31+12767*q^35+6851*q^36), q^42*(1+4785*
q^25+22647*q^42+2259*q^22+3689*q^24+1785*q^21+2850*q^23+3*q+4566*q^44+19268*q^
30+1561*q^45+90663*q^37+14403*q^29+429*q^46+77936*q^39+39592*q^41+58137*q^34+
89199*q^38+45351*q^33+4*q^2+6221*q^26+7*q^3+13*q^4+19*q^5+31*q^6+34464*q^32+45*
q^7+64*q^8+90*q^9+120*q^10+161*q^11+220*q^12+283*q^13+354*q^14+454*q^15+581*q^
16+11068*q^43+q^49+10822*q^28+725*q^17+908*q^18+1128*q^19+1408*q^20+8173*q^27+
59627*q^40+25846*q^31+71562*q^35+13*q^48+83536*q^36+90*q^47), q^49*(2+4096*q^25
+239284*q^42+2233*q^22+3355*q^24+1794*q^21+2713*q^23+2*q+6555*q^51+252274*q^44+
532*q^53+12994*q^30+2080*q^52+14*q^55+104*q^54+223163*q^45+82265*q^37+10130*q^
29+175267*q^46+138078*q^39+208668*q^41+36378*q^34+17184*q^50+107458*q^38+27763*
q^33+q^56+6*q^2+5117*q^26+8*q^3+14*q^4+20*q^5+35*q^6+21486*q^32+49*q^7+72*q^8+
95*q^9+135*q^10+175*q^11+241*q^12+305*q^13+400*q^14+500*q^15+632*q^16+256621*q^
43+38216*q^49+8081*q^28+779*q^17+990*q^18+1205*q^19+1480*q^20+6379*q^27+173089*
q^40+16619*q^31+47562*q^35+73045*q^48+62670*q^36+121072*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [3, 5, 1, 2, 4]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[3, 2, 1], [3, 5, 1, 2, 4]},

     is given by

[1, 1, 3, 7, 22, 61, 187, 547, 1654, 4921, 14803, 44287, 132982, 398581,

    1196107]

The number of ODD permutations avoiding, {[3, 2, 1], [3, 5, 1, 2, 4]},

     is given by

[0, 1, 2, 7, 19, 61, 178, 547, 1627, 4921, 14722, 44287, 132739, 398581,

    1195378]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 61, 178, 547, 1654, 4921, 14722, 44287, 132982, 398581,

    1195378]

 ODD:, [0, 1, 3, 7, 19, 61, 187, 547, 1627, 4921, 14803, 44287, 132739, 398581,

    1196107]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.048780488, 4.090163934,

    5.172602740, 6.282449726, 7.410850350, 8.551818736, 9.701270110,

    10.85642514, 12.01540337, 13.17693894, 14.34018277]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.342571604, 1.669416794,

    2.008288324, 2.354295741, 2.702577237, 3.048881594, 3.389854876,

    3.723068152, 4.046913462, 4.360452438, 4.663261746]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.80250, 2.78695, 4.03322, 5.54271,

    7.30392, 9.29568, 11.4911, 13.8612, 16.3775, 19.0135, 21.7460]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.09252531664,

    0.3011555457, 0.4853302133, 0.6481058388, 0.7917159442, 0.9181047223,

    1.029016045, 1.125941943, 1.210160648, 1.282817301, 1.344923346]

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

    2.786267339, 3.088634105, 3.433974826, 3.820369693, 4.235195088,

    4.666468644, 5.103942327, 5.538719249, 5.962922436, 6.369947524,

    6.753938596]

                                end of this data



For the equivalence class of patterns, {{[3, 1, 2], [3, 4, 5, 2, 1]},

    {[2, 1, 3], [1, 2, 5, 4, 3]}, {[2, 3, 1], [5, 4, 1, 2, 3]},

    {[1, 3, 2], [3, 2, 1, 4, 5]}}

      the member , {[3, 1, 2], [3, 4, 5, 2, 1]}, has a scheme of depth , 3

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 1+4*q+6*q^2+7*q^3+7*q^
4+5*q^5+5*q^6+2*q^7+2*q^8+q^9+q^10, 1+5*q+10*q^2+14*q^3+17*q^4+16*q^5+16*q^6+12
*q^7+9*q^8+8*q^9+4*q^10+3*q^11+2*q^12+2*q^13+q^14+q^15, 1+6*q+15*q^2+25*q^3+35*
q^4+40*q^5+43*q^6+41*q^7+34*q^8+32*q^9+22*q^10+17*q^11+12*q^12+10*q^13+6*q^14+6
*q^15+2*q^16+3*q^17+2*q^18+2*q^19+q^20+q^21, 1+7*q+21*q^2+41*q^3+65*q^4+86*q^5+
102*q^6+111*q^7+106*q^8+103*q^9+88*q^10+71*q^11+58*q^12+44*q^13+35*q^14+27*q^15
+19*q^16+11*q^17+12*q^18+9*q^19+8*q^20+4*q^21+4*q^22+2*q^23+3*q^24+2*q^25+2*q^
26+q^27+q^28, 1+8*q+28*q^2+63*q^3+112*q^4+167*q^5+219*q^6+263*q^7+283*q^8+292*q
^9+282*q^10+250*q^11+221*q^12+180*q^13+151*q^14+120*q^15+95*q^16+65*q^17+57*q^
18+43*q^19+38*q^20+25*q^21+17*q^22+13*q^23+14*q^24+10*q^25+10*q^26+7*q^27+6*q^
28+2*q^29+4*q^30+2*q^31+3*q^32+2*q^33+2*q^34+q^35+q^36, 1+9*q+36*q^2+92*q^3+182
*q^4+301*q^5+434*q^6+568*q^7+674*q^8+748*q^9+786*q^10+765*q^11+722*q^12+643*q^
13+561*q^14+479*q^15+393*q^16+312*q^17+249*q^18+203*q^19+162*q^20+135*q^21+94*q
^22+76*q^23+59*q^24+52*q^25+39*q^26+35*q^27+25*q^28+18*q^29+15*q^30+14*q^31+10*
q^32+14*q^33+8*q^34+8*q^35+5*q^36+4*q^37+2*q^38+4*q^39+2*q^40+3*q^41+2*q^42+2*q
^43+q^44+q^45, 1+10*q+45*q^2+129*q^3+282*q^4+512*q^5+806*q^6+1142*q^7+1473*q^8+
1760*q^9+1985*q^10+2094*q^11+2109*q^12+2026*q^13+1869*q^14+1690*q^15+1464*q^16+
1245*q^17+1028*q^18+855*q^19+694*q^20+586*q^21+452*q^22+364*q^23+285*q^24+239*q
^25+191*q^26+164*q^27+127*q^28+99*q^29+75*q^30+64*q^31+45*q^32+50*q^33+41*q^34+
33*q^35+25*q^36+21*q^37+13*q^38+15*q^39+14*q^40+12*q^41+10*q^42+12*q^43+6*q^44+
6*q^45+3*q^46+4*q^47+2*q^48+4*q^49+2*q^50+3*q^51+2*q^52+2*q^53+q^54+q^55, 1+11*
q+55*q^2+175*q^3+420*q^4+831*q^5+1419*q^6+2166*q^7+3009*q^8+3857*q^9+4642*q^10+
5250*q^11+5633*q^12+5774*q^13+5658*q^14+5389*q^15+4945*q^16+4420*q^17+3850*q^18
+3294*q^19+2783*q^20+2350*q^21+1943*q^22+1576*q^23+1297*q^24+1041*q^25+870*q^26
+708*q^27+595*q^28+468*q^29+388*q^30+303*q^31+249*q^32+209*q^33+189*q^34+146*q^
35+129*q^36+92*q^37+75*q^38+58*q^39+58*q^40+50*q^41+46*q^42+39*q^43+35*q^44+25*
q^45+21*q^46+15*q^47+16*q^48+12*q^49+15*q^50+12*q^51+12*q^52+8*q^53+10*q^54+4*q
^55+4*q^56+3*q^57+4*q^58+2*q^59+4*q^60+2*q^61+3*q^62+2*q^63+2*q^64+q^65+q^66, 1
+12*q+66*q^2+231*q^3+605*q^4+1297*q^5+2389*q^6+3912*q^7+5817*q^8+7963*q^9+10189
*q^10+12263*q^11+13969*q^12+15180*q^13+15763*q^14+15801*q^15+15297*q^16+14350*q
^17+13127*q^18+11699*q^19+10266*q^20+8881*q^21+7605*q^22+6374*q^23+5363*q^24+
4408*q^25+3685*q^26+3046*q^27+2551*q^28+2086*q^29+1726*q^30+1397*q^31+1152*q^32
+949*q^33+821*q^34+671*q^35+586*q^36+466*q^37+381*q^38+302*q^39+260*q^40+220*q^
41+196*q^42+166*q^43+144*q^44+116*q^45+99*q^46+72*q^47+67*q^48+54*q^49+56*q^50+
49*q^51+46*q^52+36*q^53+39*q^54+27*q^55+17*q^56+17*q^57+18*q^58+12*q^59+16*q^60
+12*q^61+13*q^62+12*q^63+10*q^64+6*q^65+8*q^66+2*q^67+4*q^68+3*q^69+4*q^70+2*q^
71+4*q^72+2*q^73+3*q^74+2*q^75+2*q^76+q^77+q^78, 1+13*q+78*q^2+298*q^3+847*q^4+
1958*q^5+3872*q^6+6776*q^7+10736*q^8+15629*q^9+21194*q^10+27009*q^11+32539*q^12
+37328*q^13+40913*q^14+43118*q^15+43868*q^16+43159*q^17+41310*q^18+38466*q^19+
35069*q^20+31396*q^21+27697*q^22+24038*q^23+20670*q^24+17548*q^25+14806*q^26+
12505*q^27+10475*q^28+8806*q^29+7306*q^30+6097*q^31+5018*q^32+4208*q^33+3489*q^
34+2948*q^35+2475*q^36+2107*q^37+1713*q^38+1459*q^39+1173*q^40+1019*q^41+850*q^
42+760*q^43+631*q^44+554*q^45+450*q^46+375*q^47+301*q^48+263*q^49+216*q^50+207*
q^51+175*q^52+160*q^53+142*q^54+129*q^55+86*q^56+77*q^57+68*q^58+61*q^59+54*q^
60+56*q^61+43*q^62+50*q^63+41*q^64+31*q^65+28*q^66+25*q^67+13*q^68+16*q^69+16*q
^70+16*q^71+12*q^72+16*q^73+10*q^74+13*q^75+10*q^76+8*q^77+4*q^78+6*q^79+2*q^80
+4*q^81+3*q^82+4*q^83+2*q^84+4*q^85+2*q^86+3*q^87+2*q^88+2*q^89+q^90+q^91, 1+14
*q+91*q^2+377*q^3+1157*q^4+2872*q^5+6073*q^6+11319*q^7+19041*q^8+29370*q^9+
42087*q^10+56584*q^11+71845*q^12+86722*q^13+99959*q^14+110551*q^15+117862*q^16+
121395*q^17+121344*q^18+117926*q^19+111801*q^20+103813*q^21+94603*q^22+84827*q^
23+74968*q^24+65475*q^25+56492*q^26+48581*q^27+41368*q^28+35239*q^29+29746*q^30
+25105*q^31+21024*q^32+17704*q^33+14812*q^34+12480*q^35+10500*q^36+8915*q^37+
7451*q^38+6317*q^39+5219*q^40+4408*q^41+3724*q^42+3210*q^43+2743*q^44+2385*q^45
+2014*q^46+1709*q^47+1422*q^48+1210*q^49+1007*q^50+901*q^51+771*q^52+689*q^53+
598*q^54+539*q^55+431*q^56+361*q^57+297*q^58+259*q^59+229*q^60+213*q^61+181*q^
62+182*q^63+163*q^64+138*q^65+114*q^66+104*q^67+76*q^68+68*q^69+60*q^70+63*q^71
+52*q^72+54*q^73+45*q^74+48*q^75+41*q^76+37*q^77+25*q^78+21*q^79+18*q^80+18*q^
81+14*q^82+16*q^83+14*q^84+16*q^85+12*q^86+14*q^87+10*q^88+11*q^89+8*q^90+6*q^
91+2*q^92+6*q^93+2*q^94+4*q^95+3*q^96+4*q^97+2*q^98+4*q^99+2*q^100+3*q^101+2*q^
102+2*q^103+q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+q+2*q^2+q^3, q^6+3*q^5+3*q^4+3*q^3+2*q^2+q+1, q^10+4*q^9+6*q^8+7*q^7
+7*q^6+5*q^5+5*q^4+2*q^3+2*q^2+q+1, q^15+5*q^14+10*q^13+14*q^12+17*q^11+16*q^10
+16*q^9+12*q^8+9*q^7+8*q^6+4*q^5+3*q^4+2*q^3+2*q^2+q+1, q^21+6*q^20+15*q^19+25*
q^18+35*q^17+40*q^16+43*q^15+41*q^14+34*q^13+32*q^12+22*q^11+17*q^10+12*q^9+10*
q^8+6*q^7+6*q^6+2*q^5+3*q^4+2*q^3+2*q^2+q+1, 1+41*q^25+102*q^22+65*q^24+111*q^
21+86*q^23+q+2*q^2+21*q^26+2*q^3+3*q^4+2*q^5+4*q^6+4*q^7+8*q^8+9*q^9+12*q^10+11
*q^11+19*q^12+27*q^13+35*q^14+44*q^15+58*q^16+q^28+71*q^17+88*q^18+103*q^19+106
*q^20+7*q^27, 1+250*q^25+151*q^22+221*q^24+120*q^21+180*q^23+q+219*q^30+263*q^
29+28*q^34+63*q^33+2*q^2+282*q^26+2*q^3+3*q^4+2*q^5+4*q^6+112*q^32+2*q^7+6*q^8+
7*q^9+10*q^10+10*q^11+14*q^12+13*q^13+17*q^14+25*q^15+38*q^16+283*q^28+43*q^17+
57*q^18+65*q^19+95*q^20+292*q^27+167*q^31+8*q^35+q^36, 1+162*q^25+92*q^42+76*q^
22+135*q^24+59*q^21+94*q^23+q+9*q^44+479*q^30+q^45+674*q^37+393*q^29+434*q^39+
182*q^41+765*q^34+568*q^38+722*q^33+2*q^2+203*q^26+2*q^3+3*q^4+2*q^5+4*q^6+643*
q^32+2*q^7+4*q^8+5*q^9+8*q^10+8*q^11+14*q^12+10*q^13+14*q^14+15*q^15+18*q^16+36
*q^43+312*q^28+25*q^17+35*q^18+39*q^19+52*q^20+249*q^27+301*q^40+561*q^31+786*q
^35+748*q^36, 1+75*q^25+2026*q^42+50*q^22+64*q^24+41*q^21+45*q^23+q+282*q^51+
2094*q^44+45*q^53+239*q^30+129*q^52+q^55+10*q^54+1985*q^45+1028*q^37+191*q^29+
1760*q^46+1464*q^39+1869*q^41+586*q^34+512*q^50+1245*q^38+452*q^33+2*q^2+99*q^
26+2*q^3+3*q^4+2*q^5+4*q^6+364*q^32+2*q^7+4*q^8+3*q^9+6*q^10+6*q^11+12*q^12+10*
q^13+12*q^14+14*q^15+15*q^16+2109*q^43+806*q^49+164*q^28+13*q^17+21*q^18+25*q^
19+33*q^20+127*q^27+1690*q^40+285*q^31+694*q^35+1142*q^48+855*q^36+1473*q^47, 1
+50*q^25+1297*q^42+35*q^22+46*q^24+25*q^21+39*q^23+q+5389*q^51+1943*q^44+5774*q
^53+3857*q^57+129*q^30+5658*q^52+3009*q^58+5250*q^55+1419*q^60+5633*q^54+2350*q
^45+468*q^37+92*q^29+2783*q^46+11*q^65+708*q^39+831*q^61+1041*q^41+249*q^34+
4945*q^50+55*q^64+595*q^38+209*q^33+4642*q^56+2*q^2+58*q^26+2*q^3+3*q^4+2*q^5+4
*q^6+189*q^32+2*q^7+4*q^8+3*q^9+420*q^62+4*q^10+4*q^11+10*q^12+8*q^13+12*q^14+
12*q^15+15*q^16+q^66+1576*q^43+4420*q^49+75*q^28+12*q^17+16*q^18+15*q^19+21*q^
20+58*q^27+870*q^40+175*q^63+146*q^31+303*q^35+3850*q^48+2166*q^59+388*q^36+
3294*q^47, 1+36*q^25+586*q^42+17*q^22+39*q^24+17*q^21+27*q^23+q+3046*q^51+821*q
^44+4408*q^53+8881*q^57+67*q^30+3685*q^52+10266*q^58+6374*q^55+13127*q^60+5363*
q^54+949*q^45+3912*q^71+605*q^74+220*q^37+12263*q^67+54*q^29+1152*q^46+15180*q^
65+7963*q^69+302*q^39+14350*q^61+466*q^41+144*q^34+2551*q^50+15763*q^64+260*q^
38+116*q^33+7605*q^56+2*q^2+10189*q^68+46*q^26+2*q^3+3*q^4+2*q^5+4*q^6+99*q^32+
2*q^7+4*q^8+3*q^9+15297*q^62+4*q^10+2*q^11+8*q^12+66*q^76+6*q^13+10*q^14+12*q^
15+13*q^16+13969*q^66+5817*q^70+12*q^77+q^78+671*q^43+2086*q^49+56*q^28+12*q^17
+16*q^18+12*q^19+18*q^20+49*q^27+2389*q^72+381*q^40+1297*q^73+15801*q^63+72*q^
31+231*q^75+166*q^35+1726*q^48+11699*q^59+196*q^36+1397*q^47, 1+28*q^25+263*q^
42+16*q^22+13*q^90+25*q^24+16*q^21+13*q^23+q+1173*q^51+375*q^44+1713*q^53+3489*
q^57+56*q^30+1459*q^52+4208*q^58+2475*q^55+6097*q^60+q^91+2107*q^54+450*q^45+
35069*q^71+43159*q^74+142*q^37+20670*q^67+43*q^29+554*q^46+14806*q^65+27697*q^
69+175*q^39+7306*q^61+216*q^41+77*q^34+32539*q^79+1019*q^50+12505*q^64+160*q^38
+68*q^33+2948*q^56+2*q^2+24038*q^68+31*q^26+2*q^3+3*q^4+2*q^5+4*q^6+61*q^32+2*q
^7+4*q^8+3*q^9+8806*q^62+4*q^10+2*q^11+6*q^12+43118*q^76+4*q^13+8*q^14+10*q^15+
13*q^16+17548*q^66+31396*q^70+40913*q^77+37328*q^78+301*q^43+850*q^49+50*q^28+
10*q^17+16*q^18+12*q^19+16*q^20+41*q^27+38466*q^72+207*q^40+41310*q^73+10475*q^
63+54*q^31+43868*q^75+86*q^35+27009*q^80+760*q^48+10736*q^83+21194*q^81+3872*q^
85+1958*q^86+847*q^87+298*q^88+78*q^89+6776*q^84+15629*q^82+5018*q^59+129*q^36+
631*q^47, 1+18*q^25+182*q^42+16*q^22+110551*q^90+18*q^24+14*q^21+14*q^23+q+
19041*q^97+598*q^51+213*q^44+771*q^53+1422*q^57+48*q^30+56584*q^94+689*q^52+
1709*q^58+1007*q^55+2385*q^60+99959*q^91+901*q^54+229*q^45+14812*q^71+25105*q^
74+29370*q^96+11319*q^98+76*q^37+7451*q^67+41*q^29+259*q^46+71845*q^93+6073*q^
99+5219*q^65+10500*q^69+114*q^39+2743*q^61+163*q^41+63*q^34+56492*q^79+539*q^50
+4408*q^64+2872*q^100+104*q^38+52*q^33+1210*q^56+2*q^2+8915*q^68+21*q^26+2*q^3+
3*q^4+2*q^5+4*q^6+54*q^32+2*q^7+4*q^8+3*q^9+42087*q^95+3210*q^62+1157*q^101+4*q
^10+2*q^11+6*q^12+35239*q^76+2*q^13+6*q^14+8*q^15+11*q^16+6317*q^66+12480*q^70+
41368*q^77+48581*q^78+181*q^43+377*q^102+431*q^49+37*q^28+10*q^17+14*q^18+12*q^
19+16*q^20+25*q^27+17704*q^72+138*q^40+21024*q^73+91*q^103+3724*q^63+45*q^31+
29746*q^75+60*q^35+14*q^104+65475*q^80+361*q^48+q^105+94603*q^83+74968*q^81+
111801*q^85+117926*q^86+121344*q^87+121395*q^88+117862*q^89+103813*q^84+84827*q
^82+2014*q^59+68*q^36+86722*q^92+297*q^47]


 The number of permutations avoiding, {[3, 1, 2], [3, 4, 5, 2, 1]}, is given by

[1, 2, 5, 14, 41, 121, 356, 1044, 3057, 8948, 26192, 76674, 224465, 657137,

    1923817]

The number of EVEN permutations avoiding, {[3, 1, 2], [3, 4, 5, 2, 1]},

     is given by

[1, 1, 2, 7, 22, 60, 173, 525, 1544, 4462, 13050, 38381, 112366, 328411, 961527

    ]

The number of ODD permutations avoiding, {[3, 1, 2], [3, 4, 5, 2, 1]},

     is given by

[0, 1, 3, 7, 19, 61, 183, 519, 1513, 4486, 13142, 38293, 112099, 328726, 962290

    ]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 22, 61, 183, 525, 1544, 4486, 13142, 38381, 112366, 328726,

    962290]

 ODD:, [0, 1, 2, 7, 19, 60, 173, 519, 1513, 4462, 13050, 38293, 112099, 328411,

    961527]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.400000000, 2.642857143, 4.121951220, 5.710743802,

    7.328651685, 8.941570881, 10.54268891, 12.13567278, 13.72525962,

    15.31420038, 16.90343261, 18.49301439, 20.08278022]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.307940925, 3.008082821,

    3.689701638, 4.323312443, 4.896464149, 5.411469933, 5.878169574,

    6.307403368, 6.707793431, 7.085251116, 7.443671027]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.32659, 9.04856, 13.6139, 18.6910,

    23.9754, 29.2840, 34.5529, 39.7833, 44.9945, 50.2008, 55.4082]

Skewness: , [Float(undefined), 0., 0.2715454176, 0.3874842230, 0.5178796979,

    0.6777113676, 0.8425017245, 0.9895817521, 1.105245039, 1.184676162,

    1.230464375, 1.249746926, 1.250784275, 1.240619050, 1.224203931]

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

    2.764755706, 3.249103237, 3.841964143, 4.498137518, 5.146256224,

    5.712302289, 6.144779456, 6.428337368, 6.578348684, 6.626072518,

    6.604249762]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [1, 4, 2, 5, 3]},

    {[3, 2, 1], [1, 3, 5, 2, 4]}, {[1, 2, 3], [4, 2, 5, 3, 1]},

    {[1, 2, 3], [3, 5, 2, 4, 1]}, {[1, 2, 3], [5, 2, 4, 1, 3]},

    {[1, 2, 3], [5, 3, 1, 4, 2]}, {[3, 2, 1], [2, 4, 1, 3, 5]},

    {[3, 2, 1], [3, 1, 4, 2, 5]}}

      the member , {[3, 2, 1], [1, 4, 2, 5, 3]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

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

%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, 1+3*q+5*q^2+4*q^3+q^4, 1+4*q+9*q^2+11*q^3+10*q^4+4*q^5+2*
q^6, 1+5*q+14*q^2+23*q^3+26*q^4+22*q^5+15*q^6+9*q^7+4*q^8+q^9, 1+6*q+20*q^2+41*
q^3+58*q^4+59*q^5+52*q^6+38*q^7+32*q^8+20*q^9+12*q^10+4*q^11+2*q^12, 1+7*q+27*q
^2+66*q^3+113*q^4+140*q^5+139*q^6+119*q^7+97*q^8+81*q^9+68*q^10+48*q^11+33*q^12
+19*q^13+9*q^14+4*q^15+q^16, 1+8*q+35*q^2+99*q^3+199*q^4+294*q^5+339*q^6+325*q^
7+285*q^8+236*q^9+204*q^10+171*q^11+150*q^12+114*q^13+90*q^14+59*q^15+42*q^16+
22*q^17+12*q^18+4*q^19+2*q^20, 1+9*q+44*q^2+141*q^3+325*q^4+559*q^5+748*q^6+814
*q^7+776*q^8+684*q^9+590*q^10+500*q^11+432*q^12+372*q^13+329*q^14+272*q^15+224*
q^16+173*q^17+133*q^18+91*q^19+61*q^20+37*q^21+19*q^22+9*q^23+4*q^24+q^25, 1+10
*q+54*q^2+193*q^3+501*q^4+983*q^5+1508*q^6+1868*q^7+1968*q^8+1863*q^9+1675*q^10
+1462*q^11+1267*q^12+1078*q^13+937*q^14+816*q^15+732*q^16+625*q^17+550*q^18+448
*q^19+375*q^20+291*q^21+228*q^22+154*q^23+114*q^24+70*q^25+44*q^26+22*q^27+12*q
^28+4*q^29+2*q^30, 1+11*q+65*q^2+256*q^3+738*q^4+1625*q^5+2818*q^6+3949*q^7+
4637*q^8+4772*q^9+4532*q^10+4114*q^11+3661*q^12+3194*q^13+2772*q^14+2374*q^15+
2073*q^16+1814*q^17+1633*q^18+1438*q^19+1281*q^20+1114*q^21+970*q^22+811*q^23+
676*q^24+536*q^25+423*q^26+313*q^27+230*q^28+158*q^29+106*q^30+65*q^31+37*q^32+
19*q^33+9*q^34+4*q^35+q^36, 1+12*q+77*q^2+331*q^3+1048*q^4+2556*q^5+4946*q^6+
7766*q^7+10163*q^8+11469*q^9+11656*q^10+11094*q^11+10195*q^12+9152*q^13+8111*q^
14+7093*q^15+6177*q^16+5334*q^17+4665*q^18+4103*q^19+3680*q^20+3275*q^21+2974*q
^22+2644*q^23+2380*q^24+2073*q^25+1826*q^26+1543*q^27+1309*q^28+1055*q^29+869*q
^30+663*q^31+522*q^32+374*q^33+272*q^34+181*q^35+126*q^36+72*q^37+44*q^38+22*q^
39+12*q^40+4*q^41+2*q^42, 1+13*q+90*q^2+419*q^3+1444*q^4+3860*q^5+8242*q^6+
14355*q^7+20834*q^8+25856*q^9+28391*q^10+28618*q^11+27370*q^12+25342*q^13+22996
*q^14+20543*q^15+18210*q^16+15987*q^17+14008*q^18+12181*q^19+10697*q^20+9402*q^
21+8419*q^22+7530*q^23+6845*q^24+6179*q^25+5636*q^26+5083*q^27+4577*q^28+4039*q
^29+3579*q^30+3085*q^31+2653*q^32+2226*q^33+1852*q^34+1490*q^35+1206*q^36+930*q
^37+706*q^38+519*q^39+376*q^40+257*q^41+175*q^42+110*q^43+65*q^44+37*q^45+19*q^
46+9*q^47+4*q^48+q^49, 1+14*q+104*q^2+521*q^3+1940*q^4+5635*q^5+13152*q^6+25173
*q^7+40242*q^8+54834*q^9+65383*q^10+70395*q^11+70677*q^12+67844*q^13+63300*q^14
+57857*q^15+52188*q^16+46588*q^17+41393*q^18+36551*q^19+32210*q^20+28232*q^21+
24853*q^22+21932*q^23+19572*q^24+17496*q^25+15859*q^26+14403*q^27+13225*q^28+
12047*q^29+11056*q^30+10011*q^31+9107*q^32+8138*q^33+7282*q^34+6368*q^35+5608*q
^36+4796*q^37+4118*q^38+3417*q^39+2855*q^40+2293*q^41+1855*q^42+1428*q^43+1117*
q^44+821*q^45+612*q^46+422*q^47+302*q^48+194*q^49+128*q^50+72*q^51+44*q^52+22*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+11*q^
3+10*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+23*q^6+26*q^5+22*q^4+15*q^3+9*q^2+4*q+1)
, q^9*(q^12+6*q^11+20*q^10+41*q^9+58*q^8+59*q^7+52*q^6+38*q^5+32*q^4+20*q^3+12*
q^2+4*q+2), q^12*(q^16+7*q^15+27*q^14+66*q^13+113*q^12+140*q^11+139*q^10+119*q^
9+97*q^8+81*q^7+68*q^6+48*q^5+33*q^4+19*q^3+9*q^2+4*q+1), q^16*(q^20+8*q^19+35*
q^18+99*q^17+199*q^16+294*q^15+339*q^14+325*q^13+285*q^12+236*q^11+204*q^10+171
*q^9+150*q^8+114*q^7+90*q^6+59*q^5+42*q^4+22*q^3+12*q^2+4*q+2), q^20*(1+q^25+
141*q^22+9*q^24+325*q^21+44*q^23+4*q+9*q^2+19*q^3+37*q^4+61*q^5+91*q^6+133*q^7+
173*q^8+224*q^9+272*q^10+329*q^11+372*q^12+432*q^13+500*q^14+590*q^15+684*q^16+
776*q^17+814*q^18+748*q^19+559*q^20), q^25*(2+983*q^25+1968*q^22+1508*q^24+1863
*q^21+1868*q^23+4*q+q^30+10*q^29+12*q^2+501*q^26+22*q^3+44*q^4+70*q^5+114*q^6+
154*q^7+228*q^8+291*q^9+375*q^10+448*q^11+550*q^12+625*q^13+732*q^14+816*q^15+
937*q^16+54*q^28+1078*q^17+1267*q^18+1462*q^19+1675*q^20+193*q^27), q^30*(1+
4114*q^25+2772*q^22+3661*q^24+2374*q^21+3194*q^23+4*q+2818*q^30+3949*q^29+65*q^
34+256*q^33+9*q^2+4532*q^26+19*q^3+37*q^4+65*q^5+106*q^6+738*q^32+158*q^7+230*q
^8+313*q^9+423*q^10+536*q^11+676*q^12+811*q^13+970*q^14+1114*q^15+1281*q^16+
4637*q^28+1438*q^17+1633*q^18+1814*q^19+2073*q^20+4772*q^27+1625*q^31+11*q^35+q
^36), q^36*(2+5334*q^25+q^42+3680*q^22+4665*q^24+3275*q^21+4103*q^23+4*q+10195*
q^30+2556*q^37+9152*q^29+331*q^39+12*q^41+10163*q^34+1048*q^38+11469*q^33+12*q^
2+6177*q^26+22*q^3+44*q^4+72*q^5+126*q^6+11656*q^32+181*q^7+272*q^8+374*q^9+522
*q^10+663*q^11+869*q^12+1055*q^13+1309*q^14+1543*q^15+1826*q^16+8111*q^28+2073*
q^17+2380*q^18+2644*q^19+2974*q^20+7093*q^27+77*q^40+11094*q^31+7766*q^35+4946*
q^36), q^42*(1+6845*q^25+14355*q^42+5083*q^22+6179*q^24+4577*q^21+5636*q^23+4*q
+3860*q^44+12181*q^30+1444*q^45+27370*q^37+10697*q^29+419*q^46+28391*q^39+20834
*q^41+20543*q^34+28618*q^38+18210*q^33+9*q^2+7530*q^26+19*q^3+37*q^4+65*q^5+110
*q^6+15987*q^32+175*q^7+257*q^8+376*q^9+519*q^10+706*q^11+930*q^12+1206*q^13+
1490*q^14+1852*q^15+2226*q^16+8242*q^43+q^49+9402*q^28+2653*q^17+3085*q^18+3579
*q^19+4039*q^20+8419*q^27+25856*q^40+14008*q^31+22996*q^35+13*q^48+25342*q^36+
90*q^47), q^49*(2+10011*q^25+63300*q^42+7282*q^22+9107*q^24+6368*q^21+8138*q^23
+4*q+5635*q^51+70677*q^44+521*q^53+15859*q^30+1940*q^52+14*q^55+104*q^54+70395*
q^45+36551*q^37+14403*q^29+65383*q^46+46588*q^39+57857*q^41+24853*q^34+13152*q^
50+41393*q^38+21932*q^33+q^56+12*q^2+11056*q^26+22*q^3+44*q^4+72*q^5+128*q^6+
19572*q^32+194*q^7+302*q^8+422*q^9+612*q^10+821*q^11+1117*q^12+1428*q^13+1855*q
^14+2293*q^15+2855*q^16+67844*q^43+25173*q^49+13225*q^28+3417*q^17+4118*q^18+
4796*q^19+5608*q^20+12047*q^27+52188*q^40+17496*q^31+28232*q^35+40242*q^48+
32210*q^36+54834*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [1, 4, 2, 5, 3]}, is given by

[1, 2, 5, 14, 41, 120, 345, 972, 2691, 7348, 19855, 53230, 141871, 376466,

    995705]

The number of EVEN permutations avoiding, {[3, 2, 1], [1, 4, 2, 5, 3]},

     is given by

 [1, 1, 3, 7, 22, 60, 177, 488, 1359, 3686, 9968, 26663, 71055, 188395, 498197]

The number of ODD permutations avoiding, {[3, 2, 1], [1, 4, 2, 5, 3]},

     is given by

 [0, 1, 2, 7, 19, 60, 168, 484, 1332, 3662, 9887, 26567, 70816, 188071, 497508]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 60, 168, 488, 1359, 3662, 9887, 26663, 71055, 188071,

    497508]

 ODD:, [0, 1, 3, 7, 19, 60, 177, 484, 1332, 3686, 9968, 26567, 70816, 188395,

    498197]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.097560976, 4.250000000,

    5.501449275, 6.827160494, 8.205128205, 9.616494284, 11.04573155,

    12.48053729, 13.91153936, 15.33191576, 16.73699138]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.375834612, 1.790018622,

    2.277965832, 2.834052734, 3.448553207, 4.110233932, 4.807565996,

    5.529344354, 6.265068150, 7.005198778, 7.741331165]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.89292, 3.20417, 5.18913, 8.03185,

    11.8925, 16.8940, 23.1127, 30.5736, 39.2511, 49.0728, 59.9282]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.04913712930,

    0.2059549210, 0.3379844336, 0.4535582023, 0.5591964260, 0.6592764567,

    0.7563376239, 0.8517033060, 0.9459443097, 1.039216699, 1.131407274]

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

    2.610582899, 2.647392612, 2.676626145, 2.731969019, 2.822008120,

    2.947553159, 3.107767933, 3.301766732, 3.528824882, 3.788222917,

    4.079034426]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [1, 4, 2, 3, 5]},

    {[3, 2, 1], [1, 3, 4, 2, 5]}, {[1, 2, 3], [5, 3, 2, 4, 1]},

    {[1, 2, 3], [5, 2, 4, 3, 1]}}

      the member , {[3, 2, 1], [1, 4, 2, 3, 5]}, has a scheme of depth , 5

                                  here it is:

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

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

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

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

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

    [1, 4, 2, 3],

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

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

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

    [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {}], [[1, 3, 2, 4, 5], {

    [0, 0, 0, 2, 1, 0], [0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0],

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

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

    [[2, 4, 3, 5, 1], {[0, 0, 0, 0, 0, 0]}, {4}],

    [[1, 4, 3, 5, 2], {[0, 0, 0, 0, 0, 0]}, {4}], [[1, 4, 2, 5, 3], {

    [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]}, {4}],

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

    [1, 3, 2, 5, 4], {[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, 3}],

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

                 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+8*q^2+12*q^3+10*q^4+4*q^5+2*
q^6, 1+5*q+12*q^2+20*q^3+26*q^4+24*q^5+16*q^6+9*q^7+4*q^8+q^9, 1+6*q+17*q^2+33*
q^3+46*q^4+54*q^5+50*q^6+42*q^7+33*q^8+22*q^9+12*q^10+4*q^11+2*q^12, 1+7*q+23*q
^2+52*q^3+84*q^4+105*q^5+108*q^6+100*q^7+85*q^8+79*q^9+65*q^10+51*q^11+36*q^12+
20*q^13+9*q^14+4*q^15+q^16, 1+8*q+30*q^2+78*q^3+146*q^4+212*q^5+237*q^6+228*q^7
+194*q^8+166*q^9+150*q^10+138*q^11+121*q^12+104*q^13+86*q^14+66*q^15+44*q^16+24
*q^17+12*q^18+4*q^19+2*q^20, 1+9*q+38*q^2+112*q^3+239*q^4+399*q^5+522*q^6+557*q
^7+498*q^8+411*q^9+326*q^10+284*q^11+248*q^12+234*q^13+208*q^14+190*q^15+169*q^
16+152*q^17+124*q^18+96*q^19+66*q^20+40*q^21+20*q^22+9*q^23+4*q^24+q^25, 1+10*q
+47*q^2+155*q^3+371*q^4+698*q^5+1044*q^6+1288*q^7+1311*q^8+1159*q^9+928*q^10+
722*q^11+565*q^12+476*q^13+412*q^14+375*q^15+343*q^16+318*q^17+298*q^18+276*q^
19+256*q^20+230*q^21+189*q^22+152*q^23+117*q^24+78*q^25+46*q^26+24*q^27+12*q^28
+4*q^29+2*q^30, 1+11*q+57*q^2+208*q^3+551*q^4+1150*q^5+1924*q^6+2671*q^7+3106*q
^8+3104*q^9+2728*q^10+2226*q^11+1707*q^12+1314*q^13+1005*q^14+818*q^15+683*q^16
+613*q^17+541*q^18+511*q^19+478*q^20+461*q^21+436*q^22+422*q^23+383*q^24+347*q^
25+300*q^26+260*q^27+208*q^28+161*q^29+113*q^30+70*q^31+40*q^32+20*q^33+9*q^34+
4*q^35+q^36, 1+12*q+68*q^2+272*q^3+789*q^4+1806*q^5+3332*q^6+5116*q^7+6615*q^8+
7379*q^9+7225*q^10+6415*q^11+5312*q^12+4216*q^13+3230*q^14+2452*q^15+1864*q^16+
1481*q^17+1197*q^18+1019*q^19+883*q^20+801*q^21+750*q^22+718*q^23+688*q^24+668*
q^25+653*q^26+618*q^27+572*q^28+528*q^29+478*q^30+434*q^31+375*q^32+306*q^33+
246*q^34+188*q^35+130*q^36+80*q^37+46*q^38+24*q^39+12*q^40+4*q^41+2*q^42, 1+13*
q+80*q^2+348*q^3+1096*q^4+2728*q^5+5497*q^6+9242*q^7+13124*q^8+16049*q^9+17204*
q^10+16593*q^11+14754*q^12+12494*q^13+10157*q^14+8032*q^15+6177*q^16+4743*q^17+
3609*q^18+2847*q^19+2236*q^20+1828*q^21+1523*q^22+1332*q^23+1178*q^24+1114*q^25
+1048*q^26+1025*q^27+984*q^28+971*q^29+932*q^30+898*q^31+843*q^32+806*q^33+742*
q^34+697*q^35+629*q^36+555*q^37+473*q^38+403*q^39+324*q^40+251*q^41+180*q^42+
117*q^43+70*q^44+40*q^45+20*q^46+9*q^47+4*q^48+q^49, 1+14*q+93*q^2+437*q^3+1484
*q^4+3990*q^5+8718*q^6+15937*q^7+24684*q^8+32910*q^9+38296*q^10+39721*q^11+
37618*q^12+33451*q^13+28582*q^14+23750*q^15+19250*q^16+15312*q^17+11963*q^18+
9325*q^19+7251*q^20+5679*q^21+4466*q^22+3581*q^23+2859*q^24+2385*q^25+2048*q^26
+1813*q^27+1652*q^28+1557*q^29+1477*q^30+1438*q^31+1407*q^32+1365*q^33+1315*q^
34+1277*q^35+1232*q^36+1186*q^37+1134*q^38+1072*q^39+997*q^40+918*q^41+829*q^42
+742*q^43+660*q^44+564*q^45+461*q^46+372*q^47+287*q^48+202*q^49+132*q^50+80*q^
51+46*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+8*q^4+12*q^
3+10*q^2+4*q+2), q^6*(q^9+5*q^8+12*q^7+20*q^6+26*q^5+24*q^4+16*q^3+9*q^2+4*q+1)
, q^9*(q^12+6*q^11+17*q^10+33*q^9+46*q^8+54*q^7+50*q^6+42*q^5+33*q^4+22*q^3+12*
q^2+4*q+2), q^12*(q^16+7*q^15+23*q^14+52*q^13+84*q^12+105*q^11+108*q^10+100*q^9
+85*q^8+79*q^7+65*q^6+51*q^5+36*q^4+20*q^3+9*q^2+4*q+1), q^16*(q^20+8*q^19+30*q
^18+78*q^17+146*q^16+212*q^15+237*q^14+228*q^13+194*q^12+166*q^11+150*q^10+138*
q^9+121*q^8+104*q^7+86*q^6+66*q^5+44*q^4+24*q^3+12*q^2+4*q+2), q^20*(1+q^25+112
*q^22+9*q^24+239*q^21+38*q^23+4*q+9*q^2+20*q^3+40*q^4+66*q^5+96*q^6+124*q^7+152
*q^8+169*q^9+190*q^10+208*q^11+234*q^12+248*q^13+284*q^14+326*q^15+411*q^16+498
*q^17+557*q^18+522*q^19+399*q^20), q^25*(2+698*q^25+1311*q^22+1044*q^24+1159*q^
21+1288*q^23+4*q+q^30+10*q^29+12*q^2+371*q^26+24*q^3+46*q^4+78*q^5+117*q^6+152*
q^7+189*q^8+230*q^9+256*q^10+276*q^11+298*q^12+318*q^13+343*q^14+375*q^15+412*q
^16+47*q^28+476*q^17+565*q^18+722*q^19+928*q^20+155*q^27), q^30*(1+2226*q^25+
1005*q^22+1707*q^24+818*q^21+1314*q^23+4*q+1924*q^30+2671*q^29+57*q^34+208*q^33
+9*q^2+2728*q^26+20*q^3+40*q^4+70*q^5+113*q^6+551*q^32+161*q^7+208*q^8+260*q^9+
300*q^10+347*q^11+383*q^12+422*q^13+436*q^14+461*q^15+478*q^16+3106*q^28+511*q^
17+541*q^18+613*q^19+683*q^20+3104*q^27+1150*q^31+11*q^35+q^36), q^36*(2+1481*q
^25+q^42+883*q^22+1197*q^24+801*q^21+1019*q^23+4*q+5312*q^30+1806*q^37+4216*q^
29+272*q^39+12*q^41+6615*q^34+789*q^38+7379*q^33+12*q^2+1864*q^26+24*q^3+46*q^4
+80*q^5+130*q^6+7225*q^32+188*q^7+246*q^8+306*q^9+375*q^10+434*q^11+478*q^12+
528*q^13+572*q^14+618*q^15+653*q^16+3230*q^28+668*q^17+688*q^18+718*q^19+750*q^
20+2452*q^27+68*q^40+6415*q^31+5116*q^35+3332*q^36), q^42*(1+1178*q^25+9242*q^
42+1025*q^22+1114*q^24+984*q^21+1048*q^23+4*q+2728*q^44+2847*q^30+1096*q^45+
14754*q^37+2236*q^29+348*q^46+17204*q^39+13124*q^41+8032*q^34+16593*q^38+6177*q
^33+9*q^2+1332*q^26+20*q^3+40*q^4+70*q^5+117*q^6+4743*q^32+180*q^7+251*q^8+324*
q^9+403*q^10+473*q^11+555*q^12+629*q^13+697*q^14+742*q^15+806*q^16+5497*q^43+q^
49+1828*q^28+843*q^17+898*q^18+932*q^19+971*q^20+1523*q^27+16049*q^40+3609*q^31
+10157*q^35+13*q^48+12494*q^36+80*q^47), q^49*(2+1438*q^25+28582*q^42+1315*q^22
+1407*q^24+1277*q^21+1365*q^23+4*q+3990*q^51+37618*q^44+437*q^53+2048*q^30+1484
*q^52+14*q^55+93*q^54+39721*q^45+9325*q^37+1813*q^29+38296*q^46+15312*q^39+
23750*q^41+4466*q^34+8718*q^50+11963*q^38+3581*q^33+q^56+12*q^2+1477*q^26+24*q^
3+46*q^4+80*q^5+132*q^6+2859*q^32+202*q^7+287*q^8+372*q^9+461*q^10+564*q^11+660
*q^12+742*q^13+829*q^14+918*q^15+997*q^16+33451*q^43+15937*q^49+1652*q^28+1072*
q^17+1134*q^18+1186*q^19+1232*q^20+1557*q^27+19250*q^40+2385*q^31+5679*q^35+
24684*q^48+7251*q^36+32910*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [1, 4, 2, 3, 5]}, is given by

[1, 2, 5, 14, 41, 118, 322, 830, 2051, 4957, 11907, 28642, 69005, 166021,

    398062]

The number of EVEN permutations avoiding, {[3, 2, 1], [1, 4, 2, 3, 5]},

     is given by

 [1, 1, 3, 7, 21, 59, 161, 412, 1023, 2463, 5942, 14271, 34468, 82885, 198956]

The number of ODD permutations avoiding, {[3, 2, 1], [1, 4, 2, 3, 5]},

     is given by

 [0, 1, 2, 7, 20, 59, 161, 418, 1028, 2494, 5965, 14371, 34537, 83136, 199106]

            For the reverse patterns and complement patterns, we get

EVEN:, [

    1, 1, 2, 7, 21, 59, 161, 412, 1023, 2494, 5965, 14271, 34468, 83136, 199106

    ]

 ODD:, [

    0, 1, 3, 7, 20, 59, 161, 418, 1028, 2463, 5942, 14371, 34537, 82885, 198956

    ]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.121951220, 4.347457627,

    5.704968944, 7.125301205, 8.517796197, 9.788581804, 10.87973461,

    11.79725578, 12.59542062, 13.33419266, 14.05031377]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.364982300, 1.777339082,

    2.298373750, 2.940979403, 3.692343232, 4.501102318, 5.280840341,

    5.943986101, 6.442149974, 6.776187371, 6.976242693]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.86318, 3.15893, 5.28252, 8.64936,

    13.6334, 20.2599, 27.8873, 35.3310, 41.5013, 45.9167, 48.6680]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.009381844257,

    0.1018846605, 0.1910271969, 0.3052962191, 0.4596338905, 0.6628948876,

    0.9149482050, 1.205483283, 1.516945783, 1.828227799, 2.117872842]

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

    2.681797978, 2.688268714, 2.600600407, 2.520203183, 2.527288008,

    2.705575210, 3.146335703, 3.931491113, 5.113222978, 6.697580863,

    8.633421656]

                                end of this data



For the equivalence class of patterns, {{[2, 1, 3], [1, 3, 4, 5, 2]},

    {[1, 3, 2], [4, 1, 2, 3, 5]}, {[1, 3, 2], [2, 3, 4, 1, 5]},

    {[2, 3, 1], [5, 3, 2, 1, 4]}, {[2, 3, 1], [5, 1, 4, 3, 2]},

    {[2, 1, 3], [1, 5, 2, 3, 4]}, {[3, 1, 2], [4, 3, 2, 5, 1]},

    {[3, 1, 2], [2, 5, 4, 3, 1]}}

      the member , {[2, 1, 3], [1, 3, 4, 5, 2]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

    [[2, 1], {[0, 0, 1]}, {1}], [[1, 2, 3], {[0, 1, 0, 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+2*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+2*q^3+4*q^4+6*q^5+6*q^6+9*q^7+12*
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+2*q^3+4*q^4+5
*q^5+7*q^6+9*q^7+13*q^8+13*q^9+18*q^10+22*q^11+28*q^12+34*q^13+41*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+2*q^3+4*q^4+5*q^5+6*q^6+
10*q^7+13*q^8+15*q^9+17*q^10+23*q^11+30*q^12+35*q^13+40*q^14+47*q^15+54*q^16+70
*q^17+82*q^18+97*q^19+106*q^20+111*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+2*q^3+4*q^4+5*q^5+6*q^6+9*q^7+14*q^8+15*q^9+19*q^10+
23*q^11+32*q^12+37*q^13+43*q^14+51*q^15+62*q^16+71*q^17+85*q^18+99*q^19+115*q^
20+127*q^21+149*q^22+172*q^23+204*q^24+239*q^25+266*q^26+284*q^27+283*q^28+263*
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+2*q^3+4*
q^4+5*q^5+6*q^6+9*q^7+13*q^8+16*q^9+19*q^10+25*q^11+32*q^12+40*q^13+46*q^14+52*
q^15+67*q^16+83*q^17+95*q^18+108*q^19+123*q^20+145*q^21+161*q^22+187*q^23+219*q
^24+251*q^25+279*q^26+316*q^27+358*q^28+407*q^29+454*q^30+526*q^31+595*q^32+674
*q^33+730*q^34+756*q^35+738*q^36+674*q^37+568*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+2*q^3+4*q^4+5*q^5+6*q^6+9*q^7+13*q^8+15*
q^9+20*q^10+25*q^11+34*q^12+40*q^13+49*q^14+56*q^15+69*q^16+86*q^17+106*q^18+
120*q^19+137*q^20+158*q^21+185*q^22+215*q^23+248*q^24+279*q^25+322*q^26+359*q^
27+398*q^28+439*q^29+499*q^30+570*q^31+644*q^32+719*q^33+807*q^34+896*q^35+999*
q^36+1111*q^37+1243*q^38+1388*q^39+1547*q^40+1726*q^41+1879*q^42+1994*q^43+2015
*q^44+1937*q^45+1748*q^46+1473*q^47+1142*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+2*q^3+4*q^4+5*q^5+6*q^6+9*q^7+13*q^8+15*q^9+
19*q^10+26*q^11+34*q^12+42*q^13+49*q^14+59*q^15+73*q^16+89*q^17+110*q^18+129*q^
19+148*q^20+172*q^21+202*q^22+240*q^23+280*q^24+319*q^25+355*q^26+407*q^27+465*
q^28+525*q^29+583*q^30+658*q^31+744*q^32+830*q^33+917*q^34+1011*q^35+1114*q^36+
1234*q^37+1356*q^38+1513*q^39+1687*q^40+1889*q^41+2088*q^42+2310*q^43+2550*q^44
+2823*q^45+3096*q^46+3409*q^47+3745*q^48+4134*q^49+4522*q^50+4914*q^51+5237*q^
52+5417*q^53+5389*q^54+5101*q^55+4572*q^56+3843*q^57+3009*q^58+2166*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+2*q^3+4*q^4+5*q^
5+6*q^6+9*q^7+13*q^8+15*q^9+19*q^10+25*q^11+35*q^12+42*q^13+51*q^14+59*q^15+76*
q^16+93*q^17+113*q^18+134*q^19+158*q^20+181*q^21+215*q^22+257*q^23+307*q^24+351
*q^25+397*q^26+451*q^27+516*q^28+588*q^29+677*q^30+765*q^31+863*q^32+964*q^33+
1077*q^34+1199*q^35+1332*q^36+1461*q^37+1610*q^38+1795*q^39+2001*q^40+2196*q^41
+2397*q^42+2624*q^43+2881*q^44+3132*q^45+3420*q^46+3738*q^47+4123*q^48+4552*q^
49+5012*q^50+5515*q^51+6058*q^52+6654*q^53+7298*q^54+7981*q^55+8705*q^56+9476*q
^57+10291*q^58+11183*q^59+12115*q^60+13068*q^61+13892*q^62+14513*q^63+14727*q^
64+14418*q^65+13500*q^66+12012*q^67+10093*q^68+7947*q^69+5817*q^70+3912*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+2*q^3+4*q
^4+5*q^5+6*q^6+9*q^7+13*q^8+15*q^9+19*q^10+25*q^11+34*q^12+43*q^13+51*q^14+61*q
^15+76*q^16+96*q^17+117*q^18+137*q^19+163*q^20+192*q^21+225*q^22+268*q^23+323*q
^24+378*q^25+431*q^26+491*q^27+561*q^28+648*q^29+745*q^30+856*q^31+975*q^32+
1096*q^33+1220*q^34+1372*q^35+1546*q^36+1730*q^37+1913*q^38+2121*q^39+2364*q^40
+2623*q^41+2891*q^42+3180*q^43+3479*q^44+3800*q^45+4133*q^46+4510*q^47+4929*q^
48+5395*q^49+5868*q^50+6387*q^51+6956*q^52+7561*q^53+8167*q^54+8831*q^55+9583*q
^56+10435*q^57+11345*q^58+12405*q^59+13569*q^60+14854*q^61+16176*q^62+17626*q^
63+19207*q^64+20924*q^65+22708*q^66+24596*q^67+26541*q^68+28603*q^69+30736*q^70
+32976*q^71+35225*q^72+37350*q^73+39081*q^74+40072*q^75+40034*q^76+38648*q^77+
35840*q^78+31707*q^79+26618*q^80+21068*q^81+15611*q^82+10736*q^83+6776*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+2*q^3+4*q
^4+5*q^5+6*q^6+9*q^7+13*q^8+15*q^9+19*q^10+25*q^11+34*q^12+42*q^13+52*q^14+61*q
^15+78*q^16+96*q^17+120*q^18+141*q^19+166*q^20+197*q^21+236*q^22+279*q^23+335*q
^24+392*q^25+457*q^26+525*q^27+603*q^28+691*q^29+804*q^30+932*q^31+1069*q^32+
1207*q^33+1360*q^34+1527*q^35+1724*q^36+1942*q^37+2194*q^38+2459*q^39+2744*q^40
+3042*q^41+3369*q^42+3732*q^43+4132*q^44+4551*q^45+4993*q^46+5459*q^47+5986*q^
48+6581*q^49+7219*q^50+7856*q^51+8514*q^52+9230*q^53+10027*q^54+10847*q^55+
11714*q^56+12626*q^57+13640*q^58+14748*q^59+15951*q^60+17233*q^61+18606*q^62+
20064*q^63+21610*q^64+23285*q^65+25124*q^66+27090*q^67+29223*q^68+31589*q^69+
34285*q^70+37247*q^71+40417*q^72+43825*q^73+47478*q^74+51438*q^75+55633*q^76+
60128*q^77+64864*q^78+69797*q^79+74859*q^80+80089*q^81+85416*q^82+90895*q^83+
96328*q^84+101575*q^85+106097*q^86+109338*q^87+110458*q^88+108781*q^89+103811*q
^90+95414*q^91+84009*q^92+70462*q^93+56009*q^94+41927*q^95+29350*q^96+19041*q^
97+11319*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+q^2+q^3, q^6+q^5+2*q^4+3*q^3+3*q^2+3*q+1, q^10+q^9+2*q^8+2*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+2*q^12+4*q^11+6*q^10+6*q^9
+9*q^8+12*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+2*q^18
+4*q^17+5*q^16+7*q^15+9*q^14+13*q^13+13*q^12+18*q^11+22*q^10+28*q^9+34*q^8+41*q
^7+43*q^6+40*q^5+35*q^4+25*q^3+15*q^2+6*q+1, 1+2*q^25+6*q^22+4*q^24+10*q^21+5*q
^23+7*q+21*q^2+2*q^26+41*q^3+65*q^4+86*q^5+102*q^6+111*q^7+106*q^8+97*q^9+82*q^
10+70*q^11+54*q^12+47*q^13+40*q^14+35*q^15+30*q^16+q^28+23*q^17+17*q^18+15*q^19
+13*q^20+q^27, 1+23*q^25+43*q^22+32*q^24+51*q^21+37*q^23+8*q+6*q^30+9*q^29+2*q^
34+2*q^33+28*q^2+19*q^26+63*q^3+112*q^4+167*q^5+219*q^6+4*q^32+263*q^7+283*q^8+
284*q^9+266*q^10+239*q^11+204*q^12+172*q^13+149*q^14+127*q^15+115*q^16+14*q^28+
99*q^17+85*q^18+71*q^19+62*q^20+15*q^27+5*q^31+q^35+q^36, 1+123*q^25+2*q^42+187
*q^22+145*q^24+219*q^21+161*q^23+9*q+q^44+52*q^30+q^45+13*q^37+67*q^29+6*q^39+4
*q^41+25*q^34+9*q^38+32*q^33+36*q^2+108*q^26+92*q^3+182*q^4+301*q^5+434*q^6+40*
q^32+568*q^7+674*q^8+738*q^9+756*q^10+730*q^11+674*q^12+595*q^13+526*q^14+454*q
^15+407*q^16+2*q^43+83*q^28+358*q^17+316*q^18+279*q^19+251*q^20+95*q^27+5*q^40+
46*q^31+19*q^35+16*q^36, 1+499*q^25+40*q^42+719*q^22+570*q^24+807*q^21+644*q^23
+10*q+4*q^51+25*q^44+2*q^53+279*q^30+2*q^52+q^55+q^54+20*q^45+106*q^37+322*q^29
+15*q^46+69*q^39+49*q^41+158*q^34+5*q^50+86*q^38+185*q^33+45*q^2+439*q^26+129*q
^3+282*q^4+512*q^5+806*q^6+215*q^32+1142*q^7+1473*q^8+1748*q^9+1937*q^10+2015*q
^11+1994*q^12+1879*q^13+1726*q^14+1547*q^15+1388*q^16+34*q^43+6*q^49+359*q^28+
1243*q^17+1111*q^18+999*q^19+896*q^20+398*q^27+56*q^40+248*q^31+137*q^35+9*q^48
+120*q^36+13*q^47, 1+1889*q^25+280*q^42+2550*q^22+2088*q^24+2823*q^21+2310*q^23
+11*q+59*q^51+202*q^44+42*q^53+15*q^57+1114*q^30+49*q^52+13*q^58+26*q^55+6*q^60
+34*q^54+172*q^45+525*q^37+1234*q^29+148*q^46+q^65+407*q^39+5*q^61+319*q^41+744
*q^34+73*q^50+2*q^64+465*q^38+830*q^33+19*q^56+55*q^2+1687*q^26+175*q^3+420*q^4
+831*q^5+1419*q^6+917*q^32+2166*q^7+3009*q^8+3843*q^9+4*q^62+4572*q^10+5101*q^
11+5389*q^12+5417*q^13+5237*q^14+4914*q^15+4522*q^16+q^66+240*q^43+89*q^49+1356
*q^28+4134*q^17+3745*q^18+3409*q^19+3096*q^20+1513*q^27+355*q^40+2*q^63+1011*q^
31+658*q^35+110*q^48+9*q^59+583*q^36+129*q^47, 1+6654*q^25+1332*q^42+8705*q^22+
7298*q^24+9476*q^21+7981*q^23+12*q+451*q^51+1077*q^44+351*q^53+181*q^57+4123*q^
30+397*q^52+158*q^58+257*q^55+113*q^60+307*q^54+964*q^45+9*q^71+4*q^74+2196*q^
37+25*q^67+4552*q^29+863*q^46+42*q^65+15*q^69+1795*q^39+93*q^61+1461*q^41+2881*
q^34+516*q^50+51*q^64+2001*q^38+3132*q^33+215*q^56+66*q^2+19*q^68+6058*q^26+231
*q^3+605*q^4+1297*q^5+2389*q^6+3420*q^32+3912*q^7+5817*q^8+7947*q^9+76*q^62+
10093*q^10+12012*q^11+13500*q^12+2*q^76+14418*q^13+14727*q^14+14513*q^15+13892*
q^16+35*q^66+13*q^70+q^77+q^78+1199*q^43+588*q^49+5012*q^28+13068*q^17+12115*q^
18+11183*q^19+10291*q^20+5515*q^27+6*q^72+1610*q^40+5*q^73+59*q^63+3738*q^31+2*
q^75+2624*q^35+677*q^48+134*q^59+2397*q^36+765*q^47, 1+22708*q^25+5395*q^42+
28603*q^22+q^90+24596*q^24+30736*q^21+26541*q^23+13*q+2364*q^51+4510*q^44+1913*
q^53+1220*q^57+14854*q^30+2121*q^52+1096*q^58+1546*q^55+856*q^60+q^91+1730*q^54
+4133*q^45+163*q^71+96*q^74+8167*q^37+323*q^67+16176*q^29+3800*q^46+431*q^65+
225*q^69+6956*q^39+745*q^61+5868*q^41+10435*q^34+34*q^79+2623*q^50+491*q^64+
7561*q^38+11345*q^33+1372*q^56+78*q^2+268*q^68+20924*q^26+298*q^3+847*q^4+1958*
q^5+3872*q^6+12405*q^32+6776*q^7+10736*q^8+15611*q^9+648*q^62+21068*q^10+26618*
q^11+31707*q^12+61*q^76+35840*q^13+38648*q^14+40034*q^15+40072*q^16+378*q^66+
192*q^70+51*q^77+43*q^78+4929*q^43+2891*q^49+17626*q^28+39081*q^17+37350*q^18+
35225*q^19+32976*q^20+19207*q^27+137*q^72+6387*q^40+117*q^73+561*q^63+13569*q^
31+76*q^75+9583*q^35+25*q^80+3180*q^48+13*q^83+19*q^81+6*q^85+5*q^86+4*q^87+2*q
^88+2*q^89+9*q^84+15*q^82+975*q^59+8831*q^36+3479*q^47, 1+74859*q^25+20064*q^42
+90895*q^22+61*q^90+80089*q^24+96328*q^21+85416*q^23+14*q+13*q^97+10027*q^51+
17233*q^44+8514*q^53+5986*q^57+51438*q^30+25*q^94+9230*q^52+5459*q^58+7219*q^55
+4551*q^60+52*q^91+7856*q^54+15951*q^45+1360*q^71+932*q^74+15*q^96+9*q^98+29223
*q^37+2194*q^67+55633*q^29+14748*q^46+34*q^93+6*q^99+2744*q^65+1724*q^69+25124*
q^39+4132*q^61+21610*q^41+37247*q^34+457*q^79+10847*q^50+3042*q^64+5*q^100+
27090*q^38+40417*q^33+6581*q^56+91*q^2+1942*q^68+69797*q^26+377*q^3+1157*q^4+
2872*q^5+6073*q^6+43825*q^32+11319*q^7+19041*q^8+29350*q^9+19*q^95+3732*q^62+4*
q^101+41927*q^10+56009*q^11+70462*q^12+691*q^76+84009*q^13+95414*q^14+103811*q^
15+108781*q^16+2459*q^66+1527*q^70+603*q^77+525*q^78+18606*q^43+2*q^102+11714*q
^49+60128*q^28+110458*q^17+109338*q^18+106097*q^19+101575*q^20+64864*q^27+1207*
q^72+23285*q^40+1069*q^73+2*q^103+3369*q^63+47478*q^31+804*q^75+34285*q^35+q^
104+392*q^80+12626*q^48+q^105+236*q^83+335*q^81+166*q^85+141*q^86+120*q^87+96*q
^88+78*q^89+197*q^84+279*q^82+4993*q^59+31589*q^36+42*q^92+13640*q^47]


 The number of permutations avoiding, {[2, 1, 3], [1, 3, 4, 5, 2]}, is given by

[1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162,

    2391485]

The number of EVEN permutations avoiding, {[2, 1, 3], [1, 3, 4, 5, 2]},

     is given by

[1, 1, 3, 7, 22, 60, 185, 544, 1645, 4912, 14768, 44265, 132863, 398531,

    1195725]

The number of ODD permutations avoiding, {[2, 1, 3], [1, 3, 4, 5, 2]},

     is given by

[0, 1, 2, 7, 19, 62, 180, 550, 1636, 4930, 14757, 44309, 132858, 398631,

    1195760]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 22, 62, 180, 544, 1645, 4930, 14757, 44265, 132863, 398631,

    1195760]

 ODD:, [0, 1, 3, 7, 19, 60, 185, 550, 1636, 4912, 14768, 44309, 132858, 398531,

    1195725]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.600000000, 3.357142857, 5.878048780, 9.229508197,

    13.44657534, 18.55027422, 24.55592807, 31.47612274, 39.32169348,

    48.10209542, 57.82559903, 68.49943550, 80.12992262]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.307940925, 3.053583988,

    3.867747998, 4.746088057, 5.681852361, 6.667753754, 7.696804387,

    8.762585987, 9.859289019, 10.98167502, 12.12502183]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.32659, 9.32438, 14.9595, 22.5254,

    32.2834, 44.4589, 59.2408, 76.7829, 97.2056, 120.597, 147.016]

Skewness: , [Float(undefined), 0., -0.2715454176, -0.3874888379, -0.5178772576,

    -0.6513904452, -0.7700568354, -0.8697675360, -0.9531242974, -1.024092955,

    -1.086090279, -1.141619989, -1.192420692, -1.239674189, -1.284167448]

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

    2.764752181, 3.096910734, 3.379710242, 3.623596822, 3.840434844,

    4.039651467, 4.227900485, 4.409803827, 4.588329694, 4.765458792,

    4.942419805]

                                end of this data



For the equivalence class of patterns, {{[3, 1, 2], [2, 1, 4, 3, 5]},

    {[2, 3, 1], [1, 3, 2, 5, 4]}, {[1, 3, 2], [5, 3, 4, 1, 2]},

    {[1, 3, 2], [4, 5, 2, 3, 1]}, {[2, 3, 1], [2, 1, 4, 3, 5]},

    {[3, 1, 2], [1, 3, 2, 5, 4]}, {[2, 1, 3], [5, 3, 4, 1, 2]},

    {[2, 1, 3], [4, 5, 2, 3, 1]}}

      the member , {[3, 1, 2], [2, 1, 4, 3, 5]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+q^2+q^3, 1+3*q+3*q^2+3*q^3+2*q^4+q^5+q^6, 1+4*q+5*q^2+7*q^3+7*q^
4+5*q^5+5*q^6+3*q^7+2*q^8+q^9+q^10, 1+5*q+7*q^2+11*q^3+15*q^4+15*q^5+15*q^6+13*
q^7+11*q^8+9*q^9+7*q^10+5*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+9*q^2+15*q^3+23*q
^4+29*q^5+35*q^6+34*q^7+34*q^8+31*q^9+30*q^10+26*q^11+21*q^12+17*q^13+13*q^14+
11*q^15+7*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+11*q^2+19*q^3+31*q^4+43*q^
5+60*q^6+70*q^7+75*q^8+78*q^9+80*q^10+79*q^11+75*q^12+70*q^13+60*q^14+55*q^15+
47*q^16+39*q^17+32*q^18+25*q^19+19*q^20+15*q^21+11*q^22+7*q^23+5*q^24+3*q^25+2*
q^26+q^27+q^28, 1+8*q+13*q^2+23*q^3+39*q^4+57*q^5+85*q^6+112*q^7+137*q^8+152*q^
9+170*q^10+180*q^11+187*q^12+190*q^13+184*q^14+179*q^15+168*q^16+156*q^17+141*q
^18+128*q^19+111*q^20+99*q^21+82*q^22+69*q^23+56*q^24+45*q^25+36*q^26+27*q^27+
21*q^28+15*q^29+11*q^30+7*q^31+5*q^32+3*q^33+2*q^34+q^35+q^36, 1+9*q+15*q^2+27*
q^3+47*q^4+71*q^5+110*q^6+154*q^7+206*q^8+254*q^9+298*q^10+340*q^11+376*q^12+
408*q^13+427*q^14+441*q^15+444*q^16+442*q^17+429*q^18+414*q^19+392*q^20+370*q^
21+344*q^22+314*q^23+285*q^24+251*q^25+223*q^26+193*q^27+167*q^28+141*q^29+119*
q^30+96*q^31+78*q^32+62*q^33+49*q^34+38*q^35+29*q^36+21*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+17*q^2+31*q^3+55*q^4+85*q^5+135*q^6
+196*q^7+275*q^8+364*q^9+462*q^10+551*q^11+647*q^12+740*q^13+822*q^14+899*q^15+
956*q^16+1000*q^17+1029*q^18+1050*q^19+1044*q^20+1038*q^21+1017*q^22+987*q^23+
949*q^24+903*q^25+846*q^26+793*q^27+732*q^28+670*q^29+610*q^30+547*q^31+487*q^
32+427*q^33+376*q^34+321*q^35+278*q^36+233*q^37+195*q^38+161*q^39+133*q^40+105*
q^41+84*q^42+66*q^43+51*q^44+40*q^45+29*q^46+21*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+19*q^2+35*q^3+63*q^4+99*q^5+160*q^6+238*q^7
+344*q^8+474*q^9+635*q^10+807*q^11+984*q^12+1181*q^13+1376*q^14+1577*q^15+1766*
q^16+1934*q^17+2081*q^18+2216*q^19+2316*q^20+2398*q^21+2450*q^22+2478*q^23+2488
*q^24+2473*q^25+2437*q^26+2378*q^27+2311*q^28+2222*q^29+2130*q^30+2018*q^31+
1902*q^32+1777*q^33+1652*q^34+1522*q^35+1397*q^36+1267*q^37+1144*q^38+1023*q^39
+909*q^40+802*q^41+699*q^42+608*q^43+521*q^44+449*q^45+377*q^46+316*q^47+261*q^
48+215*q^49+175*q^50+142*q^51+111*q^52+88*q^53+68*q^54+53*q^55+40*q^56+29*q^57+
21*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+21*q^2+39
*q^3+71*q^4+113*q^5+185*q^6+280*q^7+413*q^8+584*q^9+808*q^10+1073*q^11+1376*q^
12+1705*q^13+2070*q^14+2467*q^15+2878*q^16+3293*q^17+3691*q^18+4083*q^19+4439*q
^20+4777*q^21+5066*q^22+5321*q^23+5526*q^24+5693*q^25+5813*q^26+5887*q^27+5916*
q^28+5906*q^29+5858*q^30+5776*q^31+5657*q^32+5511*q^33+5331*q^34+5128*q^35+4917
*q^36+4675*q^37+4432*q^38+4164*q^39+3901*q^40+3626*q^41+3361*q^42+3085*q^43+
2823*q^44+2569*q^45+2324*q^46+2087*q^47+1864*q^48+1651*q^49+1458*q^50+1279*q^51
+1110*q^52+962*q^53+823*q^54+705*q^55+595*q^56+503*q^57+415*q^58+344*q^59+281*q
^60+229*q^61+184*q^62+148*q^63+115*q^64+90*q^65+70*q^66+53*q^67+40*q^68+29*q^69
+21*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+23*q^2+
43*q^3+79*q^4+127*q^5+210*q^6+322*q^7+482*q^8+694*q^9+981*q^10+1339*q^11+1779*q
^12+2295*q^13+2866*q^14+3532*q^15+4263*q^16+5050*q^17+5878*q^18+6732*q^19+7581*
q^20+8438*q^21+9265*q^22+10050*q^23+10790*q^24+11474*q^25+12080*q^26+12617*q^27
+13079*q^28+13438*q^29+13739*q^30+13945*q^31+14070*q^32+14120*q^33+14099*q^34+
13986*q^35+13831*q^36+13595*q^37+13307*q^38+12966*q^39+12576*q^40+12135*q^41+
11664*q^42+11160*q^43+10620*q^44+10076*q^45+9515*q^46+8937*q^47+8359*q^48+7785*
q^49+7209*q^50+6660*q^51+6107*q^52+5584*q^53+5074*q^54+4601*q^55+4138*q^56+3713
*q^57+3310*q^58+2933*q^59+2589*q^60+2273*q^61+1982*q^62+1723*q^63+1488*q^64+
1273*q^65+1089*q^66+921*q^67+777*q^68+649*q^69+541*q^70+443*q^71+364*q^72+295*q
^73+238*q^74+190*q^75+152*q^76+117*q^77+92*q^78+70*q^79+53*q^80+40*q^81+29*q^82
+21*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+25*q^2+
47*q^3+87*q^4+141*q^5+235*q^6+364*q^7+551*q^8+804*q^9+1154*q^10+1605*q^11+2182*
q^12+2897*q^13+3740*q^14+4720*q^15+5862*q^16+7149*q^17+8573*q^18+10135*q^19+
11771*q^20+13505*q^21+15287*q^22+17099*q^23+18900*q^24+20693*q^25+22423*q^26+
24092*q^27+25665*q^28+27113*q^29+28442*q^30+29642*q^31+30688*q^32+31585*q^33+
32333*q^34+32915*q^35+33353*q^36+33637*q^37+33770*q^38+33765*q^39+33620*q^40+
33332*q^41+32933*q^42+32393*q^43+31762*q^44+31008*q^45+30185*q^46+29253*q^47+
28269*q^48+27198*q^49+26084*q^50+24920*q^51+23721*q^52+22499*q^53+21258*q^54+
20019*q^55+18774*q^56+17551*q^57+16336*q^58+15155*q^59+14000*q^60+12886*q^61+
11807*q^62+10791*q^63+9810*q^64+8894*q^65+8017*q^66+7208*q^67+6445*q^68+5750*q^
69+5095*q^70+4500*q^71+3956*q^72+3460*q^73+3015*q^74+2613*q^75+2255*q^76+1935*q
^77+1654*q^78+1401*q^79+1185*q^80+993*q^81+831*q^82+687*q^83+569*q^84+463*q^85+
378*q^86+304*q^87+244*q^88+194*q^89+154*q^90+119*q^91+92*q^92+70*q^93+53*q^94+
40*q^95+29*q^96+21*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+q+2*q^2+q^3, 1+q+2*q^2+3*q^3+3*q^4+3*q^5+q^6, q^10+4*q^9+5*q^8+7*q^7
+7*q^6+5*q^5+5*q^4+3*q^3+2*q^2+q+1, q^15+5*q^14+7*q^13+11*q^12+15*q^11+15*q^10+
15*q^9+13*q^8+11*q^7+9*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+9*q^19+15*q
^18+23*q^17+29*q^16+35*q^15+34*q^14+34*q^13+31*q^12+30*q^11+26*q^10+21*q^9+17*q
^8+13*q^7+11*q^6+7*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+19*q^25+60*q^22+31*q^24+70*q^21
+43*q^23+q+2*q^2+11*q^26+3*q^3+5*q^4+7*q^5+11*q^6+15*q^7+19*q^8+25*q^9+32*q^10+
39*q^11+47*q^12+55*q^13+60*q^14+70*q^15+75*q^16+q^28+79*q^17+80*q^18+78*q^19+75
*q^20+7*q^27, 1+180*q^25+184*q^22+187*q^24+179*q^21+190*q^23+q+85*q^30+112*q^29
+13*q^34+23*q^33+2*q^2+170*q^26+3*q^3+5*q^4+7*q^5+11*q^6+39*q^32+15*q^7+21*q^8+
27*q^9+36*q^10+45*q^11+56*q^12+69*q^13+82*q^14+99*q^15+111*q^16+137*q^28+128*q^
17+141*q^18+156*q^19+168*q^20+152*q^27+57*q^31+8*q^35+q^36, 1+392*q^25+27*q^42+
314*q^22+370*q^24+285*q^21+344*q^23+q+9*q^44+441*q^30+q^45+206*q^37+444*q^29+
110*q^39+47*q^41+340*q^34+154*q^38+376*q^33+2*q^2+414*q^26+3*q^3+5*q^4+7*q^5+11
*q^6+408*q^32+15*q^7+21*q^8+29*q^9+38*q^10+49*q^11+62*q^12+78*q^13+96*q^14+119*
q^15+141*q^16+15*q^43+442*q^28+167*q^17+193*q^18+223*q^19+251*q^20+429*q^27+71*
q^40+427*q^31+298*q^35+254*q^36, 1+610*q^25+740*q^42+427*q^22+547*q^24+376*q^21
+487*q^23+q+55*q^51+551*q^44+17*q^53+903*q^30+31*q^52+q^55+10*q^54+462*q^45+
1029*q^37+846*q^29+364*q^46+956*q^39+822*q^41+1038*q^34+85*q^50+1000*q^38+1017*
q^33+2*q^2+670*q^26+3*q^3+5*q^4+7*q^5+11*q^6+987*q^32+15*q^7+21*q^8+29*q^9+40*q
^10+51*q^11+66*q^12+84*q^13+105*q^14+133*q^15+161*q^16+647*q^43+135*q^49+793*q^
28+195*q^17+233*q^18+278*q^19+321*q^20+732*q^27+899*q^40+949*q^31+1044*q^35+196
*q^48+1050*q^36+275*q^47, 1+802*q^25+2488*q^42+521*q^22+699*q^24+449*q^21+608*q
^23+q+1577*q^51+2450*q^44+1181*q^53+474*q^57+1397*q^30+1376*q^52+344*q^58+807*q
^55+160*q^60+984*q^54+2398*q^45+2222*q^37+1267*q^29+2316*q^46+11*q^65+2378*q^39
+99*q^61+2473*q^41+1902*q^34+1766*q^50+19*q^64+2311*q^38+1777*q^33+635*q^56+2*q
^2+909*q^26+3*q^3+5*q^4+7*q^5+11*q^6+1652*q^32+15*q^7+21*q^8+29*q^9+63*q^62+40*
q^10+53*q^11+68*q^12+88*q^13+111*q^14+142*q^15+175*q^16+q^66+2478*q^43+1934*q^
49+1144*q^28+215*q^17+261*q^18+316*q^19+377*q^20+1023*q^27+2437*q^40+35*q^63+
1522*q^31+2018*q^35+2081*q^48+238*q^59+2130*q^36+2216*q^47, 1+962*q^25+4917*q^
42+595*q^22+823*q^24+503*q^21+705*q^23+q+5887*q^51+5331*q^44+5693*q^53+4777*q^
57+1864*q^30+5813*q^52+4439*q^58+5321*q^55+3691*q^60+5526*q^54+5511*q^45+280*q^
71+71*q^74+3626*q^37+1073*q^67+1651*q^29+5657*q^46+1705*q^65+584*q^69+4164*q^39
+3293*q^61+4675*q^41+2823*q^34+5916*q^50+2070*q^64+3901*q^38+2569*q^33+5066*q^
56+2*q^2+808*q^68+1110*q^26+3*q^3+5*q^4+7*q^5+11*q^6+2324*q^32+15*q^7+21*q^8+29
*q^9+2878*q^62+40*q^10+53*q^11+70*q^12+21*q^76+90*q^13+115*q^14+148*q^15+184*q^
16+1376*q^66+413*q^70+12*q^77+q^78+5128*q^43+5906*q^49+1458*q^28+229*q^17+281*q
^18+344*q^19+415*q^20+1279*q^27+185*q^72+4432*q^40+113*q^73+2467*q^63+2087*q^31
+39*q^75+3085*q^35+5858*q^48+4083*q^59+3361*q^36+5776*q^47, 1+1089*q^25+7785*q^
42+649*q^22+13*q^90+921*q^24+541*q^21+777*q^23+q+12576*q^51+8937*q^44+13307*q^
53+14099*q^57+2273*q^30+12966*q^52+14120*q^58+13831*q^55+13945*q^60+q^91+13595*
q^54+9515*q^45+7581*q^71+5050*q^74+5074*q^37+10790*q^67+1982*q^29+10076*q^46+
12080*q^65+9265*q^69+6107*q^39+13739*q^61+7209*q^41+3713*q^34+1779*q^79+12135*q
^50+12617*q^64+5584*q^38+3310*q^33+13986*q^56+2*q^2+10050*q^68+1273*q^26+3*q^3+
5*q^4+7*q^5+11*q^6+2933*q^32+15*q^7+21*q^8+29*q^9+13438*q^62+40*q^10+53*q^11+70
*q^12+3532*q^76+92*q^13+117*q^14+152*q^15+190*q^16+11474*q^66+8438*q^70+2866*q^
77+2295*q^78+8359*q^43+11664*q^49+1723*q^28+238*q^17+295*q^18+364*q^19+443*q^20
+1488*q^27+6732*q^72+6660*q^40+5878*q^73+13079*q^63+2589*q^31+4263*q^75+4138*q^
35+1339*q^80+11160*q^48+482*q^83+981*q^81+210*q^85+127*q^86+79*q^87+43*q^88+23*
q^89+322*q^84+694*q^82+14070*q^59+4601*q^36+10620*q^47, 1+1185*q^25+10791*q^42+
687*q^22+4720*q^90+993*q^24+569*q^21+831*q^23+q+551*q^97+21258*q^51+12886*q^44+
23721*q^53+28269*q^57+2613*q^30+1605*q^94+22499*q^52+29253*q^58+26084*q^55+
31008*q^60+3740*q^91+24920*q^54+14000*q^45+32333*q^71+29642*q^74+804*q^96+364*q
^98+6445*q^37+33770*q^67+2255*q^29+15155*q^46+2182*q^93+235*q^99+33620*q^65+
33353*q^69+8017*q^39+31762*q^61+9810*q^41+4500*q^34+22423*q^79+20019*q^50+33332
*q^64+141*q^100+7208*q^38+3956*q^33+27198*q^56+2*q^2+33637*q^68+1401*q^26+3*q^3
+5*q^4+7*q^5+11*q^6+3460*q^32+15*q^7+21*q^8+29*q^9+1154*q^95+32393*q^62+87*q^
101+40*q^10+53*q^11+70*q^12+27113*q^76+92*q^13+119*q^14+154*q^15+194*q^16+33765
*q^66+32915*q^70+25665*q^77+24092*q^78+11807*q^43+47*q^102+18774*q^49+1935*q^28
+244*q^17+304*q^18+378*q^19+463*q^20+1654*q^27+31585*q^72+8894*q^40+30688*q^73+
25*q^103+32933*q^63+3015*q^31+28442*q^75+5095*q^35+14*q^104+20693*q^80+17551*q^
48+q^105+15287*q^83+18900*q^81+11771*q^85+10135*q^86+8573*q^87+7149*q^88+5862*q
^89+13505*q^84+17099*q^82+30185*q^59+5750*q^36+2897*q^92+16336*q^47]


 The number of permutations avoiding, {[3, 1, 2], [2, 1, 4, 3, 5]}, is given by

[1, 2, 5, 14, 41, 121, 354, 1021, 2901, 8130, 22513, 61713, 167746, 452789,

    1215197]

The number of EVEN permutations avoiding, {[3, 1, 2], [2, 1, 4, 3, 5]},

     is given by

[1, 1, 2, 7, 21, 60, 177, 510, 1450, 4065, 11256, 30857, 83873, 226395, 607599]

The number of ODD permutations avoiding, {[3, 1, 2], [2, 1, 4, 3, 5]},

     is given by

[0, 1, 3, 7, 20, 61, 177, 511, 1451, 4065, 11257, 30856, 83873, 226394, 607598]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 7, 21, 61, 177, 510, 1450, 4065, 11257, 30857, 83873, 226394,

    607598]

 ODD:, [0, 1, 2, 7, 20, 60, 177, 511, 1451, 4065, 11256, 30856, 83873, 226395,

    607599]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.400000000, 2.642857143, 4.243902439, 6.223140496,

    8.590395480, 11.34867777, 14.49810410, 18.03800738, 21.96788522,

    26.28778377, 30.99842023, 36.10118179, 41.59806352]

                 The standard deviation for each n is given by

[0., 0.5000000000, 1.019803903, 1.630387459, 2.324634645, 3.105547165,

    3.976487158, 4.937004798, 5.984100487, 7.113845408, 8.322257710,

    9.605648553, 10.96071433, 12.38452685, 13.87448879]

                          The centralized moments are

Second: , [0., 0.250000, 1.04000, 2.65816, 5.40393, 9.64442, 15.8125, 24.3740,

    35.8095, 50.6068, 69.2600, 92.2685, 120.137, 153.377, 192.501]

Skewness: , [Float(undefined), 0., 0.2715454176, 0.3874842230, 0.4141164409,

    0.4107114756, 0.3995607238, 0.3876817900, 0.3768096515, 0.3670380826,

    0.3580362850, 0.3494566157, 0.3410212650, 0.3325372246, 0.3238845410]

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

    2.623166044, 2.742250676, 2.796806700, 2.821651108, 2.833597837,

    2.839790357, 2.843037101, 2.844490281, 2.844639920, 2.843711376,

    2.842001169]

                                end of this data



For the equivalence class of patterns, {{[3, 2, 1], [3, 4, 1, 2, 5]},

    {[3, 2, 1], [1, 4, 5, 2, 3]}, {[1, 2, 3], [3, 2, 5, 4, 1]},

    {[1, 2, 3], [5, 2, 1, 4, 3]}}

      the member , {[3, 2, 1], [1, 4, 5, 2, 3]}, has a scheme of depth , 4

                                  here it is:

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

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

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

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

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

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

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

    [[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, 1+3*q+5*q^2+4*q^3+q^4, 1+4*q+9*q^2+12*q^3+9*q^4+4*q^5+2*q
^6, 1+5*q+14*q^2+25*q^3+29*q^4+21*q^5+12*q^6+7*q^7+4*q^8+q^9, 1+6*q+20*q^2+44*q
^3+67*q^4+70*q^5+50*q^6+31*q^7+20*q^8+13*q^9+8*q^10+4*q^11+2*q^12, 1+7*q+27*q^2
+70*q^3+130*q^4+175*q^5+169*q^6+119*q^7+77*q^8+51*q^9+35*q^10+24*q^11+17*q^12+
10*q^13+7*q^14+4*q^15+q^16, 1+8*q+35*q^2+104*q^3+226*q^4+368*q^5+449*q^6+407*q^
7+284*q^8+187*q^9+126*q^10+88*q^11+63*q^12+46*q^13+33*q^14+24*q^15+17*q^16+12*q
^17+8*q^18+4*q^19+2*q^20, 1+9*q+44*q^2+147*q^3+364*q^4+690*q^5+1010*q^6+1136*q^
7+979*q^8+678*q^9+451*q^10+307*q^11+217*q^12+157*q^13+117*q^14+87*q^15+67*q^16+
49*q^17+38*q^18+29*q^19+22*q^20+15*q^21+10*q^22+7*q^23+4*q^24+q^25, 1+10*q+54*q
^2+200*q^3+554*q^4+1192*q^5+2022*q^6+2707*q^7+2844*q^8+2352*q^9+1620*q^10+1085*
q^11+745*q^12+530*q^13+387*q^14+290*q^15+219*q^16+170*q^17+130*q^18+101*q^19+80
*q^20+63*q^21+48*q^22+39*q^23+30*q^24+21*q^25+16*q^26+12*q^27+8*q^28+4*q^29+2*q
^30, 1+11*q+65*q^2+264*q^3+807*q^4+1936*q^5+3716*q^6+5741*q^7+7122*q^8+7061*q^9
+5646*q^10+3874*q^11+2610*q^12+1805*q^13+1292*q^14+949*q^15+714*q^16+543*q^17+
423*q^18+327*q^19+259*q^20+205*q^21+163*q^22+129*q^23+107*q^24+85*q^25+67*q^26+
54*q^27+43*q^28+34*q^29+27*q^30+20*q^31+15*q^32+10*q^33+7*q^34+4*q^35+q^36, 1+
12*q+77*q^2+340*q^3+1135*q^4+2996*q^5+6396*q^6+11151*q^7+15897*q^8+18461*q^9+
17416*q^10+13546*q^11+9273*q^12+6282*q^13+4374*q^14+3148*q^15+2324*q^16+1754*q^
17+1341*q^18+1045*q^19+814*q^20+647*q^21+516*q^22+411*q^23+330*q^24+273*q^25+
220*q^26+178*q^27+147*q^28+122*q^29+99*q^30+80*q^31+65*q^32+54*q^33+45*q^34+36*
q^35+27*q^36+20*q^37+16*q^38+12*q^39+8*q^40+4*q^41+2*q^42, 1+13*q+90*q^2+429*q^
3+1551*q^4+4459*q^5+10452*q^6+20227*q^7+32458*q^8+43133*q^9+47277*q^10+42732*q^
11+32491*q^12+22218*q^13+15135*q^14+10606*q^15+7673*q^16+5690*q^17+4308*q^18+
3306*q^19+2579*q^20+2018*q^21+1607*q^22+1284*q^23+1029*q^24+830*q^25+685*q^26+
555*q^27+453*q^28+377*q^29+315*q^30+259*q^31+215*q^32+177*q^33+149*q^34+127*q^
35+105*q^36+85*q^37+72*q^38+61*q^39+50*q^40+39*q^41+32*q^42+25*q^43+20*q^44+15*
q^45+10*q^46+7*q^47+4*q^48+q^49, 1+14*q+104*q^2+532*q^3+2069*q^4+6426*q^5+16374
*q^6+34735*q^7+61761*q^8+92152*q^9+115082*q^10+119870*q^11+104409*q^12+77927*q^
13+53289*q^14+36504*q^15+25739*q^16+18714*q^17+13936*q^18+10583*q^19+8148*q^20+
6364*q^21+4998*q^22+3984*q^23+3188*q^24+2562*q^25+2078*q^26+1709*q^27+1389*q^28
+1140*q^29+951*q^30+794*q^31+657*q^32+548*q^33+457*q^34+390*q^35+331*q^36+274*q
^37+228*q^38+197*q^39+170*q^40+143*q^41+120*q^42+101*q^43+84*q^44+71*q^45+60*q^
46+51*q^47+42*q^48+33*q^49+26*q^50+20*q^51+16*q^52+12*q^53+8*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+9*q^2+4*q+2), q^6*(q^9+5*q^8+14*q^7+25*q^6+29*q^5+21*q^4+12*q^3+7*q^2+4*q+1),
q^9*(q^12+6*q^11+20*q^10+44*q^9+67*q^8+70*q^7+50*q^6+31*q^5+20*q^4+13*q^3+8*q^2
+4*q+2), q^12*(q^16+7*q^15+27*q^14+70*q^13+130*q^12+175*q^11+169*q^10+119*q^9+
77*q^8+51*q^7+35*q^6+24*q^5+17*q^4+10*q^3+7*q^2+4*q+1), q^16*(q^20+8*q^19+35*q^
18+104*q^17+226*q^16+368*q^15+449*q^14+407*q^13+284*q^12+187*q^11+126*q^10+88*q
^9+63*q^8+46*q^7+33*q^6+24*q^5+17*q^4+12*q^3+8*q^2+4*q+2), q^20*(1+q^25+147*q^
22+9*q^24+364*q^21+44*q^23+4*q+7*q^2+10*q^3+15*q^4+22*q^5+29*q^6+38*q^7+49*q^8+
67*q^9+87*q^10+117*q^11+157*q^12+217*q^13+307*q^14+451*q^15+678*q^16+979*q^17+
1136*q^18+1010*q^19+690*q^20), q^25*(2+1192*q^25+2844*q^22+2022*q^24+2352*q^21+
2707*q^23+4*q+q^30+10*q^29+8*q^2+554*q^26+12*q^3+16*q^4+21*q^5+30*q^6+39*q^7+48
*q^8+63*q^9+80*q^10+101*q^11+130*q^12+170*q^13+219*q^14+290*q^15+387*q^16+54*q^
28+530*q^17+745*q^18+1085*q^19+1620*q^20+200*q^27), q^30*(1+3874*q^25+1292*q^22
+2610*q^24+949*q^21+1805*q^23+4*q+3716*q^30+5741*q^29+65*q^34+264*q^33+7*q^2+
5646*q^26+10*q^3+15*q^4+20*q^5+27*q^6+807*q^32+34*q^7+43*q^8+54*q^9+67*q^10+85*
q^11+107*q^12+129*q^13+163*q^14+205*q^15+259*q^16+7122*q^28+327*q^17+423*q^18+
543*q^19+714*q^20+7061*q^27+1936*q^31+11*q^35+q^36), q^36*(2+1754*q^25+q^42+814
*q^22+1341*q^24+647*q^21+1045*q^23+4*q+9273*q^30+2996*q^37+6282*q^29+340*q^39+
12*q^41+15897*q^34+1135*q^38+18461*q^33+8*q^2+2324*q^26+12*q^3+16*q^4+20*q^5+27
*q^6+17416*q^32+36*q^7+45*q^8+54*q^9+65*q^10+80*q^11+99*q^12+122*q^13+147*q^14+
178*q^15+220*q^16+4374*q^28+273*q^17+330*q^18+411*q^19+516*q^20+3148*q^27+77*q^
40+13546*q^31+11151*q^35+6396*q^36), q^42*(1+1029*q^25+20227*q^42+555*q^22+830*
q^24+453*q^21+685*q^23+4*q+4459*q^44+3306*q^30+1551*q^45+32491*q^37+2579*q^29+
429*q^46+47277*q^39+32458*q^41+10606*q^34+42732*q^38+7673*q^33+7*q^2+1284*q^26+
10*q^3+15*q^4+20*q^5+25*q^6+5690*q^32+32*q^7+39*q^8+50*q^9+61*q^10+72*q^11+85*q
^12+105*q^13+127*q^14+149*q^15+177*q^16+10452*q^43+q^49+2018*q^28+215*q^17+259*
q^18+315*q^19+377*q^20+1607*q^27+43133*q^40+4308*q^31+15135*q^35+13*q^48+22218*
q^36+90*q^47), q^49*(2+794*q^25+53289*q^42+457*q^22+657*q^24+390*q^21+548*q^23+
4*q+6426*q^51+104409*q^44+532*q^53+2078*q^30+2069*q^52+14*q^55+104*q^54+119870*
q^45+10583*q^37+1709*q^29+115082*q^46+18714*q^39+36504*q^41+4998*q^34+16374*q^
50+13936*q^38+3984*q^33+q^56+8*q^2+951*q^26+12*q^3+16*q^4+20*q^5+26*q^6+3188*q^
32+33*q^7+42*q^8+51*q^9+60*q^10+71*q^11+84*q^12+101*q^13+120*q^14+143*q^15+170*
q^16+77927*q^43+34735*q^49+1389*q^28+197*q^17+228*q^18+274*q^19+331*q^20+1140*q
^27+25739*q^40+2562*q^31+6364*q^35+61761*q^48+8148*q^36+92152*q^47)]


 The number of permutations avoiding, {[3, 2, 1], [1, 4, 5, 2, 3]}, is given by

[1, 2, 5, 14, 41, 119, 336, 924, 2492, 6636, 17536, 46137, 121095, 317434,

    831571]

The number of EVEN permutations avoiding, {[3, 2, 1], [1, 4, 5, 2, 3]},

     is given by

 [1, 1, 3, 7, 21, 60, 168, 464, 1244, 3324, 8760, 23085, 60523, 158761, 415717]

The number of ODD permutations avoiding, {[3, 2, 1], [1, 4, 5, 2, 3]},

     is given by

 [0, 1, 2, 7, 20, 59, 168, 460, 1248, 3312, 8776, 23052, 60572, 158673, 415854]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 2, 7, 21, 59, 168, 464, 1244, 3312, 8776, 23085, 60523, 158673,

    415854]

 ODD:, [0, 1, 3, 7, 20, 60, 168, 460, 1248, 3324, 8760, 23052, 60572, 158761,

    415717]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.200000000, 2.071428571, 3.073170732, 4.126050420,

    5.172619048, 6.179653680, 7.131621188, 8.026220615, 8.869867701,

    9.673104016, 10.44698790, 11.20100871, 11.94234527]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.7483314774, 1.032630878, 1.368464404, 1.741990661,

    2.139473188, 2.539339292, 2.917046556, 3.252565120, 3.534558259,

    3.760747306, 3.935754595, 4.068066337, 4.167329448]

                          The centralized moments are

Second: , [0., 0.250000, 0.560000, 1.06633, 1.87269, 3.03453, 4.57735, 6.44824,

    8.50916, 10.5792, 12.4931, 14.1432, 15.4902, 16.5492, 17.3666]

Skewness: , [Float(undefined), 0., -0.3436215967, -0.1429626094, 0.09640199491,

    0.3405256483, 0.5937718096, 0.8468960764, 1.092926469, 1.327026186,

    1.544149171, 1.738531729, 1.904595653, 2.038238840, 2.137793901]

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

    2.667632108, 2.917084482, 3.324993209, 3.918441251, 4.707412688,

    5.688956401, 6.841470900, 8.119927699, 9.456407695, 10.76859520,

    11.97362209]

                                end of this data



                         Out of a total of , 42, cases

                      34, were successful and , 8, failed

                             Success Rate: , 0.810

                             Here are the failures

{{{[2, 1, 3], [1, 5, 3, 4, 2]}, {[1, 3, 2], [4, 2, 3, 1, 5]},

    {[3, 1, 2], [2, 4, 3, 5, 1]}, {[2, 3, 1], [5, 1, 3, 2, 4]}}, {

    {[3, 2, 1], [3, 1, 2, 5, 4]}, {[3, 2, 1], [2, 1, 5, 3, 4]},

    {[3, 2, 1], [2, 1, 4, 5, 3]}, {[3, 2, 1], [2, 3, 1, 5, 4]},

    {[1, 2, 3], [4, 3, 5, 1, 2]}, {[1, 2, 3], [3, 5, 4, 1, 2]},

    {[1, 2, 3], [4, 5, 1, 3, 2]}, {[1, 2, 3], [4, 5, 2, 1, 3]}}, {

    {[3, 2, 1], [3, 1, 5, 2, 4]}, {[3, 2, 1], [2, 4, 1, 5, 3]},

    {[1, 2, 3], [4, 2, 5, 1, 3]}, {[1, 2, 3], [3, 5, 1, 4, 2]}}, {

    {[3, 2, 1], [1, 2, 4, 3, 5]}, {[3, 2, 1], [1, 3, 2, 4, 5]},

    {[1, 2, 3], [5, 3, 4, 2, 1]}, {[1, 2, 3], [5, 4, 2, 3, 1]}},

    {{[3, 2, 1], [2, 1, 3, 5, 4]}, {[1, 2, 3], [4, 5, 3, 1, 2]}},

    {{[3, 2, 1], [1, 2, 3, 4, 5]}, {[1, 2, 3], [5, 4, 3, 2, 1]}}, {

    {[3, 2, 1], [1, 3, 2, 5, 4]}, {[1, 2, 3], [5, 3, 4, 1, 2]},

    {[1, 2, 3], [4, 5, 2, 3, 1]}, {[3, 2, 1], [2, 1, 4, 3, 5]}}, {

    {[3, 2, 1], [2, 1, 3, 4, 5]}, {[3, 2, 1], [1, 2, 3, 5, 4]},

    {[1, 2, 3], [4, 5, 3, 2, 1]}, {[1, 2, 3], [5, 4, 3, 1, 2]}}}