Pair of 4-patterns

             There all together, 56, different equivalence classes

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}

%2 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3
+13*q^4+14*q^5+13*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+26*q^4+32*q^
5+36*q^6+39*q^7+39*q^8+36*q^9+32*q^10+26*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*
q+16*q^2+31*q^3+48*q^4+63*q^5+79*q^6+91*q^7+100*q^8+110*q^9+114*q^10+114*q^11+
110*q^12+100*q^13+91*q^14+79*q^15+63*q^16+48*q^17+31*q^18+16*q^19+6*q^20+q^21,
1+7*q+22*q^2+48*q^3+83*q^4+119*q^5+157*q^6+193*q^7+226*q^8+258*q^9+287*q^10+312
*q^11+330*q^12+338*q^13+344*q^14+338*q^15+330*q^16+312*q^17+287*q^18+258*q^19+
226*q^20+193*q^21+157*q^22+119*q^23+83*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+
29*q^2+71*q^3+136*q^4+214*q^5+300*q^6+388*q^7+475*q^8+562*q^9+653*q^10+742*q^11
+822*q^12+890*q^13+949*q^14+996*q^15+1036*q^16+1055*q^17+1064*q^18+1055*q^19+
1036*q^20+996*q^21+949*q^22+890*q^23+822*q^24+742*q^25+653*q^26+562*q^27+475*q^
28+388*q^29+300*q^30+214*q^31+136*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^
2+101*q^3+213*q^4+367*q^5+551*q^6+753*q^7+961*q^8+1176*q^9+1401*q^10+1639*q^11+
1889*q^12+2130*q^13+2350*q^14+2559*q^15+2744*q^16+2908*q^17+3062*q^18+3187*q^19
+3285*q^20+3343*q^21+3367*q^22+3367*q^23+3343*q^24+3285*q^25+3187*q^26+3062*q^
27+2908*q^28+2744*q^29+2559*q^30+2350*q^31+2130*q^32+1889*q^33+1639*q^34+1401*q
^35+1176*q^36+961*q^37+753*q^38+551*q^39+367*q^40+213*q^41+101*q^42+37*q^43+9*q
^44+q^45, 1+10*q+46*q^2+139*q^3+321*q^4+603*q^5+973*q^6+1410*q^7+1886*q^8+2391*
q^9+2924*q^10+3493*q^11+4112*q^12+4766*q^13+5436*q^14+6106*q^15+6748*q^16+7358*
q^17+7925*q^18+8463*q^19+8980*q^20+9458*q^21+9882*q^22+10241*q^23+10523*q^24+
10728*q^25+10860*q^26+10916*q^27+10916*q^28+10860*q^29+10728*q^30+10523*q^31+
10241*q^32+9882*q^33+9458*q^34+8980*q^35+8463*q^36+7925*q^37+7358*q^38+6748*q^
39+6106*q^40+5436*q^41+4766*q^42+4112*q^43+3493*q^44+2924*q^45+2391*q^46+1886*q
^47+1410*q^48+973*q^49+603*q^50+321*q^51+139*q^52+46*q^53+10*q^54+q^55, 1+11*q+
56*q^2+186*q^3+468*q^4+954*q^5+1655*q^6+2549*q^7+3587*q^8+4732*q^9+5965*q^10+
7289*q^11+8729*q^12+10300*q^13+11987*q^14+13776*q^15+15634*q^16+17506*q^17+
19339*q^18+21124*q^19+22847*q^20+24530*q^21+26177*q^22+27780*q^23+29306*q^24+
30723*q^25+31976*q^26+33037*q^27+33920*q^28+34654*q^29+35257*q^30+35733*q^31+
36036*q^32+36146*q^33+36036*q^34+35733*q^35+35257*q^36+34654*q^37+33920*q^38+
33037*q^39+31976*q^40+30723*q^41+29306*q^42+27780*q^43+26177*q^44+24530*q^45+
22847*q^46+21124*q^47+19339*q^48+17506*q^49+15634*q^50+13776*q^51+11987*q^52+
10300*q^53+8729*q^54+7289*q^55+5965*q^56+4732*q^57+3587*q^58+2549*q^59+1655*q^
60+954*q^61+468*q^62+186*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+243*q^3+663*q
^4+1460*q^5+2719*q^6+4455*q^7+6611*q^8+9108*q^9+11885*q^10+14913*q^11+18217*q^
12+21839*q^13+25797*q^14+30110*q^15+34773*q^16+39720*q^17+44865*q^18+50094*q^19
+55313*q^20+60520*q^21+65664*q^22+70780*q^23+75901*q^24+80993*q^25+85988*q^26+
90796*q^27+95319*q^28+99505*q^29+103351*q^30+106851*q^31+109992*q^32+112776*q^
33+115160*q^34+117146*q^35+118720*q^36+119874*q^37+120575*q^38+120822*q^39+
120575*q^40+119874*q^41+118720*q^42+117146*q^43+115160*q^44+112776*q^45+109992*
q^46+106851*q^47+103351*q^48+99505*q^49+95319*q^50+90796*q^51+85988*q^52+80993*
q^53+75901*q^54+70780*q^55+65664*q^56+60520*q^57+55313*q^58+50094*q^59+44865*q^
60+39720*q^61+34773*q^62+30110*q^63+25797*q^64+21839*q^65+18217*q^66+14913*q^67
+11885*q^68+9108*q^69+6611*q^70+4455*q^71+2719*q^72+1460*q^73+663*q^74+243*q^75
+67*q^76+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+916*q^4+2170*q^5+4328*q^6+7542*q^7
+11816*q^8+17045*q^9+23100*q^10+29877*q^11+37361*q^12+45609*q^13+54681*q^14+
64655*q^15+75605*q^16+87560*q^17+100469*q^18+114152*q^19+128411*q^20+143056*q^
21+157919*q^22+172942*q^23+188104*q^24+203425*q^25+218916*q^26+234488*q^27+
250040*q^28+265438*q^29+280524*q^30+295131*q^31+309110*q^32+322302*q^33+334673*
q^34+346183*q^35+356872*q^36+366795*q^37+375944*q^38+384179*q^39+391347*q^40+
397284*q^41+401873*q^42+405177*q^43+407287*q^44+408299*q^45+408299*q^46+407287*
q^47+405177*q^48+401873*q^49+397284*q^50+391347*q^51+384179*q^52+375944*q^53+
366795*q^54+356872*q^55+346183*q^56+334673*q^57+322302*q^58+309110*q^59+295131*
q^60+280524*q^61+265438*q^62+250040*q^63+234488*q^64+218916*q^65+203425*q^66+
188104*q^67+172942*q^68+157919*q^69+143056*q^70+128411*q^71+114152*q^72+100469*
q^73+87560*q^74+75605*q^75+64655*q^76+54681*q^77+45609*q^78+37361*q^79+29877*q^
80+23100*q^81+17045*q^82+11816*q^83+7542*q^84+4328*q^85+2170*q^86+916*q^87+311*
q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1238*q^4+3143*q^5+6695*q^6+
12395*q^7+20507*q^8+31024*q^9+43775*q^10+58539*q^11+75189*q^12+93747*q^13+
114316*q^14+137050*q^15+162153*q^16+189834*q^17+220261*q^18+253427*q^19+289120*
q^20+327001*q^21+366673*q^22+407717*q^23+449887*q^24+493083*q^25+537221*q^26+
582255*q^27+628126*q^28+674741*q^29+722014*q^30+769734*q^31+817535*q^32+864845*
q^33+911224*q^34+956221*q^35+999652*q^36+1041572*q^37+1082000*q^38+1120935*q^39
+1158225*q^40+1193545*q^41+1226553*q^42+1256927*q^43+1284484*q^44+1309121*q^45+
1330824*q^46+1349548*q^47+1365309*q^48+1378062*q^49+1387767*q^50+1394307*q^51+
1397608*q^52+1397608*q^53+1394307*q^54+1387767*q^55+1378062*q^56+1365309*q^57+
1349548*q^58+1330824*q^59+1309121*q^60+1284484*q^61+1256927*q^62+1226553*q^63+
1193545*q^64+1158225*q^65+1120935*q^66+1082000*q^67+1041572*q^68+999652*q^69+
956221*q^70+911224*q^71+864845*q^72+817535*q^73+769734*q^74+722014*q^75+674741*
q^76+628126*q^77+582255*q^78+537221*q^79+493083*q^80+449887*q^81+407717*q^82+
366673*q^83+327001*q^84+289120*q^85+253427*q^86+220261*q^87+189834*q^88+162153*
q^89+137050*q^90+114316*q^91+93747*q^92+75189*q^93+58539*q^94+43775*q^95+31024*
q^96+20507*q^97+12395*q^98+6695*q^99+3143*q^100+1238*q^101+391*q^102+92*q^103+
14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3
+13*q^4+14*q^5+13*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+26*q^4+32*q^
5+36*q^6+39*q^7+39*q^8+36*q^9+32*q^10+26*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*
q+16*q^2+31*q^3+48*q^4+63*q^5+79*q^6+91*q^7+100*q^8+110*q^9+114*q^10+114*q^11+
110*q^12+100*q^13+91*q^14+79*q^15+63*q^16+48*q^17+31*q^18+16*q^19+6*q^20+q^21,
1+7*q+22*q^2+48*q^3+83*q^4+119*q^5+157*q^6+193*q^7+226*q^8+258*q^9+287*q^10+312
*q^11+330*q^12+338*q^13+344*q^14+338*q^15+330*q^16+312*q^17+287*q^18+258*q^19+
226*q^20+193*q^21+157*q^22+119*q^23+83*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+
29*q^2+71*q^3+136*q^4+214*q^5+300*q^6+388*q^7+475*q^8+562*q^9+653*q^10+742*q^11
+822*q^12+890*q^13+949*q^14+996*q^15+1036*q^16+1055*q^17+1064*q^18+1055*q^19+
1036*q^20+996*q^21+949*q^22+890*q^23+822*q^24+742*q^25+653*q^26+562*q^27+475*q^
28+388*q^29+300*q^30+214*q^31+136*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+9*q+37*q^
2+101*q^3+213*q^4+367*q^5+551*q^6+753*q^7+961*q^8+1176*q^9+1401*q^10+1639*q^11+
1889*q^12+2130*q^13+2350*q^14+2559*q^15+2744*q^16+2908*q^17+3062*q^18+3187*q^19
+3285*q^20+3343*q^21+3367*q^22+3367*q^23+3343*q^24+3285*q^25+3187*q^26+3062*q^
27+2908*q^28+2744*q^29+2559*q^30+2350*q^31+2130*q^32+1889*q^33+1639*q^34+1401*q
^35+1176*q^36+961*q^37+753*q^38+551*q^39+367*q^40+213*q^41+101*q^42+37*q^43+9*q
^44+q^45, 1+10*q+46*q^2+139*q^3+321*q^4+603*q^5+973*q^6+1410*q^7+1886*q^8+2391*
q^9+2924*q^10+3493*q^11+4112*q^12+4766*q^13+5436*q^14+6106*q^15+6748*q^16+7358*
q^17+7925*q^18+8463*q^19+8980*q^20+9458*q^21+9882*q^22+10241*q^23+10523*q^24+
10728*q^25+10860*q^26+10916*q^27+10916*q^28+10860*q^29+10728*q^30+10523*q^31+
10241*q^32+9882*q^33+9458*q^34+8980*q^35+8463*q^36+7925*q^37+7358*q^38+6748*q^
39+6106*q^40+5436*q^41+4766*q^42+4112*q^43+3493*q^44+2924*q^45+2391*q^46+1886*q
^47+1410*q^48+973*q^49+603*q^50+321*q^51+139*q^52+46*q^53+10*q^54+q^55, 1+11*q+
56*q^2+186*q^3+468*q^4+954*q^5+1655*q^6+2549*q^7+3587*q^8+4732*q^9+5965*q^10+
7289*q^11+8729*q^12+10300*q^13+11987*q^14+13776*q^15+15634*q^16+17506*q^17+
19339*q^18+21124*q^19+22847*q^20+24530*q^21+26177*q^22+27780*q^23+29306*q^24+
30723*q^25+31976*q^26+33037*q^27+33920*q^28+34654*q^29+35257*q^30+35733*q^31+
36036*q^32+36146*q^33+36036*q^34+35733*q^35+35257*q^36+34654*q^37+33920*q^38+
33037*q^39+31976*q^40+30723*q^41+29306*q^42+27780*q^43+26177*q^44+24530*q^45+
22847*q^46+21124*q^47+19339*q^48+17506*q^49+15634*q^50+13776*q^51+11987*q^52+
10300*q^53+8729*q^54+7289*q^55+5965*q^56+4732*q^57+3587*q^58+2549*q^59+1655*q^
60+954*q^61+468*q^62+186*q^63+56*q^64+11*q^65+q^66, 1+12*q+67*q^2+243*q^3+663*q
^4+1460*q^5+2719*q^6+4455*q^7+6611*q^8+9108*q^9+11885*q^10+14913*q^11+18217*q^
12+21839*q^13+25797*q^14+30110*q^15+34773*q^16+39720*q^17+44865*q^18+50094*q^19
+55313*q^20+60520*q^21+65664*q^22+70780*q^23+75901*q^24+80993*q^25+85988*q^26+
90796*q^27+95319*q^28+99505*q^29+103351*q^30+106851*q^31+109992*q^32+112776*q^
33+115160*q^34+117146*q^35+118720*q^36+119874*q^37+120575*q^38+120822*q^39+
120575*q^40+119874*q^41+118720*q^42+117146*q^43+115160*q^44+112776*q^45+109992*
q^46+106851*q^47+103351*q^48+99505*q^49+95319*q^50+90796*q^51+85988*q^52+80993*
q^53+75901*q^54+70780*q^55+65664*q^56+60520*q^57+55313*q^58+50094*q^59+44865*q^
60+39720*q^61+34773*q^62+30110*q^63+25797*q^64+21839*q^65+18217*q^66+14913*q^67
+11885*q^68+9108*q^69+6611*q^70+4455*q^71+2719*q^72+1460*q^73+663*q^74+243*q^75
+67*q^76+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+916*q^4+2170*q^5+4328*q^6+7542*q^7
+11816*q^8+17045*q^9+23100*q^10+29877*q^11+37361*q^12+45609*q^13+54681*q^14+
64655*q^15+75605*q^16+87560*q^17+100469*q^18+114152*q^19+128411*q^20+143056*q^
21+157919*q^22+172942*q^23+188104*q^24+203425*q^25+218916*q^26+234488*q^27+
250040*q^28+265438*q^29+280524*q^30+295131*q^31+309110*q^32+322302*q^33+334673*
q^34+346183*q^35+356872*q^36+366795*q^37+375944*q^38+384179*q^39+391347*q^40+
397284*q^41+401873*q^42+405177*q^43+407287*q^44+408299*q^45+408299*q^46+407287*
q^47+405177*q^48+401873*q^49+397284*q^50+391347*q^51+384179*q^52+375944*q^53+
366795*q^54+356872*q^55+346183*q^56+334673*q^57+322302*q^58+309110*q^59+295131*
q^60+280524*q^61+265438*q^62+250040*q^63+234488*q^64+218916*q^65+203425*q^66+
188104*q^67+172942*q^68+157919*q^69+143056*q^70+128411*q^71+114152*q^72+100469*
q^73+87560*q^74+75605*q^75+64655*q^76+54681*q^77+45609*q^78+37361*q^79+29877*q^
80+23100*q^81+17045*q^82+11816*q^83+7542*q^84+4328*q^85+2170*q^86+916*q^87+311*
q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1238*q^4+3143*q^5+6695*q^6+
12395*q^7+20507*q^8+31024*q^9+43775*q^10+58539*q^11+75189*q^12+93747*q^13+
114316*q^14+137050*q^15+162153*q^16+189834*q^17+220261*q^18+253427*q^19+289120*
q^20+327001*q^21+366673*q^22+407717*q^23+449887*q^24+493083*q^25+537221*q^26+
582255*q^27+628126*q^28+674741*q^29+722014*q^30+769734*q^31+817535*q^32+864845*
q^33+911224*q^34+956221*q^35+999652*q^36+1041572*q^37+1082000*q^38+1120935*q^39
+1158225*q^40+1193545*q^41+1226553*q^42+1256927*q^43+1284484*q^44+1309121*q^45+
1330824*q^46+1349548*q^47+1365309*q^48+1378062*q^49+1387767*q^50+1394307*q^51+
1397608*q^52+1397608*q^53+1394307*q^54+1387767*q^55+1378062*q^56+1365309*q^57+
1349548*q^58+1330824*q^59+1309121*q^60+1284484*q^61+1256927*q^62+1226553*q^63+
1193545*q^64+1158225*q^65+1120935*q^66+1082000*q^67+1041572*q^68+999652*q^69+
956221*q^70+911224*q^71+864845*q^72+817535*q^73+769734*q^74+722014*q^75+674741*
q^76+628126*q^77+582255*q^78+537221*q^79+493083*q^80+449887*q^81+407717*q^82+
366673*q^83+327001*q^84+289120*q^85+253427*q^86+220261*q^87+189834*q^88+162153*
q^89+137050*q^90+114316*q^91+93747*q^92+75189*q^93+58539*q^94+43775*q^95+31024*
q^96+20507*q^97+12395*q^98+6695*q^99+3143*q^100+1238*q^101+391*q^102+92*q^103+
14*q^104+q^105]


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

[1, 2, 6, 22, 86, 338, 1318, 5106, 19718, 76066, 293398, 1131794, 4366374,

    16846018, 64995254]

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

     is given by

[1, 1, 3, 10, 42, 169, 659, 2556, 9866, 38033, 146699, 565880, 2183162, 8423009,

    32497627]

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

     is given by

[0, 1, 3, 12, 44, 169, 659, 2550, 9852, 38033, 146699, 565914, 2183212, 8423009,

    32497627]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 42, 169, 659, 2556, 9866, 38033, 146699, 565880, 2183162,

    8423009, 32497627]

 ODD:, [0, 1, 3, 12, 44, 169, 659, 2550, 9852, 38033, 146699, 565914, 2183212,

    8423009, 32497627]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.500000000,

    10.50000000, 14.00000000, 18.00000000, 22.50000000, 27.50000000,

    33.00000000, 39.00000000, 45.50000000, 52.50000000]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.507556723, 2.204646256, 3.071389254,

    4.104246663, 5.289984920, 6.614644712, 8.066814452, 9.638033768,

    11.32199642, 13.11360843, 15.00836388, 17.00206843]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.27273, 4.86047, 9.43343, 16.8448, 27.9839,

    43.7535, 65.0735, 92.8917, 128.188, 171.967, 225.251, 289.070]

                                                      -5
Skewness: , [Float(undefined), 0., 0., 0.2918624564 10  , 0., 0.,

                    -5                  -6                  -5
    -0.1446442626 10  , -0.6755191106 10  , -0.3455260541 10  , 0., 0.,

                   -6                 -6                      -6
    0.6890154208 10  , 0.4434374080 10  , 0., -0.2034676955 10  ]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.340792622,

    2.367486488, 2.346026910, 2.334277825, 2.337393863, 2.350487581,

    2.368888160, 2.389502327, 2.410571218, 2.431276947, 2.451219365,

    2.470349224]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+2*q^5+q^6, 1+4*q+9*q^2+15*q^3
+18*q^4+16*q^5+12*q^6+8*q^7+4*q^8+2*q^9+q^10, 1+5*q+14*q^2+29*q^3+46*q^4+58*q^5
+60*q^6+54*q^7+44*q^8+32*q^9+22*q^10+14*q^11+8*q^12+4*q^13+2*q^14+q^15, 1+6*q+
20*q^2+49*q^3+94*q^4+147*q^5+193*q^6+218*q^7+220*q^8+204*q^9+175*q^10+142*q^11+
110*q^12+80*q^13+56*q^14+38*q^15+24*q^16+14*q^17+8*q^18+4*q^19+2*q^20+q^21, 1+7
*q+27*q^2+76*q^3+169*q^4+310*q^5+483*q^6+653*q^7+783*q^8+854*q^9+864*q^10+820*q
^11+741*q^12+642*q^13+534*q^14+431*q^15+338*q^16+256*q^17+188*q^18+134*q^19+92*
q^20+62*q^21+40*q^22+24*q^23+14*q^24+8*q^25+4*q^26+2*q^27+q^28, 1+8*q+35*q^2+
111*q^3+279*q^4+582*q^5+1038*q^6+1616*q^7+2236*q^8+2799*q^9+3227*q^10+3478*q^11
+3544*q^12+3450*q^13+3232*q^14+2928*q^15+2580*q^16+2218*q^17+1862*q^18+1528*q^
19+1228*q^20+967*q^21+746*q^22+564*q^23+416*q^24+300*q^25+212*q^26+146*q^27+98*
q^28+64*q^29+40*q^30+24*q^31+14*q^32+8*q^33+4*q^34+2*q^35+q^36, 1+9*q+44*q^2+
155*q^3+433*q^4+1007*q^5+2010*q^6+3516*q^7+5480*q^8+7724*q^9+9986*q^10+12004*q^
11+13577*q^12+14589*q^13+15020*q^14+14917*q^15+14364*q^16+13475*q^17+12358*q^18
+11102*q^19+9789*q^20+8486*q^21+7238*q^22+6080*q^23+5032*q^24+4100*q^25+3292*q^
26+2606*q^27+2031*q^28+1560*q^29+1180*q^30+876*q^31+640*q^32+460*q^33+324*q^34+
224*q^35+152*q^36+100*q^37+64*q^38+40*q^39+24*q^40+14*q^41+8*q^42+4*q^43+2*q^44
+q^45, 1+10*q+54*q^2+209*q^3+641*q^4+1639*q^5+3605*q^6+6967*q^7+12022*q^8+18774
*q^9+26859*q^10+35610*q^11+44228*q^12+51956*q^13+58204*q^14+62627*q^15+65111*q^
16+65728*q^17+64700*q^18+62312*q^19+58863*q^20+54659*q^21+49978*q^22+45055*q^23
+40090*q^24+35234*q^25+30596*q^26+26268*q^27+22304*q^28+18730*q^29+15562*q^30+
12790*q^31+10394*q^32+8356*q^33+6644*q^34+5222*q^35+4059*q^36+3118*q^37+2364*q^
38+1770*q^39+1308*q^40+952*q^41+684*q^42+484*q^43+336*q^44+230*q^45+154*q^46+
100*q^47+64*q^48+40*q^49+24*q^50+14*q^51+8*q^52+4*q^53+2*q^54+q^55, 1+11*q+65*q
^2+274*q^3+914*q^4+2543*q^5+6094*q^6+12853*q^7+24243*q^8+41422*q^9+64829*q^10+
93888*q^11+127022*q^12+161960*q^13+196176*q^14+227315*q^15+253522*q^16+273586*q
^17+286949*q^18+293644*q^19+294124*q^20+289110*q^21+279494*q^22+266224*q^23+
250222*q^24+232344*q^25+213342*q^26+193852*q^27+174417*q^28+155472*q^29+137344*
q^30+120280*q^31+104444*q^32+89932*q^33+76798*q^34+65048*q^35+54646*q^36+45534*
q^37+37632*q^38+30842*q^39+25064*q^40+20196*q^41+16132*q^42+12772*q^43+10022*q^
44+7791*q^45+5998*q^46+4572*q^47+3448*q^48+2572*q^49+1898*q^50+1384*q^51+996*q^
52+708*q^53+496*q^54+342*q^55+232*q^56+154*q^57+100*q^58+64*q^59+40*q^60+24*q^
61+14*q^62+8*q^63+4*q^64+2*q^65+q^66, 1+12*q+77*q^2+351*q^3+1264*q^4+3796*q^5+
9825*q^6+22405*q^7+45744*q^8+84677*q^9+143619*q^10+225276*q^11+329597*q^12+
453388*q^13+590707*q^14+733830*q^15+874454*q^16+1004846*q^17+1118666*q^18+
1211408*q^19+1280530*q^20+1325249*q^21+1346205*q^22+1345143*q^23+1324538*q^24+
1287256*q^25+1236320*q^26+1174701*q^27+1105180*q^28+1030318*q^29+952381*q^30+
873298*q^31+794704*q^32+717924*q^33+643996*q^34+573742*q^35+507755*q^36+446418*
q^37+389969*q^38+338492*q^39+291944*q^40+250208*q^41+213088*q^42+180324*q^43+
151632*q^44+126696*q^45+105176*q^46+86744*q^47+71070*q^48+57832*q^49+46740*q^50
+37514*q^51+29892*q^52+23646*q^53+18566*q^54+14463*q^55+11178*q^56+8568*q^57+
6508*q^58+4900*q^59+3656*q^60+2700*q^61+1974*q^62+1428*q^63+1020*q^64+720*q^65+
502*q^66+344*q^67+232*q^68+154*q^69+100*q^70+64*q^71+40*q^72+24*q^73+14*q^74+8*
q^75+4*q^76+2*q^77+q^78, 1+13*q+90*q^2+441*q^3+1704*q^4+5488*q^5+15236*q^6+
37291*q^7+81782*q^8+162728*q^9+296794*q^10+500558*q^11+786836*q^12+1161172*q^13
+1619650*q^14+2148677*q^15+2726644*q^16+3326873*q^17+3921076*q^18+4482511*q^19+
4988371*q^20+5421262*q^21+5769660*q^22+6027670*q^23+6194420*q^24+6273104*q^25+
6269941*q^26+6193237*q^27+6052497*q^28+5857728*q^29+5618984*q^30+5345925*q^31+
5047518*q^32+4731930*q^33+4406383*q^34+4077146*q^35+3749618*q^36+3428296*q^37+
3116836*q^38+2818168*q^39+2534506*q^40+2267440*q^41+2018055*q^42+1786952*q^43+
1574326*q^44+1380065*q^45+1203756*q^46+1044751*q^47+902254*q^48+775316*q^49+
662902*q^50+563950*q^51+477348*q^52+401984*q^53+336784*q^54+280694*q^55+232710*
q^56+191902*q^57+157390*q^58+128364*q^59+104104*q^60+83944*q^61+67286*q^62+
53612*q^63+42452*q^64+33398*q^65+26105*q^66+20266*q^67+15620*q^68+11952*q^69+
9076*q^70+6836*q^71+5108*q^72+3784*q^73+2776*q^74+2018*q^75+1452*q^76+1032*q^77
+726*q^78+504*q^79+344*q^80+232*q^81+154*q^82+100*q^83+64*q^84+40*q^85+24*q^86+
14*q^87+8*q^88+4*q^89+2*q^90+q^91, 1+14*q+104*q^2+545*q^3+2248*q^4+7723*q^5+
22869*q^6+59720*q^7+139810*q^8+297127*q^9+579034*q^10+1043542*q^11+1752251*q^12
+2759910*q^13+4103101*q^14+5791144*q^15+7801615*q^16+10081342*q^17+12552248*q^
18+15120390*q^19+17686061*q^20+20153157*q^21+22436630*q^22+24467299*q^23+
26194194*q^24+27584947*q^25+28624512*q^26+29312862*q^27+29662244*q^28+29694234*
q^29+29436993*q^30+28922940*q^31+28186660*q^32+27263200*q^33+26186866*q^34+
24990254*q^35+23703643*q^36+22354710*q^37+20968264*q^38+19566160*q^39+18167396*
q^40+16788160*q^41+15441980*q^42+14139976*q^43+12891024*q^44+11701950*q^45+
10577784*q^46+9521920*q^47+8536290*q^48+7621604*q^49+6777500*q^50+6002714*q^51+
5295276*q^52+4652616*q^53+4071686*q^54+3549111*q^55+3081276*q^56+2664407*q^57+
2294678*q^58+1968268*q^59+1681408*q^60+1430458*q^61+1211926*q^62+1022480*q^63+
858996*q^64+718560*q^65+598468*q^66+496246*q^67+409640*q^68+336602*q^69+275296*
q^70+224088*q^71+181520*q^72+146308*q^73+117330*q^74+93600*q^75+74268*q^76+
58606*q^77+45983*q^78+35866*q^79+27806*q^80+21420*q^81+16392*q^82+12460*q^83+
9404*q^84+7044*q^85+5236*q^86+3860*q^87+2820*q^88+2042*q^89+1464*q^90+1038*q^91
+728*q^92+504*q^93+344*q^94+232*q^95+154*q^96+100*q^97+64*q^98+40*q^99+24*q^100
+14*q^101+8*q^102+4*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+4*q^2+2*q+1, q^10+4*q^9+9*q^8+
15*q^7+18*q^6+16*q^5+12*q^4+8*q^3+4*q^2+2*q+1, q^15+5*q^14+14*q^13+29*q^12+46*q
^11+58*q^10+60*q^9+54*q^8+44*q^7+32*q^6+22*q^5+14*q^4+8*q^3+4*q^2+2*q+1, q^21+6
*q^20+20*q^19+49*q^18+94*q^17+147*q^16+193*q^15+218*q^14+220*q^13+204*q^12+175*
q^11+142*q^10+110*q^9+80*q^8+56*q^7+38*q^6+24*q^5+14*q^4+8*q^3+4*q^2+2*q+1, 1+
76*q^25+483*q^22+169*q^24+653*q^21+310*q^23+2*q+4*q^2+27*q^26+8*q^3+14*q^4+24*q
^5+40*q^6+62*q^7+92*q^8+134*q^9+188*q^10+256*q^11+338*q^12+431*q^13+534*q^14+
642*q^15+741*q^16+q^28+820*q^17+864*q^18+854*q^19+783*q^20+7*q^27, 1+3478*q^25+
3232*q^22+3544*q^24+2928*q^21+3450*q^23+2*q+1038*q^30+1616*q^29+35*q^34+111*q^
33+4*q^2+3227*q^26+8*q^3+14*q^4+24*q^5+40*q^6+279*q^32+64*q^7+98*q^8+146*q^9+
212*q^10+300*q^11+416*q^12+564*q^13+746*q^14+967*q^15+1228*q^16+2236*q^28+1528*
q^17+1862*q^18+2218*q^19+2580*q^20+2799*q^27+582*q^31+8*q^35+q^36, 1+9789*q^25+
155*q^42+6080*q^22+8486*q^24+5032*q^21+7238*q^23+2*q+9*q^44+14917*q^30+q^45+
5480*q^37+14364*q^29+2010*q^39+433*q^41+12004*q^34+3516*q^38+13577*q^33+4*q^2+
11102*q^26+8*q^3+14*q^4+24*q^5+40*q^6+14589*q^32+64*q^7+100*q^8+152*q^9+224*q^
10+324*q^11+460*q^12+640*q^13+876*q^14+1180*q^15+1560*q^16+44*q^43+13475*q^28+
2031*q^17+2606*q^18+3292*q^19+4100*q^20+12358*q^27+1007*q^40+15020*q^31+9986*q^
35+7724*q^36, 1+15562*q^25+51956*q^42+8356*q^22+12790*q^24+6644*q^21+10394*q^23
+2*q+641*q^51+35610*q^44+54*q^53+35234*q^30+209*q^52+q^55+10*q^54+26859*q^45+
64700*q^37+30596*q^29+18774*q^46+65111*q^39+58204*q^41+54659*q^34+1639*q^50+
65728*q^38+49978*q^33+4*q^2+18730*q^26+8*q^3+14*q^4+24*q^5+40*q^6+45055*q^32+64
*q^7+100*q^8+154*q^9+230*q^10+336*q^11+484*q^12+684*q^13+952*q^14+1308*q^15+
1770*q^16+44228*q^43+3605*q^49+26268*q^28+2364*q^17+3118*q^18+4059*q^19+5222*q^
20+22304*q^27+62627*q^40+40090*q^31+58863*q^35+6967*q^48+62312*q^36+12022*q^47,
1+20196*q^25+250222*q^42+10022*q^22+16132*q^24+7791*q^21+12772*q^23+2*q+227315*
q^51+279494*q^44+161960*q^53+41422*q^57+54646*q^30+196176*q^52+24243*q^58+93888
*q^55+6094*q^60+127022*q^54+289110*q^45+155472*q^37+45534*q^29+294124*q^46+11*q
^65+193852*q^39+2543*q^61+232344*q^41+104444*q^34+253522*q^50+65*q^64+174417*q^
38+89932*q^33+64829*q^56+4*q^2+25064*q^26+8*q^3+14*q^4+24*q^5+40*q^6+76798*q^32
+64*q^7+100*q^8+154*q^9+914*q^62+232*q^10+342*q^11+496*q^12+708*q^13+996*q^14+
1384*q^15+1898*q^16+q^66+266224*q^43+273586*q^49+37632*q^28+2572*q^17+3448*q^18
+4572*q^19+5998*q^20+30842*q^27+213342*q^40+274*q^63+65048*q^31+120280*q^35+
286949*q^48+12853*q^59+137344*q^36+293644*q^47, 1+23646*q^25+507755*q^42+11178*
q^22+18566*q^24+8568*q^21+14463*q^23+2*q+1174701*q^51+643996*q^44+1287256*q^53+
1325249*q^57+71070*q^30+1236320*q^52+1280530*q^58+1345143*q^55+1118666*q^60+
1324538*q^54+717924*q^45+22405*q^71+1264*q^74+250208*q^37+225276*q^67+57832*q^
29+794704*q^46+453388*q^65+84677*q^69+338492*q^39+1004846*q^61+446418*q^41+
151632*q^34+1105180*q^50+590707*q^64+291944*q^38+126696*q^33+1346205*q^56+4*q^2
+143619*q^68+29892*q^26+8*q^3+14*q^4+24*q^5+40*q^6+105176*q^32+64*q^7+100*q^8+
154*q^9+874454*q^62+232*q^10+344*q^11+502*q^12+77*q^76+720*q^13+1020*q^14+1428*
q^15+1974*q^16+329597*q^66+45744*q^70+12*q^77+q^78+573742*q^43+1030318*q^49+
46740*q^28+2700*q^17+3656*q^18+4900*q^19+6508*q^20+37514*q^27+9825*q^72+389969*
q^40+3796*q^73+733830*q^63+86744*q^31+351*q^75+180324*q^35+952381*q^48+1211408*
q^59+213088*q^36+873298*q^47, 1+26105*q^25+775316*q^42+11952*q^22+13*q^90+20266
*q^24+9076*q^21+15620*q^23+2*q+2534506*q^51+1044751*q^44+3116836*q^53+4406383*q
^57+83944*q^30+2818168*q^52+4731930*q^58+3749618*q^55+5345925*q^60+q^91+3428296
*q^54+1203756*q^45+4988371*q^71+3326873*q^74+336784*q^37+6194420*q^67+67286*q^
29+1380065*q^46+6269941*q^65+5769660*q^69+477348*q^39+5618984*q^61+662902*q^41+
191902*q^34+786836*q^79+2267440*q^50+6193237*q^64+401984*q^38+157390*q^33+
4077146*q^56+4*q^2+6027670*q^68+33398*q^26+8*q^3+14*q^4+24*q^5+40*q^6+128364*q^
32+64*q^7+100*q^8+154*q^9+5857728*q^62+232*q^10+344*q^11+504*q^12+2148677*q^76+
726*q^13+1032*q^14+1452*q^15+2018*q^16+6273104*q^66+5421262*q^70+1619650*q^77+
1161172*q^78+902254*q^43+2018055*q^49+53612*q^28+2776*q^17+3784*q^18+5108*q^19+
6836*q^20+42452*q^27+4482511*q^72+563950*q^40+3921076*q^73+6052497*q^63+104104*
q^31+2726644*q^75+232710*q^35+500558*q^80+1786952*q^48+81782*q^83+296794*q^81+
15236*q^85+5488*q^86+1704*q^87+441*q^88+90*q^89+37291*q^84+162728*q^82+5047518*
q^59+280694*q^36+1574326*q^47, 1+27806*q^25+1022480*q^42+12460*q^22+5791144*q^
90+21420*q^24+9404*q^21+16392*q^23+2*q+139810*q^97+4071686*q^51+1430458*q^44+
5295276*q^53+8536290*q^57+93600*q^30+1043542*q^94+4652616*q^52+9521920*q^58+
6777500*q^55+11701950*q^60+4103101*q^91+6002714*q^54+1681408*q^45+26186866*q^71
+28922940*q^74+297127*q^96+59720*q^98+409640*q^37+20968264*q^67+74268*q^29+
1968268*q^46+1752251*q^93+22869*q^99+18167396*q^65+23703643*q^69+598468*q^39+
12891024*q^61+858996*q^41+224088*q^34+28624512*q^79+3549111*q^50+16788160*q^64+
7723*q^100+496246*q^38+181520*q^33+7621604*q^56+4*q^2+22354710*q^68+35866*q^26+
8*q^3+14*q^4+24*q^5+40*q^6+146308*q^32+64*q^7+100*q^8+154*q^9+579034*q^95+
14139976*q^62+2248*q^101+232*q^10+344*q^11+504*q^12+29694234*q^76+728*q^13+1038
*q^14+1464*q^15+2042*q^16+19566160*q^66+24990254*q^70+29662244*q^77+29312862*q^
78+1211926*q^43+545*q^102+3081276*q^49+58606*q^28+2820*q^17+3860*q^18+5236*q^19
+7044*q^20+45983*q^27+27263200*q^72+718560*q^40+28186660*q^73+104*q^103+
15441980*q^63+117330*q^31+29436993*q^75+275296*q^35+14*q^104+27584947*q^80+
2664407*q^48+q^105+22436630*q^83+26194194*q^81+17686061*q^85+15120390*q^86+
12552248*q^87+10081342*q^88+7801615*q^89+20153157*q^84+24467299*q^82+10577784*q
^59+336602*q^36+2759910*q^92+2294678*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.863636364, 4.522222222, 6.439086294,

    8.589700997, 10.95571395, 13.52260376, 16.27847432, 19.21333638,

    22.31864215, 25.58696662, 29.01178033, 32.58728204]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.455255509, 2.012338483, 2.629893587,

    3.304663676, 4.032965000, 4.811473059, 5.637309898, 6.507994624,

    7.421376085, 8.375573109, 9.368925986, 10.39995810]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.11777, 4.04951, 6.91634, 10.9208, 16.2648,

    23.1503, 31.7793, 42.3540, 55.0768, 70.1502, 87.7768, 108.159]

Skewness: , [Float(undefined), 0., 0., 0.1491970460, 0.2825172660, 0.3784325686,

    0.4457770208, 0.4938825271, 0.5291492000, 0.5556811268, 0.5761218138,

    0.5922021498, 0.6050926652, 0.6155906127, 0.6242690131]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.547029639,

    2.797768857, 2.954003510, 3.062603889, 3.142342608, 3.202898761,

    3.250056736, 3.287567410, 3.317901092, 3.342843118, 3.363593122,

    3.381103729]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}

%2 := {[0, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0]}

%3 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+9*q^2+13*q^3
+15*q^4+16*q^5+14*q^6+8*q^7+5*q^8+2*q^9+q^10, 1+5*q+14*q^2+26*q^3+37*q^4+44*q^5
+52*q^6+51*q^7+44*q^8+34*q^9+25*q^10+16*q^11+8*q^12+5*q^13+2*q^14+q^15, 1+6*q+
20*q^2+45*q^3+77*q^4+107*q^5+134*q^6+159*q^7+173*q^8+169*q^9+154*q^10+135*q^11+
113*q^12+87*q^13+61*q^14+40*q^15+27*q^16+16*q^17+8*q^18+5*q^19+2*q^20+q^21, 1+7
*q+27*q^2+71*q^3+142*q^4+229*q^5+319*q^6+405*q^7+490*q^8+553*q^9+586*q^10+584*q
^11+561*q^12+523*q^13+473*q^14+405*q^15+328*q^16+258*q^17+195*q^18+142*q^19+101
*q^20+67*q^21+42*q^22+27*q^23+16*q^24+8*q^25+5*q^26+2*q^27+q^28, 1+8*q+35*q^2+
105*q^3+240*q^4+442*q^5+691*q^6+959*q^7+1234*q^8+1505*q^9+1744*q^10+1925*q^11+
2036*q^12+2080*q^13+2067*q^14+2011*q^15+1899*q^16+1732*q^17+1534*q^18+1319*q^19
+1108*q^20+910*q^21+730*q^22+563*q^23+425*q^24+311*q^25+222*q^26+156*q^27+107*q
^28+69*q^29+42*q^30+27*q^31+16*q^32+8*q^33+5*q^34+2*q^35+q^36, 1+9*q+44*q^2+148
*q^3+380*q^4+787*q^5+1374*q^6+2099*q^7+2911*q^8+3769*q^9+4640*q^10+5460*q^11+
6190*q^12+6783*q^13+7231*q^14+7526*q^15+7673*q^16+7655*q^17+7468*q^18+7118*q^19
+6626*q^20+6048*q^21+5425*q^22+4790*q^23+4149*q^24+3525*q^25+2931*q^26+2388*q^
27+1909*q^28+1503*q^29+1162*q^30+883*q^31+656*q^32+476*q^33+338*q^34+236*q^35+
162*q^36+109*q^37+69*q^38+42*q^39+27*q^40+16*q^41+8*q^42+5*q^43+2*q^44+q^45, 1+
10*q+54*q^2+201*q^3+572*q^4+1315*q^5+2542*q^6+4268*q^7+6419*q^8+8884*q^9+11561*
q^10+14335*q^11+17080*q^12+19696*q^13+22101*q^14+24234*q^15+26050*q^16+27504*q^
17+28505*q^18+29017*q^19+29000*q^20+28460*q^21+27456*q^22+26083*q^23+24443*q^24
+22610*q^25+20645*q^26+18576*q^27+16470*q^28+14384*q^29+12387*q^30+10524*q^31+
8829*q^32+7308*q^33+5972*q^34+4813*q^35+3819*q^36+2989*q^37+2306*q^38+1753*q^39
+1311*q^40+974*q^41+707*q^42+503*q^43+352*q^44+242*q^45+164*q^46+109*q^47+69*q^
48+42*q^49+27*q^50+16*q^51+8*q^52+5*q^53+2*q^54+q^55, 1+11*q+65*q^2+265*q^3+827
*q^4+2088*q^5+4430*q^6+8134*q^7+13273*q^8+19719*q^9+27245*q^10+35576*q^11+44411
*q^12+53431*q^13+62419*q^14+71181*q^15+79592*q^16+87496*q^17+94728*q^18+100992*
q^19+106011*q^20+109568*q^21+111523*q^22+111878*q^23+110726*q^24+108248*q^25+
104627*q^26+100094*q^27+94748*q^28+88708*q^29+82111*q^30+75154*q^31+68011*q^32+
60902*q^33+53979*q^34+47378*q^35+41187*q^36+35444*q^37+30181*q^38+25428*q^39+
21196*q^40+17475*q^41+14271*q^42+11536*q^43+9227*q^44+7303*q^45+5719*q^46+4424*
q^47+3384*q^48+2552*q^49+1900*q^50+1402*q^51+1025*q^52+734*q^53+517*q^54+358*q^
55+244*q^56+164*q^57+109*q^58+69*q^59+42*q^60+27*q^61+16*q^62+8*q^63+5*q^64+2*q
^65+q^66, 1+12*q+77*q^2+341*q^3+1157*q^4+3180*q^5+7346*q^6+14662*q^7+25891*q^8+
41327*q^9+60810*q^10+83865*q^11+109827*q^12+137897*q^13+167302*q^14+197456*q^15
+227948*q^16+258475*q^17+288734*q^18+318281*q^19+346354*q^20+372024*q^21+394276
*q^22+412368*q^23+425788*q^24+434389*q^25+438227*q^26+437601*q^27+432904*q^28+
424495*q^29+412663*q^30+397638*q^31+379722*q^32+359309*q^33+336930*q^34+313159*
q^35+288656*q^36+263949*q^37+239503*q^38+215661*q^39+192690*q^40+170763*q^41+
150099*q^42+130873*q^43+113187*q^44+97111*q^45+82662*q^46+69832*q^47+58546*q^48
+48699*q^49+40170*q^50+32852*q^51+26637*q^52+21422*q^53+17081*q^54+13502*q^55+
10578*q^56+8213*q^57+6322*q^58+4815*q^59+3628*q^60+2699*q^61+1991*q^62+1453*q^
63+1052*q^64+748*q^65+523*q^66+360*q^67+244*q^68+164*q^69+109*q^70+69*q^71+42*q
^72+27*q^73+16*q^74+8*q^75+5*q^76+2*q^77+q^78, 1+13*q+90*q^2+430*q^3+1575*q^4+
4678*q^5+11684*q^6+25199*q^7+47964*q^8+82145*q^9+128856*q^10+188101*q^11+258996
*q^12+339977*q^13+429086*q^14+524352*q^15+624264*q^16+727753*q^17+834192*q^18+
942982*q^19+1053295*q^20+1163461*q^21+1271032*q^22+1372887*q^23+1465949*q^24+
1547694*q^25+1616267*q^26+1670646*q^27+1710520*q^28+1736181*q^29+1748052*q^30+
1746786*q^31+1732771*q^32+1706386*q^33+1667933*q^34+1618124*q^35+1557995*q^36+
1489118*q^37+1413165*q^38+1331979*q^39+1247278*q^40+1160613*q^41+1073205*q^42+
986172*q^43+900456*q^44+816872*q^45+736238*q^46+659270*q^47+586612*q^48+518728*
q^49+455919*q^50+398265*q^51+345800*q^52+298390*q^53+255864*q^54+218007*q^55+
184566*q^56+155234*q^57+129716*q^58+107699*q^59+88855*q^60+72841*q^61+59307*q^
62+47951*q^63+38497*q^64+30691*q^65+24281*q^66+19067*q^67+14857*q^68+11486*q^69
+8812*q^70+6711*q^71+5059*q^72+3775*q^73+2790*q^74+2042*q^75+1480*q^76+1066*q^
77+754*q^78+525*q^79+360*q^80+244*q^81+164*q^82+109*q^83+69*q^84+42*q^85+27*q^
86+16*q^87+8*q^88+5*q^89+2*q^90+q^91, 1+14*q+104*q^2+533*q^3+2095*q^4+6683*q^5+
17938*q^6+41573*q^7+84924*q^8+155649*q^9+260123*q^10+402294*q^11+583375*q^12+
802065*q^13+1055033*q^14+1337641*q^15+1645068*q^16+1973315*q^17+2319640*q^18+
2682476*q^19+3060860*q^20+3453480*q^21+3857410*q^22+4267324*q^23+4675268*q^24+
5071937*q^25+5447869*q^26+5794865*q^27+6106216*q^28+6377541*q^29+6606280*q^30+
6791396*q^31+6932658*q^32+7030324*q^33+7084399*q^34+7094535*q^35+7060361*q^36+
6982008*q^37+6860871*q^38+6699679*q^39+6502324*q^40+6273434*q^41+6018215*q^42+
5741733*q^43+5448893*q^44+5144070*q^45+4831150*q^46+4513559*q^47+4194645*q^48+
3877578*q^49+3565545*q^50+3261564*q^51+2968238*q^52+2687691*q^53+2421605*q^54+
2171160*q^55+1937070*q^56+1719734*q^57+1519144*q^58+1335151*q^59+1167466*q^60+
1015668*q^61+879132*q^62+757146*q^63+648802*q^64+553152*q^65+469233*q^66+396017
*q^67+332466*q^68+277623*q^69+230568*q^70+190429*q^71+156417*q^72+127762*q^73+
103771*q^74+83810*q^75+67296*q^76+53705*q^77+42600*q^78+33570*q^79+26271*q^80+
20420*q^81+15761*q^82+12083*q^83+9201*q^84+6955*q^85+5206*q^86+3866*q^87+2841*q
^88+2069*q^89+1494*q^90+1072*q^91+756*q^92+525*q^93+360*q^94+244*q^95+164*q^96+
109*q^97+69*q^98+42*q^99+27*q^100+16*q^101+8*q^102+5*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+5*q^3+5*q^2+2*q+1, q^10+4*q^9+9*q^8+
13*q^7+15*q^6+16*q^5+14*q^4+8*q^3+5*q^2+2*q+1, q^15+5*q^14+14*q^13+26*q^12+37*q
^11+44*q^10+52*q^9+51*q^8+44*q^7+34*q^6+25*q^5+16*q^4+8*q^3+5*q^2+2*q+1, q^21+6
*q^20+20*q^19+45*q^18+77*q^17+107*q^16+134*q^15+159*q^14+173*q^13+169*q^12+154*
q^11+135*q^10+113*q^9+87*q^8+61*q^7+40*q^6+27*q^5+16*q^4+8*q^3+5*q^2+2*q+1, 1+
71*q^25+319*q^22+142*q^24+405*q^21+229*q^23+2*q+5*q^2+27*q^26+8*q^3+16*q^4+27*q
^5+42*q^6+67*q^7+101*q^8+142*q^9+195*q^10+258*q^11+328*q^12+405*q^13+473*q^14+
523*q^15+561*q^16+q^28+584*q^17+586*q^18+553*q^19+490*q^20+7*q^27, 1+1925*q^25+
2067*q^22+2036*q^24+2011*q^21+2080*q^23+2*q+691*q^30+959*q^29+35*q^34+105*q^33+
5*q^2+1744*q^26+8*q^3+16*q^4+27*q^5+42*q^6+240*q^32+69*q^7+107*q^8+156*q^9+222*
q^10+311*q^11+425*q^12+563*q^13+730*q^14+910*q^15+1108*q^16+1234*q^28+1319*q^17
+1534*q^18+1732*q^19+1899*q^20+1505*q^27+442*q^31+8*q^35+q^36, 1+6626*q^25+148*
q^42+4790*q^22+6048*q^24+4149*q^21+5425*q^23+2*q+9*q^44+7526*q^30+q^45+2911*q^
37+7673*q^29+1374*q^39+380*q^41+5460*q^34+2099*q^38+6190*q^33+5*q^2+7118*q^26+8
*q^3+16*q^4+27*q^5+42*q^6+6783*q^32+69*q^7+109*q^8+162*q^9+236*q^10+338*q^11+
476*q^12+656*q^13+883*q^14+1162*q^15+1503*q^16+44*q^43+7655*q^28+1909*q^17+2388
*q^18+2931*q^19+3525*q^20+7468*q^27+787*q^40+7231*q^31+4640*q^35+3769*q^36, 1+
12387*q^25+19696*q^42+7308*q^22+10524*q^24+5972*q^21+8829*q^23+2*q+572*q^51+
14335*q^44+54*q^53+22610*q^30+201*q^52+q^55+10*q^54+11561*q^45+28505*q^37+20645
*q^29+8884*q^46+26050*q^39+22101*q^41+28460*q^34+1315*q^50+27504*q^38+27456*q^
33+5*q^2+14384*q^26+8*q^3+16*q^4+27*q^5+42*q^6+26083*q^32+69*q^7+109*q^8+164*q^
9+242*q^10+352*q^11+503*q^12+707*q^13+974*q^14+1311*q^15+1753*q^16+17080*q^43+
2542*q^49+18576*q^28+2306*q^17+2989*q^18+3819*q^19+4813*q^20+16470*q^27+24234*q
^40+24443*q^31+29000*q^35+4268*q^48+29017*q^36+6419*q^47, 1+17475*q^25+110726*q
^42+9227*q^22+14271*q^24+7303*q^21+11536*q^23+2*q+71181*q^51+111523*q^44+53431*
q^53+19719*q^57+41187*q^30+62419*q^52+13273*q^58+35576*q^55+4430*q^60+44411*q^
54+109568*q^45+88708*q^37+35444*q^29+106011*q^46+11*q^65+100094*q^39+2088*q^61+
108248*q^41+68011*q^34+79592*q^50+65*q^64+94748*q^38+60902*q^33+27245*q^56+5*q^
2+21196*q^26+8*q^3+16*q^4+27*q^5+42*q^6+53979*q^32+69*q^7+109*q^8+164*q^9+827*q
^62+244*q^10+358*q^11+517*q^12+734*q^13+1025*q^14+1402*q^15+1900*q^16+q^66+
111878*q^43+87496*q^49+30181*q^28+2552*q^17+3384*q^18+4424*q^19+5719*q^20+25428
*q^27+104627*q^40+265*q^63+47378*q^31+75154*q^35+94728*q^48+8134*q^59+82111*q^
36+100992*q^47, 1+21422*q^25+288656*q^42+10578*q^22+17081*q^24+8213*q^21+13502*
q^23+2*q+437601*q^51+336930*q^44+434389*q^53+372024*q^57+58546*q^30+438227*q^52
+346354*q^58+412368*q^55+288734*q^60+425788*q^54+359309*q^45+14662*q^71+1157*q^
74+170763*q^37+83865*q^67+48699*q^29+379722*q^46+137897*q^65+41327*q^69+215661*
q^39+258475*q^61+263949*q^41+113187*q^34+432904*q^50+167302*q^64+192690*q^38+
97111*q^33+394276*q^56+5*q^2+60810*q^68+26637*q^26+8*q^3+16*q^4+27*q^5+42*q^6+
82662*q^32+69*q^7+109*q^8+164*q^9+227948*q^62+244*q^10+360*q^11+523*q^12+77*q^
76+748*q^13+1052*q^14+1453*q^15+1991*q^16+109827*q^66+25891*q^70+12*q^77+q^78+
313159*q^43+424495*q^49+40170*q^28+2699*q^17+3628*q^18+4815*q^19+6322*q^20+
32852*q^27+7346*q^72+239503*q^40+3180*q^73+197456*q^63+69832*q^31+341*q^75+
130873*q^35+412663*q^48+318281*q^59+150099*q^36+397638*q^47, 1+24281*q^25+
518728*q^42+11486*q^22+13*q^90+19067*q^24+8812*q^21+14857*q^23+2*q+1247278*q^51
+659270*q^44+1413165*q^53+1667933*q^57+72841*q^30+1331979*q^52+1706386*q^58+
1557995*q^55+1746786*q^60+q^91+1489118*q^54+736238*q^45+1053295*q^71+727753*q^
74+255864*q^37+1465949*q^67+59307*q^29+816872*q^46+1616267*q^65+1271032*q^69+
345800*q^39+1748052*q^61+455919*q^41+155234*q^34+258996*q^79+1160613*q^50+
1670646*q^64+298390*q^38+129716*q^33+1618124*q^56+5*q^2+1372887*q^68+30691*q^26
+8*q^3+16*q^4+27*q^5+42*q^6+107699*q^32+69*q^7+109*q^8+164*q^9+1736181*q^62+244
*q^10+360*q^11+525*q^12+524352*q^76+754*q^13+1066*q^14+1480*q^15+2042*q^16+
1547694*q^66+1163461*q^70+429086*q^77+339977*q^78+586612*q^43+1073205*q^49+
47951*q^28+2790*q^17+3775*q^18+5059*q^19+6711*q^20+38497*q^27+942982*q^72+
398265*q^40+834192*q^73+1710520*q^63+88855*q^31+624264*q^75+184566*q^35+188101*
q^80+986172*q^48+47964*q^83+128856*q^81+11684*q^85+4678*q^86+1575*q^87+430*q^88
+90*q^89+25199*q^84+82145*q^82+1732771*q^59+218007*q^36+900456*q^47, 1+26271*q^
25+757146*q^42+12083*q^22+1337641*q^90+20420*q^24+9201*q^21+15761*q^23+2*q+
84924*q^97+2421605*q^51+1015668*q^44+2968238*q^53+4194645*q^57+83810*q^30+
402294*q^94+2687691*q^52+4513559*q^58+3565545*q^55+5144070*q^60+1055033*q^91+
3261564*q^54+1167466*q^45+7084399*q^71+6791396*q^74+155649*q^96+41573*q^98+
332466*q^37+6860871*q^67+67296*q^29+1335151*q^46+583375*q^93+17938*q^99+6502324
*q^65+7060361*q^69+469233*q^39+5448893*q^61+648802*q^41+190429*q^34+5447869*q^
79+2171160*q^50+6273434*q^64+6683*q^100+396017*q^38+156417*q^33+3877578*q^56+5*
q^2+6982008*q^68+33570*q^26+8*q^3+16*q^4+27*q^5+42*q^6+127762*q^32+69*q^7+109*q
^8+164*q^9+260123*q^95+5741733*q^62+2095*q^101+244*q^10+360*q^11+525*q^12+
6377541*q^76+756*q^13+1072*q^14+1494*q^15+2069*q^16+6699679*q^66+7094535*q^70+
6106216*q^77+5794865*q^78+879132*q^43+533*q^102+1937070*q^49+53705*q^28+2841*q^
17+3866*q^18+5206*q^19+6955*q^20+42600*q^27+7030324*q^72+553152*q^40+6932658*q^
73+104*q^103+6018215*q^63+103771*q^31+6606280*q^75+230568*q^35+14*q^104+5071937
*q^80+1719734*q^48+q^105+3857410*q^83+4675268*q^81+3060860*q^85+2682476*q^86+
2319640*q^87+1973315*q^88+1645068*q^89+3453480*q^84+4267324*q^82+4831150*q^59+
277623*q^36+802065*q^92+1519144*q^47]


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

[1, 2, 6, 22, 88, 365, 1540, 6568, 28269, 122752, 537708, 2375500, 10579400,

    47469377, 214454528]

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

     is given by

[1, 1, 3, 12, 45, 183, 770, 3287, 14137, 61376, 268852, 1187751, 5289699,

    23734663, 107227233]

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

     is given by

[0, 1, 3, 10, 43, 182, 770, 3281, 14132, 61376, 268856, 1187749, 5289701,

    23734714, 107227295]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 45, 182, 770, 3287, 14137, 61376, 268856, 1187751, 5289699,

    23734714, 107227295]

 ODD:, [0, 1, 3, 10, 43, 183, 770, 3281, 14132, 61376, 268852, 1187749, 5289701,

    23734663, 107227233]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.909090909, 4.647727273, 6.673972603,

    8.971428571, 11.53395250, 14.35897980, 17.44536953, 20.79264582,

    24.40059651, 28.26900552, 32.39749239, 36.78544666]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.474297704, 2.061772035, 2.729946931,

    3.481373709, 4.311882511, 5.214413181, 6.181759662, 7.207829678,

    8.287955127, 9.418739638, 10.59774658, 11.82318660]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.17355, 4.25090, 7.45261, 12.1200, 18.5923,

    27.1901, 38.2142, 51.9528, 68.6902, 88.7127, 112.312, 139.788]

Skewness: , [Float(undefined), 0., 0., 0.07174070299, 0.1584138951,

    0.2324574421, 0.2843368900, 0.3168643376, 0.3356444113, 0.3452889304,

    0.3489493975, 0.3487046348, 0.3459228518, 0.3415358249, 0.3361808979]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.419438345,

    2.626928823, 2.744565522, 2.807731268, 2.843717151, 2.867409157,

    2.885371245, 2.900073750, 2.912314413, 2.922473465, 2.930778735,

    2.937461004]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+3*q^5, 1+4*q+9*q^2+15*q^3+18*
q^4+18*q^5+13*q^6+8*q^7+3*q^8, 1+5*q+14*q^2+29*q^3+46*q^4+61*q^5+65*q^6+60*q^7+
46*q^8+30*q^9+14*q^10+8*q^11+q^12, 1+6*q+20*q^2+49*q^3+94*q^4+151*q^5+204*q^6+
238*q^7+243*q^8+218*q^9+172*q^10+125*q^11+78*q^12+43*q^13+24*q^14+9*q^15+3*q^16
, 1+7*q+27*q^2+76*q^3+169*q^4+315*q^5+502*q^6+698*q^7+858*q^8+942*q^9+931*q^10+
841*q^11+697*q^12+536*q^13+384*q^14+260*q^15+161*q^16+94*q^17+47*q^18+26*q^19+9
*q^20+3*q^21, 1+8*q+35*q^2+111*q^3+279*q^4+588*q^5+1067*q^6+1700*q^7+2412*q^8+
3077*q^9+3564*q^10+3781*q^11+3708*q^12+3388*q^13+2915*q^14+2381*q^15+1854*q^16+
1378*q^17+975*q^18+664*q^19+431*q^20+256*q^21+150*q^22+83*q^23+43*q^24+17*q^25+
8*q^26+q^27, 1+9*q+44*q^2+155*q^3+433*q^4+1014*q^5+2051*q^6+3656*q^7+5828*q^8+
8394*q^9+11022*q^10+13299*q^11+14860*q^12+15488*q^13+15183*q^14+14106*q^15+
12526*q^16+10692*q^17+8799*q^18+7022*q^19+5418*q^20+4037*q^21+2912*q^22+2041*q^
23+1377*q^24+896*q^25+559*q^26+341*q^27+193*q^28+111*q^29+54*q^30+27*q^31+9*q^
32+3*q^33, 1+10*q+54*q^2+209*q^3+641*q^4+1647*q^5+3660*q^6+7183*q^7+12639*q^8+
20156*q^9+29392*q^10+39481*q^11+49192*q^12+57214*q^13+62534*q^14+64648*q^15+
63666*q^16+60110*q^17+54744*q^18+48349*q^19+41544*q^20+34794*q^21+28428*q^22+
22665*q^23+17646*q^24+13416*q^25+9962*q^26+7215*q^27+5101*q^28+3522*q^29+2375*q
^30+1549*q^31+984*q^32+604*q^33+370*q^34+208*q^35+116*q^36+56*q^37+27*q^38+9*q^
39+3*q^40, 1+11*q+65*q^2+274*q^3+914*q^4+2552*q^5+6165*q^6+13168*q^7+25256*q^8+
43991*q^9+70209*q^10+103421*q^11+141514*q^12+180922*q^13+217342*q^14+246701*q^
15+266121*q^16+274373*q^17+271925*q^18+260475*q^19+242335*q^20+219845*q^21+
195040*q^22+169521*q^23+144488*q^24+120872*q^25+99227*q^26+79999*q^27+63346*q^
28+49296*q^29+37706*q^30+28329*q^31+20860*q^32+15099*q^33+10760*q^34+7521*q^35+
5135*q^36+3443*q^37+2255*q^38+1454*q^39+905*q^40+560*q^41+320*q^42+183*q^43+95*
q^44+46*q^45+17*q^46+8*q^47+q^48, 1+12*q+77*q^2+351*q^3+1264*q^4+3806*q^5+9914*
q^6+22845*q^7+47314*q^8+89105*q^9+153981*q^10+245955*q^11+365398*q^12+507668*q^
13+662997*q^14+817865*q^15+957651*q^16+1069558*q^17+1145017*q^18+1180755*q^19+
1178454*q^20+1143346*q^21+1082590*q^22+1003574*q^23+913036*q^24+816680*q^25+
718894*q^26+623214*q^27+532277*q^28+448041*q^29+371893*q^30+304492*q^31+245870*
q^32+195832*q^33+153967*q^34+119393*q^35+91284*q^36+68849*q^37+51239*q^38+37609
*q^39+27223*q^40+19417*q^41+13654*q^42+9433*q^43+6430*q^44+4288*q^45+2826*q^46+
1818*q^47+1151*q^48+694*q^49+414*q^50+228*q^51+123*q^52+57*q^53+27*q^54+9*q^55+
3*q^56, 1+13*q+90*q^2+441*q^3+1704*q^4+5499*q^5+15345*q^6+37885*q^7+84108*q^8+
169931*q^9+315347*q^10+541495*q^11+865693*q^12+1295407*q^13+1823049*q^14+
2423676*q^15+3057035*q^16+3673788*q^17+4224291*q^18+4667311*q^19+4976126*q^20+
5140699*q^21+5166181*q^22+5068665*q^23+4870430*q^24+4595564*q^25+4266844*q^26+
3904342*q^27+3524642*q^28+3141383*q^29+2765572*q^30+2406024*q^31+2069009*q^32+
1759279*q^33+1479631*q^34+1231152*q^35+1013414*q^36+825368*q^37+665115*q^38+
530276*q^39+418212*q^40+326432*q^41+252156*q^42+192750*q^43+145712*q^44+108933*
q^45+80547*q^46+58945*q^47+42650*q^48+30483*q^49+21492*q^50+14975*q^51+10280*q^
52+6975*q^53+4643*q^54+3054*q^55+1943*q^56+1219*q^57+730*q^58+430*q^59+233*q^60
+125*q^61+57*q^62+27*q^63+9*q^64+3*q^65, 1+14*q+104*q^2+545*q^3+2248*q^4+7735*q
^5+23000*q^6+60500*q^7+143133*q^8+308317*q^9+610412*q^10+1119113*q^11+1911754*q
^12+3058986*q^13+4605906*q^14+6553386*q^15+8845802*q^16+11370349*q^17+13969834*
q^18+16466430*q^19+18689910*q^20+20502917*q^21+21817126*q^22+22597764*q^23+
22857960*q^24+22646490*q^25+22033648*q^26+21098460*q^27+19919006*q^28+18566812*
q^29+17104375*q^30+15584904*q^31+14052447*q^32+12543860*q^33+11088845*q^34+
9710684*q^35+8426344*q^36+7246916*q^37+6178343*q^38+5222016*q^39+4375804*q^40+
3635453*q^41+2994898*q^42+2446369*q^43+1981405*q^44+1591295*q^45+1267323*q^46+
1000972*q^47+784190*q^48+609111*q^49+468928*q^50+357861*q^51+270839*q^52+203192
*q^53+151143*q^54+111370*q^55+81285*q^56+58711*q^57+42012*q^58+29700*q^59+20778
*q^60+14349*q^61+9809*q^62+6564*q^63+4340*q^64+2794*q^65+1767*q^66+1070*q^67+
641*q^68+356*q^69+195*q^70+98*q^71+46*q^72+17*q^73+8*q^74+q^75]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+6*q^2+4*q+3), q^2*(q^8+4*q^7+9*q^6
+15*q^5+18*q^4+18*q^3+13*q^2+8*q+3), q^3*(q^12+5*q^11+14*q^10+29*q^9+46*q^8+61*
q^7+65*q^6+60*q^5+46*q^4+30*q^3+14*q^2+8*q+1), q^5*(q^16+6*q^15+20*q^14+49*q^13
+94*q^12+151*q^11+204*q^10+238*q^9+243*q^8+218*q^7+172*q^6+125*q^5+78*q^4+43*q^
3+24*q^2+9*q+3), q^7*(q^21+7*q^20+27*q^19+76*q^18+169*q^17+315*q^16+502*q^15+
698*q^14+858*q^13+942*q^12+931*q^11+841*q^10+697*q^9+536*q^8+384*q^7+260*q^6+
161*q^5+94*q^4+47*q^3+26*q^2+9*q+3), q^9*(1+35*q^25+588*q^22+111*q^24+1067*q^21
+279*q^23+8*q+17*q^2+8*q^26+43*q^3+83*q^4+150*q^5+256*q^6+431*q^7+664*q^8+975*q
^9+1378*q^10+1854*q^11+2381*q^12+2915*q^13+3388*q^14+3708*q^15+3781*q^16+3564*q
^17+3077*q^18+2412*q^19+1700*q^20+q^27), q^12*(3+5828*q^25+13299*q^22+8394*q^24
+14860*q^21+11022*q^23+9*q+155*q^30+433*q^29+q^33+27*q^2+3656*q^26+54*q^3+111*q
^4+193*q^5+341*q^6+9*q^32+559*q^7+896*q^8+1377*q^9+2041*q^10+2912*q^11+4037*q^
12+5418*q^13+7022*q^14+8799*q^15+10692*q^16+1014*q^28+12526*q^17+14106*q^18+
15183*q^19+15488*q^20+2051*q^27+44*q^31), q^15*(3+64648*q^25+54744*q^22+63666*q
^24+48349*q^21+60110*q^23+9*q+29392*q^30+209*q^37+39481*q^29+10*q^39+3660*q^34+
54*q^38+7183*q^33+27*q^2+62534*q^26+56*q^3+116*q^4+208*q^5+370*q^6+12639*q^32+
604*q^7+984*q^8+1549*q^9+2375*q^10+3522*q^11+5101*q^12+7215*q^13+9962*q^14+
13416*q^15+17646*q^16+49192*q^28+22665*q^17+28428*q^18+34794*q^19+41544*q^20+
57214*q^27+q^40+20156*q^31+1647*q^35+641*q^36), q^18*(1+169521*q^25+6165*q^42+
99227*q^22+144488*q^24+79999*q^21+120872*q^23+8*q+914*q^44+271925*q^30+274*q^45
+103421*q^37+260475*q^29+65*q^46+43991*q^39+13168*q^41+217342*q^34+70209*q^38+
246701*q^33+17*q^2+195040*q^26+46*q^3+95*q^4+183*q^5+320*q^6+266121*q^32+560*q^
7+905*q^8+1454*q^9+2255*q^10+3443*q^11+5135*q^12+7521*q^13+10760*q^14+15099*q^
15+20860*q^16+2552*q^43+242335*q^28+28329*q^17+37706*q^18+49296*q^19+63346*q^20
+219845*q^27+25256*q^40+274373*q^31+180922*q^35+q^48+141514*q^36+11*q^47), q^22
*(3+304492*q^25+662997*q^42+153967*q^22+245870*q^24+119393*q^21+195832*q^23+9*q
+3806*q^51+365398*q^44+351*q^53+718894*q^30+1264*q^52+12*q^55+77*q^54+245955*q^
45+1180755*q^37+623214*q^29+153981*q^46+1069558*q^39+817865*q^41+1082590*q^34+
9914*q^50+1145017*q^38+1003574*q^33+q^56+27*q^2+371893*q^26+57*q^3+123*q^4+228*
q^5+414*q^6+913036*q^32+694*q^7+1151*q^8+1818*q^9+2826*q^10+4288*q^11+6430*q^12
+9433*q^13+13654*q^14+19417*q^15+27223*q^16+507668*q^43+22845*q^49+532277*q^28+
37609*q^17+51239*q^18+68849*q^19+91284*q^20+448041*q^27+957651*q^40+816680*q^31
+1143346*q^35+47314*q^48+1178454*q^36+89105*q^47), q^26*(3+418212*q^25+5068665*
q^42+192750*q^22+326432*q^24+145712*q^21+252156*q^23+9*q+1823049*q^51+5140699*q
^44+865693*q^53+84108*q^57+1231152*q^30+1295407*q^52+37885*q^58+315347*q^55+
5499*q^60+541495*q^54+4976126*q^45+3524642*q^37+1013414*q^29+4667311*q^46+q^65+
4266844*q^39+1704*q^61+4870430*q^41+2406024*q^34+2423676*q^50+13*q^64+3904342*q
^38+2069009*q^33+169931*q^56+27*q^2+530276*q^26+57*q^3+125*q^4+233*q^5+430*q^6+
1759279*q^32+730*q^7+1219*q^8+1943*q^9+441*q^62+3054*q^10+4643*q^11+6975*q^12+
10280*q^13+14975*q^14+21492*q^15+30483*q^16+5166181*q^43+3057035*q^49+825368*q^
28+42650*q^17+58945*q^18+80547*q^19+108933*q^20+665115*q^27+4595564*q^40+90*q^
63+1479631*q^31+2765572*q^35+3673788*q^48+15345*q^59+3141383*q^36+4224291*q^47)
, q^30*(1+468928*q^25+12543860*q^42+203192*q^22+357861*q^24+151143*q^21+270839*
q^23+8*q+22857960*q^51+15584904*q^44+21817126*q^53+13969834*q^57+1591295*q^30+
22597764*q^52+11370349*q^58+18689910*q^55+6553386*q^60+20502917*q^54+17104375*q
^45+2248*q^71+14*q^74+6178343*q^37+143133*q^67+1267323*q^29+18566812*q^46+
610412*q^65+23000*q^69+8426344*q^39+4605906*q^61+11088845*q^41+3635453*q^34+
22646490*q^50+1119113*q^64+7246916*q^38+2994898*q^33+16466430*q^56+17*q^2+60500
*q^68+609111*q^26+46*q^3+98*q^4+195*q^5+356*q^6+2446369*q^32+641*q^7+1070*q^8+
1767*q^9+3058986*q^62+2794*q^10+4340*q^11+6564*q^12+9809*q^13+14349*q^14+20778*
q^15+29700*q^16+308317*q^66+7735*q^70+14052447*q^43+22033648*q^49+1000972*q^28+
42012*q^17+58711*q^18+81285*q^19+111370*q^20+784190*q^27+545*q^72+9710684*q^40+
104*q^73+1911754*q^63+1981405*q^31+q^75+4375804*q^35+21098460*q^48+8845802*q^59
+5222016*q^36+19919006*q^47)]


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

[1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863,

    84324840, 409471090]

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

     is given by

[1, 1, 3, 10, 44, 187, 839, 3786, 17442, 81269, 383079, 1822002, 8734969,

    42162291, 204735609]

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

     is given by

[0, 1, 3, 12, 45, 193, 839, 3798, 17433, 81291, 383045, 1822064, 8734894,

    42162549, 204735481]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 44, 193, 839, 3786, 17442, 81291, 383045, 1822002, 8734969,

    42162549, 204735481]

 ODD:, [0, 1, 3, 12, 45, 187, 839, 3798, 17433, 81269, 383079, 1822064, 8734894,

    42162291, 204735609]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.818181818, 4.348314607, 6.044736842,

    7.883790226, 9.850474684, 11.93425090, 14.12731914, 16.42369773,

    18.81865532, 21.30833928, 23.88952514, 26.55944460]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.369683561, 1.780677333, 2.214925134,

    2.679585461, 3.175435581, 3.701408190, 4.256071881, 4.838075720,

    5.446254664, 6.079625786, 6.737354333, 7.418717161]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.87603, 3.17081, 4.90589, 7.18018, 10.0834,

    13.7004, 18.1141, 23.4070, 29.6617, 36.9618, 45.3919, 55.0374]

Skewness: , [Float(undefined), 0., 0., -0.09473975796, -0.05501765157,

    0.03651698845, 0.1352969027, 0.2251305381, 0.3017842517, 0.3654733706,

    0.4178307558, 0.4607546384, 0.4959909858, 0.5250323287, 0.5490745447]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.245837673,

    2.472917329, 2.646893968, 2.783940301, 2.897970284, 2.994694323,

    3.076703938, 3.145790936, 3.203679597, 3.251992687, 3.292238913,

    3.325756425]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+9*q^2+13*q^3
+17*q^4+15*q^5+11*q^6+8*q^7+5*q^8+2*q^9+q^10, 1+5*q+14*q^2+26*q^3+41*q^4+50*q^5
+51*q^6+45*q^7+35*q^8+27*q^9+15*q^10+12*q^11+8*q^12+5*q^13+2*q^14+q^15, 1+6*q+
20*q^2+45*q^3+83*q^4+124*q^5+157*q^6+170*q^7+165*q^8+145*q^9+112*q^10+84*q^11+
65*q^12+47*q^13+31*q^14+20*q^15+15*q^16+12*q^17+8*q^18+5*q^19+2*q^20+q^21, 1+7*
q+27*q^2+71*q^3+150*q^4+261*q^5+388*q^6+497*q^7+569*q^8+584*q^9+542*q^10+469*q^
11+385*q^12+302*q^13+234*q^14+167*q^15+126*q^16+88*q^17+70*q^18+50*q^19+36*q^20
+23*q^21+20*q^22+15*q^23+12*q^24+8*q^25+5*q^26+2*q^27+q^28, 1+8*q+35*q^2+105*q^
3+250*q^4+493*q^5+834*q^6+1225*q^7+1605*q^8+1889*q^9+2021*q^10+1993*q^11+1832*q
^12+1603*q^13+1336*q^14+1078*q^15+844*q^16+640*q^17+490*q^18+374*q^19+280*q^20+
204*q^21+145*q^22+114*q^23+90*q^24+72*q^25+55*q^26+40*q^27+28*q^28+23*q^29+20*q
^30+15*q^31+12*q^32+8*q^33+5*q^34+2*q^35+q^36, 1+9*q+44*q^2+148*q^3+392*q^4+861
*q^5+1625*q^6+2677*q^7+3933*q^8+5204*q^9+6272*q^10+6949*q^11+7149*q^12+6920*q^
13+6333*q^14+5561*q^15+4719*q^16+3863*q^17+3093*q^18+2452*q^19+1920*q^20+1491*q
^21+1109*q^22+844*q^23+643*q^24+511*q^25+385*q^26+299*q^27+219*q^28+170*q^29+
134*q^30+116*q^31+92*q^32+79*q^33+58*q^34+45*q^35+32*q^36+28*q^37+23*q^38+20*q^
39+15*q^40+12*q^41+8*q^42+5*q^43+2*q^44+q^45, 1+10*q+54*q^2+201*q^3+586*q^4+
1416*q^5+2941*q^6+5349*q^7+8681*q^8+12713*q^9+16981*q^10+20853*q^11+23751*q^12+
25316*q^13+25419*q^14+24285*q^15+22238*q^16+19606*q^17+16783*q^18+13988*q^19+
11497*q^20+9272*q^21+7364*q^22+5782*q^23+4514*q^24+3526*q^25+2754*q^26+2134*q^
27+1645*q^28+1260*q^29+979*q^30+772*q^31+609*q^32+491*q^33+400*q^34+309*q^35+
243*q^36+194*q^37+161*q^38+137*q^39+120*q^40+97*q^41+82*q^42+65*q^43+48*q^44+37
*q^45+32*q^46+28*q^47+23*q^48+20*q^49+15*q^50+12*q^51+8*q^52+5*q^53+2*q^54+q^55
, 1+11*q+65*q^2+265*q^3+843*q^4+2220*q^5+5023*q^6+9970*q^7+17670*q^8+28298*q^9+
41377*q^10+55667*q^11+69450*q^12+80927*q^13+88644*q^14+91949*q^15+90906*q^16+
86201*q^17+78843*q^18+69871*q^19+60431*q^20+51144*q^21+42526*q^22+34815*q^23+
28144*q^24+22556*q^25+17980*q^26+14269*q^27+11308*q^28+8894*q^29+7002*q^30+5458
*q^31+4294*q^32+3395*q^33+2732*q^34+2162*q^35+1731*q^36+1348*q^37+1077*q^38+859
*q^39+719*q^40+594*q^41+503*q^42+410*q^43+341*q^44+265*q^45+222*q^46+185*q^47+
165*q^48+139*q^49+126*q^50+102*q^51+87*q^52+68*q^53+55*q^54+40*q^55+37*q^56+32*
q^57+28*q^58+23*q^59+20*q^60+15*q^61+12*q^62+8*q^63+5*q^64+2*q^65+q^66, 1+12*q+
77*q^2+341*q^3+1175*q^4+3347*q^5+8185*q^6+17574*q^7+33710*q^8+58484*q^9+92714*q
^10+135349*q^11+183279*q^12+231677*q^13+275003*q^14+308370*q^15+328505*q^16+
334398*q^17+326900*q^18+308402*q^19+282255*q^20+251544*q^21+219224*q^22+187468*
q^23+157635*q^24+130864*q^25+107291*q^26+87365*q^27+70652*q^28+56893*q^29+45572
*q^30+36404*q^31+28969*q^32+23098*q^33+18467*q^34+14828*q^35+11886*q^36+9498*q^
37+7535*q^38+6015*q^39+4827*q^40+3949*q^41+3247*q^42+2672*q^43+2178*q^44+1767*q
^45+1431*q^46+1163*q^47+966*q^48+825*q^49+709*q^50+612*q^51+522*q^52+437*q^53+
363*q^54+299*q^55+242*q^56+215*q^57+189*q^58+167*q^59+146*q^60+129*q^61+108*q^
62+92*q^63+73*q^64+58*q^65+47*q^66+40*q^67+37*q^68+32*q^69+28*q^70+23*q^71+20*q
^72+15*q^73+12*q^74+8*q^75+5*q^76+2*q^77+q^78, 1+13*q+90*q^2+430*q^3+1595*q^4+
4884*q^5+12827*q^6+29585*q^7+60980*q^8+113754*q^9+194015*q^10+304940*q^11+
444752*q^12+605543*q^13+773900*q^14+933257*q^15+1067251*q^16+1163227*q^17+
1213934*q^18+1218560*q^19+1181814*q^20+1111718*q^21+1018456*q^22+911630*q^23+
799851*q^24+689782*q^25+585930*q^26+491778*q^27+408502*q^28+336608*q^29+275540*
q^30+224355*q^31+181856*q^32+146863*q^33+118422*q^34+95649*q^35+77265*q^36+
62530*q^37+50476*q^38+40701*q^39+32738*q^40+26554*q^41+21585*q^42+17711*q^43+
14474*q^44+11850*q^45+9653*q^46+7873*q^47+6406*q^48+5314*q^49+4414*q^50+3730*q^
51+3129*q^52+2644*q^53+2212*q^54+1858*q^55+1512*q^56+1275*q^57+1085*q^58+948*q^
59+819*q^60+733*q^61+624*q^62+552*q^63+465*q^64+390*q^65+322*q^66+280*q^67+233*
q^68+217*q^69+193*q^70+174*q^71+149*q^72+136*q^73+111*q^74+98*q^75+78*q^76+63*q
^77+50*q^78+47*q^79+40*q^80+37*q^81+32*q^82+28*q^83+23*q^84+20*q^85+15*q^86+12*
q^87+8*q^88+5*q^89+2*q^90+q^91, 1+14*q+104*q^2+533*q^3+2117*q^4+6932*q^5+19449*
q^6+47916*q^7+105506*q^8+210331*q^9+383534*q^10+644843*q^11+1006490*q^12+
1466861*q^13+2006589*q^14+2588769*q^15+3164178*q^16+3680299*q^17+4090521*q^18+
4362564*q^19+4482251*q^20+4453194*q^21+4294002*q^22+4031710*q^23+3697960*q^24+
3322645*q^25+2931782*q^26+2546862*q^27+2182763*q^28+1849436*q^29+1552257*q^30+
1292281*q^31+1069131*q^32+879165*q^33+719933*q^34+587557*q^35+479123*q^36+
390457*q^37+318134*q^38+258723*q^39+210310*q^40+171012*q^41+139472*q^42+114058*
q^43+93623*q^44+76902*q^45+63260*q^46+51911*q^47+42650*q^48+35008*q^49+28924*q^
50+23982*q^51+20016*q^52+16728*q^53+14091*q^54+11783*q^55+9838*q^56+8165*q^57+
6806*q^58+5716*q^59+4885*q^60+4189*q^61+3607*q^62+3124*q^63+2703*q^64+2305*q^65
+1947*q^66+1644*q^67+1395*q^68+1206*q^69+1063*q^70+948*q^71+843*q^72+745*q^73+
658*q^74+574*q^75+497*q^76+420*q^77+350*q^78+299*q^79+272*q^80+239*q^81+219*q^
82+198*q^83+179*q^84+156*q^85+139*q^86+118*q^87+101*q^88+84*q^89+68*q^90+55*q^
91+50*q^92+47*q^93+40*q^94+37*q^95+32*q^96+28*q^97+23*q^98+20*q^99+15*q^100+12*
q^101+8*q^102+5*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+5*q^3+5*q^2+2*q+1, q^10+4*q^9+9*q^8+
13*q^7+17*q^6+15*q^5+11*q^4+8*q^3+5*q^2+2*q+1, q^15+5*q^14+14*q^13+26*q^12+41*q
^11+50*q^10+51*q^9+45*q^8+35*q^7+27*q^6+15*q^5+12*q^4+8*q^3+5*q^2+2*q+1, q^21+6
*q^20+20*q^19+45*q^18+83*q^17+124*q^16+157*q^15+170*q^14+165*q^13+145*q^12+112*
q^11+84*q^10+65*q^9+47*q^8+31*q^7+20*q^6+15*q^5+12*q^4+8*q^3+5*q^2+2*q+1, 1+71*
q^25+388*q^22+150*q^24+497*q^21+261*q^23+2*q+5*q^2+27*q^26+8*q^3+12*q^4+15*q^5+
20*q^6+23*q^7+36*q^8+50*q^9+70*q^10+88*q^11+126*q^12+167*q^13+234*q^14+302*q^15
+385*q^16+q^28+469*q^17+542*q^18+584*q^19+569*q^20+7*q^27, 1+1993*q^25+1336*q^
22+1832*q^24+1078*q^21+1603*q^23+2*q+834*q^30+1225*q^29+35*q^34+105*q^33+5*q^2+
2021*q^26+8*q^3+12*q^4+15*q^5+20*q^6+250*q^32+23*q^7+28*q^8+40*q^9+55*q^10+72*q
^11+90*q^12+114*q^13+145*q^14+204*q^15+280*q^16+1605*q^28+374*q^17+490*q^18+640
*q^19+844*q^20+1889*q^27+493*q^31+8*q^35+q^36, 1+1920*q^25+148*q^42+844*q^22+
1491*q^24+643*q^21+1109*q^23+2*q+9*q^44+5561*q^30+q^45+3933*q^37+4719*q^29+1625
*q^39+392*q^41+6949*q^34+2677*q^38+7149*q^33+5*q^2+2452*q^26+8*q^3+12*q^4+15*q^
5+20*q^6+6920*q^32+23*q^7+28*q^8+32*q^9+45*q^10+58*q^11+79*q^12+92*q^13+116*q^
14+134*q^15+170*q^16+44*q^43+3863*q^28+219*q^17+299*q^18+385*q^19+511*q^20+3093
*q^27+861*q^40+6333*q^31+6272*q^35+5204*q^36, 1+979*q^25+25316*q^42+491*q^22+
772*q^24+400*q^21+609*q^23+2*q+586*q^51+20853*q^44+54*q^53+3526*q^30+201*q^52+q
^55+10*q^54+16981*q^45+16783*q^37+2754*q^29+12713*q^46+22238*q^39+25419*q^41+
9272*q^34+1416*q^50+19606*q^38+7364*q^33+5*q^2+1260*q^26+8*q^3+12*q^4+15*q^5+20
*q^6+5782*q^32+23*q^7+28*q^8+32*q^9+37*q^10+48*q^11+65*q^12+82*q^13+97*q^14+120
*q^15+137*q^16+23751*q^43+2941*q^49+2134*q^28+161*q^17+194*q^18+243*q^19+309*q^
20+1645*q^27+24285*q^40+4514*q^31+11497*q^35+5349*q^48+13988*q^36+8681*q^47, 1+
594*q^25+28144*q^42+341*q^22+503*q^24+265*q^21+410*q^23+2*q+91949*q^51+42526*q^
44+80927*q^53+28298*q^57+1731*q^30+88644*q^52+17670*q^58+55667*q^55+5023*q^60+
69450*q^54+51144*q^45+8894*q^37+1348*q^29+60431*q^46+11*q^65+14269*q^39+2220*q^
61+22556*q^41+4294*q^34+90906*q^50+65*q^64+11308*q^38+3395*q^33+41377*q^56+5*q^
2+719*q^26+8*q^3+12*q^4+15*q^5+20*q^6+2732*q^32+23*q^7+28*q^8+32*q^9+843*q^62+
37*q^10+40*q^11+55*q^12+68*q^13+87*q^14+102*q^15+126*q^16+q^66+34815*q^43+86201
*q^49+1077*q^28+139*q^17+165*q^18+185*q^19+222*q^20+859*q^27+17980*q^40+265*q^
63+2162*q^31+5458*q^35+78843*q^48+9970*q^59+7002*q^36+69871*q^47, 1+437*q^25+
11886*q^42+242*q^22+363*q^24+215*q^21+299*q^23+2*q+87365*q^51+18467*q^44+130864
*q^53+251544*q^57+966*q^30+107291*q^52+282255*q^58+187468*q^55+326900*q^60+
157635*q^54+23098*q^45+17574*q^71+1175*q^74+3949*q^37+135349*q^67+825*q^29+
28969*q^46+231677*q^65+58484*q^69+6015*q^39+334398*q^61+9498*q^41+2178*q^34+
70652*q^50+275003*q^64+4827*q^38+1767*q^33+219224*q^56+5*q^2+92714*q^68+522*q^
26+8*q^3+12*q^4+15*q^5+20*q^6+1431*q^32+23*q^7+28*q^8+32*q^9+328505*q^62+37*q^
10+40*q^11+47*q^12+77*q^76+58*q^13+73*q^14+92*q^15+108*q^16+183279*q^66+33710*q
^70+12*q^77+q^78+14828*q^43+56893*q^49+709*q^28+129*q^17+146*q^18+167*q^19+189*
q^20+612*q^27+8185*q^72+7535*q^40+3347*q^73+308370*q^63+1163*q^31+341*q^75+2672
*q^35+45572*q^48+308402*q^59+3247*q^36+36404*q^47, 1+322*q^25+5314*q^42+217*q^
22+13*q^90+280*q^24+193*q^21+233*q^23+2*q+32738*q^51+7873*q^44+50476*q^53+
118422*q^57+733*q^30+40701*q^52+146863*q^58+77265*q^55+224355*q^60+q^91+62530*q
^54+9653*q^45+1181814*q^71+1163227*q^74+2212*q^37+799851*q^67+624*q^29+11850*q^
46+585930*q^65+1018456*q^69+3129*q^39+275540*q^61+4414*q^41+1275*q^34+444752*q^
79+26554*q^50+491778*q^64+2644*q^38+1085*q^33+95649*q^56+5*q^2+911630*q^68+390*
q^26+8*q^3+12*q^4+15*q^5+20*q^6+948*q^32+23*q^7+28*q^8+32*q^9+336608*q^62+37*q^
10+40*q^11+47*q^12+933257*q^76+50*q^13+63*q^14+78*q^15+98*q^16+689782*q^66+
1111718*q^70+773900*q^77+605543*q^78+6406*q^43+21585*q^49+552*q^28+111*q^17+136
*q^18+149*q^19+174*q^20+465*q^27+1218560*q^72+3730*q^40+1213934*q^73+408502*q^
63+819*q^31+1067251*q^75+1512*q^35+304940*q^80+17711*q^48+60980*q^83+194015*q^
81+12827*q^85+4884*q^86+1595*q^87+430*q^88+90*q^89+29585*q^84+113754*q^82+
181856*q^59+1858*q^36+14474*q^47, 1+272*q^25+3124*q^42+198*q^22+2588769*q^90+
239*q^24+179*q^21+219*q^23+2*q+105506*q^97+14091*q^51+4189*q^44+20016*q^53+
42650*q^57+574*q^30+644843*q^94+16728*q^52+51911*q^58+28924*q^55+76902*q^60+
2006589*q^91+23982*q^54+4885*q^45+719933*q^71+1292281*q^74+210331*q^96+47916*q^
98+1395*q^37+318134*q^67+497*q^29+5716*q^46+1006490*q^93+19449*q^99+210310*q^65
+479123*q^69+1947*q^39+93623*q^61+2703*q^41+948*q^34+2931782*q^79+11783*q^50+
171012*q^64+6932*q^100+1644*q^38+843*q^33+35008*q^56+5*q^2+390457*q^68+299*q^26
+8*q^3+12*q^4+15*q^5+20*q^6+745*q^32+23*q^7+28*q^8+32*q^9+383534*q^95+114058*q^
62+2117*q^101+37*q^10+40*q^11+47*q^12+1849436*q^76+50*q^13+55*q^14+68*q^15+84*q
^16+258723*q^66+587557*q^70+2182763*q^77+2546862*q^78+3607*q^43+533*q^102+9838*
q^49+420*q^28+101*q^17+118*q^18+139*q^19+156*q^20+350*q^27+879165*q^72+2305*q^
40+1069131*q^73+104*q^103+139472*q^63+658*q^31+1552257*q^75+1063*q^35+14*q^104+
3322645*q^80+8165*q^48+q^105+4294002*q^83+3697960*q^81+4482251*q^85+4362564*q^
86+4090521*q^87+3680299*q^88+3164178*q^89+4453194*q^84+4031710*q^82+63260*q^59+
1206*q^36+1466861*q^92+6806*q^47]


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

[1, 2, 6, 22, 86, 338, 1318, 5110, 19770, 76466, 295810, 1144530, 4428622,

    17136186, 66306722]

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

     is given by

[1, 1, 3, 12, 44, 167, 659, 2566, 9884, 38201, 147931, 572368, 2214186, 8567829,

    33153911]

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

     is given by

[0, 1, 3, 10, 42, 171, 659, 2544, 9886, 38265, 147879, 572162, 2214436, 8568357,

    33152811]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 44, 171, 659, 2566, 9884, 38265, 147879, 572368, 2214186,

    8568357, 33152811]

 ODD:, [0, 1, 3, 10, 42, 167, 659, 2544, 9886, 38201, 147931, 572162, 2214436,

    8567829, 33153911]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.909090909, 4.581395349, 6.375739645,

    8.196509863, 10.00195695, 11.78674760, 13.55934664, 15.32790981,

    17.09636619, 18.86561147, 20.63540107, 22.40538501]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.474297704, 2.071204101, 2.717073492,

    3.356431546, 3.944229120, 4.462165617, 4.915370922, 5.319307049,

    5.688839451, 6.034009107, 6.360479831, 6.671362284]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.17355, 4.28989, 7.38249, 11.2656, 15.5569,

    19.9109, 24.1609, 28.2950, 32.3629, 36.4093, 40.4557, 44.5071]

Skewness: , [Float(undefined), 0., 0., 0.07174070299, 0.2455307502,

    0.4516827276, 0.6475482967, 0.8109694785, 0.9294261233, 1.000161201,

    1.030285620, 1.032462806, 1.019066732, 0.9987888425, 0.9763505324]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.419438345,

    2.658549405, 3.004780159, 3.491757379, 4.068939125, 4.642623634,

    5.114594707, 5.425907923, 5.574926553, 5.600356126, 5.550859478,

    5.464838528]

                                end of this data



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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+6*q^2+11*q^3
+14*q^4+14*q^5+14*q^6+11*q^7+6*q^8+4*q^9+q^10, 1+5*q+8*q^2+16*q^3+24*q^4+33*q^5
+39*q^6+44*q^7+44*q^8+39*q^9+33*q^10+24*q^11+16*q^12+8*q^13+5*q^14+q^15, 1+6*q+
10*q^2+21*q^3+34*q^4+52*q^5+74*q^6+93*q^7+114*q^8+128*q^9+137*q^10+137*q^11+128
*q^12+114*q^13+93*q^14+74*q^15+52*q^16+34*q^17+21*q^18+10*q^19+6*q^20+q^21, 1+7
*q+12*q^2+26*q^3+44*q^4+71*q^5+109*q^6+154*q^7+204*q^8+259*q^9+317*q^10+364*q^
11+408*q^12+431*q^13+440*q^14+431*q^15+408*q^16+364*q^17+317*q^18+259*q^19+204*
q^20+154*q^21+109*q^22+71*q^23+44*q^24+26*q^25+12*q^26+7*q^27+q^28, 1+8*q+14*q^
2+31*q^3+54*q^4+90*q^5+144*q^6+215*q^7+308*q^8+414*q^9+549*q^10+689*q^11+844*q^
12+996*q^13+1139*q^14+1263*q^15+1358*q^16+1422*q^17+1440*q^18+1422*q^19+1358*q^
20+1263*q^21+1139*q^22+996*q^23+844*q^24+689*q^25+549*q^26+414*q^27+308*q^28+
215*q^29+144*q^30+90*q^31+54*q^32+31*q^33+14*q^34+8*q^35+q^36, 1+9*q+16*q^2+36*
q^3+64*q^4+109*q^5+179*q^6+276*q^7+412*q^8+585*q^9+809*q^10+1076*q^11+1400*q^12
+1759*q^13+2166*q^14+2589*q^15+3030*q^16+3451*q^17+3854*q^18+4201*q^19+4482*q^
20+4680*q^21+4782*q^22+4782*q^23+4680*q^24+4482*q^25+4201*q^26+3854*q^27+3451*q
^28+3030*q^29+2589*q^30+2166*q^31+1759*q^32+1400*q^33+1076*q^34+809*q^35+585*q^
36+412*q^37+276*q^38+179*q^39+109*q^40+64*q^41+36*q^42+16*q^43+9*q^44+q^45, 1+
10*q+18*q^2+41*q^3+74*q^4+128*q^5+214*q^6+337*q^7+516*q^8+756*q^9+1087*q^10+
1495*q^11+2028*q^12+2664*q^13+3433*q^14+4323*q^15+5338*q^16+6450*q^17+7652*q^18
+8913*q^19+10180*q^20+11447*q^21+12632*q^22+13720*q^23+14644*q^24+15383*q^25+
15882*q^26+16148*q^27+16148*q^28+15882*q^29+15383*q^30+14644*q^31+13720*q^32+
12632*q^33+11447*q^34+10180*q^35+8913*q^36+7652*q^37+6450*q^38+5338*q^39+4323*q
^40+3433*q^41+2664*q^42+2028*q^43+1495*q^44+1087*q^45+756*q^46+516*q^47+337*q^
48+214*q^49+128*q^50+74*q^51+41*q^52+18*q^53+10*q^54+q^55, 1+11*q+20*q^2+46*q^3
+84*q^4+147*q^5+249*q^6+398*q^7+620*q^8+927*q^9+1365*q^10+1934*q^11+2692*q^12+
3651*q^13+4864*q^14+6339*q^15+8134*q^16+10227*q^17+12668*q^18+15423*q^19+18494*
q^20+21837*q^21+25411*q^22+29154*q^23+32976*q^24+36808*q^25+40516*q^26+44027*q^
27+47209*q^28+49986*q^29+52234*q^30+53921*q^31+54937*q^32+55296*q^33+54937*q^34
+53921*q^35+52234*q^36+49986*q^37+47209*q^38+44027*q^39+40516*q^40+36808*q^41+
32976*q^42+29154*q^43+25411*q^44+21837*q^45+18494*q^46+15423*q^47+12668*q^48+
10227*q^49+8134*q^50+6339*q^51+4864*q^52+3651*q^53+2692*q^54+1934*q^55+1365*q^
56+927*q^57+620*q^58+398*q^59+249*q^60+147*q^61+84*q^62+46*q^63+20*q^64+11*q^65
+q^66, 1+12*q+22*q^2+51*q^3+94*q^4+166*q^5+284*q^6+459*q^7+724*q^8+1098*q^9+
1643*q^10+2373*q^11+3378*q^12+4678*q^13+6387*q^14+8541*q^15+11254*q^16+14572*q^
17+18604*q^18+23399*q^19+29020*q^20+35527*q^21+42908*q^22+51199*q^23+60322*q^24
+70273*q^25+80877*q^26+92080*q^27+103630*q^28+115402*q^29+127108*q^30+138554*q^
31+149442*q^32+159543*q^33+168590*q^34+176346*q^35+182635*q^36+187227*q^37+
190063*q^38+190992*q^39+190063*q^40+187227*q^41+182635*q^42+176346*q^43+168590*
q^44+159543*q^45+149442*q^46+138554*q^47+127108*q^48+115402*q^49+103630*q^50+
92080*q^51+80877*q^52+70273*q^53+60322*q^54+51199*q^55+42908*q^56+35527*q^57+
29020*q^58+23399*q^59+18604*q^60+14572*q^61+11254*q^62+8541*q^63+6387*q^64+4678
*q^65+3378*q^66+2373*q^67+1643*q^68+1098*q^69+724*q^70+459*q^71+284*q^72+166*q^
73+94*q^74+51*q^75+22*q^76+12*q^77+q^78, 1+13*q+24*q^2+56*q^3+104*q^4+185*q^5+
319*q^6+520*q^7+828*q^8+1269*q^9+1921*q^10+2812*q^11+4064*q^12+5729*q^13+7954*q
^14+10845*q^15+14582*q^16+19283*q^17+25188*q^18+32437*q^19+41260*q^20+51843*q^
21+64389*q^22+79058*q^23+96012*q^24+115385*q^25+137209*q^26+161585*q^27+188383*
q^28+217576*q^29+248907*q^30+282200*q^31+316993*q^32+353012*q^33+389626*q^34+
426394*q^35+462636*q^36+497798*q^37+531126*q^38+562083*q^39+589937*q^40+614176*
q^41+634235*q^42+649704*q^43+660188*q^44+665515*q^45+665515*q^46+660188*q^47+
649704*q^48+634235*q^49+614176*q^50+589937*q^51+562083*q^52+531126*q^53+497798*
q^54+462636*q^55+426394*q^56+389626*q^57+353012*q^58+316993*q^59+282200*q^60+
248907*q^61+217576*q^62+188383*q^63+161585*q^64+137209*q^65+115385*q^66+96012*q
^67+79058*q^68+64389*q^69+51843*q^70+41260*q^71+32437*q^72+25188*q^73+19283*q^
74+14582*q^75+10845*q^76+7954*q^77+5729*q^78+4064*q^79+2812*q^80+1921*q^81+1269
*q^82+828*q^83+520*q^84+319*q^85+185*q^86+104*q^87+56*q^88+24*q^89+13*q^90+q^91
, 1+14*q+26*q^2+61*q^3+114*q^4+204*q^5+354*q^6+581*q^7+932*q^8+1440*q^9+2199*q^
10+3251*q^11+4750*q^12+6780*q^13+9547*q^14+13197*q^15+18022*q^16+24224*q^17+
32180*q^18+42203*q^19+54704*q^20+70121*q^21+88910*q^22+111585*q^23+138612*q^24+
170575*q^25+207877*q^26+251119*q^27+300602*q^28+356828*q^29+419929*q^30+490254*
q^31+567658*q^32+652217*q^33+743527*q^34+841254*q^35+944699*q^36+1053112*q^37+
1165470*q^38+1280608*q^39+1397247*q^40+1513828*q^41+1628898*q^42+1740662*q^43+
1847584*q^44+1947779*q^45+2039815*q^46+2121898*q^47+2192799*q^48+2251078*q^49+
2295817*q^50+2326072*q^51+2341370*q^52+2341370*q^53+2326072*q^54+2295817*q^55+
2251078*q^56+2192799*q^57+2121898*q^58+2039815*q^59+1947779*q^60+1847584*q^61+
1740662*q^62+1628898*q^63+1513828*q^64+1397247*q^65+1280608*q^66+1165470*q^67+
1053112*q^68+944699*q^69+841254*q^70+743527*q^71+652217*q^72+567658*q^73+490254
*q^74+419929*q^75+356828*q^76+300602*q^77+251119*q^78+207877*q^79+170575*q^80+
138612*q^81+111585*q^82+88910*q^83+70121*q^84+54704*q^85+42203*q^86+32180*q^87+
24224*q^88+18022*q^89+13197*q^90+9547*q^91+6780*q^92+4750*q^93+3251*q^94+2199*q
^95+1440*q^96+932*q^97+581*q^98+354*q^99+204*q^100+114*q^101+61*q^102+26*q^103+
14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+6*q^2+11*q^3
+14*q^4+14*q^5+14*q^6+11*q^7+6*q^8+4*q^9+q^10, 1+5*q+8*q^2+16*q^3+24*q^4+33*q^5
+39*q^6+44*q^7+44*q^8+39*q^9+33*q^10+24*q^11+16*q^12+8*q^13+5*q^14+q^15, 1+6*q+
10*q^2+21*q^3+34*q^4+52*q^5+74*q^6+93*q^7+114*q^8+128*q^9+137*q^10+137*q^11+128
*q^12+114*q^13+93*q^14+74*q^15+52*q^16+34*q^17+21*q^18+10*q^19+6*q^20+q^21, 1+7
*q+12*q^2+26*q^3+44*q^4+71*q^5+109*q^6+154*q^7+204*q^8+259*q^9+317*q^10+364*q^
11+408*q^12+431*q^13+440*q^14+431*q^15+408*q^16+364*q^17+317*q^18+259*q^19+204*
q^20+154*q^21+109*q^22+71*q^23+44*q^24+26*q^25+12*q^26+7*q^27+q^28, 1+8*q+14*q^
2+31*q^3+54*q^4+90*q^5+144*q^6+215*q^7+308*q^8+414*q^9+549*q^10+689*q^11+844*q^
12+996*q^13+1139*q^14+1263*q^15+1358*q^16+1422*q^17+1440*q^18+1422*q^19+1358*q^
20+1263*q^21+1139*q^22+996*q^23+844*q^24+689*q^25+549*q^26+414*q^27+308*q^28+
215*q^29+144*q^30+90*q^31+54*q^32+31*q^33+14*q^34+8*q^35+q^36, 1+9*q+16*q^2+36*
q^3+64*q^4+109*q^5+179*q^6+276*q^7+412*q^8+585*q^9+809*q^10+1076*q^11+1400*q^12
+1759*q^13+2166*q^14+2589*q^15+3030*q^16+3451*q^17+3854*q^18+4201*q^19+4482*q^
20+4680*q^21+4782*q^22+4782*q^23+4680*q^24+4482*q^25+4201*q^26+3854*q^27+3451*q
^28+3030*q^29+2589*q^30+2166*q^31+1759*q^32+1400*q^33+1076*q^34+809*q^35+585*q^
36+412*q^37+276*q^38+179*q^39+109*q^40+64*q^41+36*q^42+16*q^43+9*q^44+q^45, 1+
10*q+18*q^2+41*q^3+74*q^4+128*q^5+214*q^6+337*q^7+516*q^8+756*q^9+1087*q^10+
1495*q^11+2028*q^12+2664*q^13+3433*q^14+4323*q^15+5338*q^16+6450*q^17+7652*q^18
+8913*q^19+10180*q^20+11447*q^21+12632*q^22+13720*q^23+14644*q^24+15383*q^25+
15882*q^26+16148*q^27+16148*q^28+15882*q^29+15383*q^30+14644*q^31+13720*q^32+
12632*q^33+11447*q^34+10180*q^35+8913*q^36+7652*q^37+6450*q^38+5338*q^39+4323*q
^40+3433*q^41+2664*q^42+2028*q^43+1495*q^44+1087*q^45+756*q^46+516*q^47+337*q^
48+214*q^49+128*q^50+74*q^51+41*q^52+18*q^53+10*q^54+q^55, 1+11*q+20*q^2+46*q^3
+84*q^4+147*q^5+249*q^6+398*q^7+620*q^8+927*q^9+1365*q^10+1934*q^11+2692*q^12+
3651*q^13+4864*q^14+6339*q^15+8134*q^16+10227*q^17+12668*q^18+15423*q^19+18494*
q^20+21837*q^21+25411*q^22+29154*q^23+32976*q^24+36808*q^25+40516*q^26+44027*q^
27+47209*q^28+49986*q^29+52234*q^30+53921*q^31+54937*q^32+55296*q^33+54937*q^34
+53921*q^35+52234*q^36+49986*q^37+47209*q^38+44027*q^39+40516*q^40+36808*q^41+
32976*q^42+29154*q^43+25411*q^44+21837*q^45+18494*q^46+15423*q^47+12668*q^48+
10227*q^49+8134*q^50+6339*q^51+4864*q^52+3651*q^53+2692*q^54+1934*q^55+1365*q^
56+927*q^57+620*q^58+398*q^59+249*q^60+147*q^61+84*q^62+46*q^63+20*q^64+11*q^65
+q^66, 1+12*q+22*q^2+51*q^3+94*q^4+166*q^5+284*q^6+459*q^7+724*q^8+1098*q^9+
1643*q^10+2373*q^11+3378*q^12+4678*q^13+6387*q^14+8541*q^15+11254*q^16+14572*q^
17+18604*q^18+23399*q^19+29020*q^20+35527*q^21+42908*q^22+51199*q^23+60322*q^24
+70273*q^25+80877*q^26+92080*q^27+103630*q^28+115402*q^29+127108*q^30+138554*q^
31+149442*q^32+159543*q^33+168590*q^34+176346*q^35+182635*q^36+187227*q^37+
190063*q^38+190992*q^39+190063*q^40+187227*q^41+182635*q^42+176346*q^43+168590*
q^44+159543*q^45+149442*q^46+138554*q^47+127108*q^48+115402*q^49+103630*q^50+
92080*q^51+80877*q^52+70273*q^53+60322*q^54+51199*q^55+42908*q^56+35527*q^57+
29020*q^58+23399*q^59+18604*q^60+14572*q^61+11254*q^62+8541*q^63+6387*q^64+4678
*q^65+3378*q^66+2373*q^67+1643*q^68+1098*q^69+724*q^70+459*q^71+284*q^72+166*q^
73+94*q^74+51*q^75+22*q^76+12*q^77+q^78, 1+13*q+24*q^2+56*q^3+104*q^4+185*q^5+
319*q^6+520*q^7+828*q^8+1269*q^9+1921*q^10+2812*q^11+4064*q^12+5729*q^13+7954*q
^14+10845*q^15+14582*q^16+19283*q^17+25188*q^18+32437*q^19+41260*q^20+51843*q^
21+64389*q^22+79058*q^23+96012*q^24+115385*q^25+137209*q^26+161585*q^27+188383*
q^28+217576*q^29+248907*q^30+282200*q^31+316993*q^32+353012*q^33+389626*q^34+
426394*q^35+462636*q^36+497798*q^37+531126*q^38+562083*q^39+589937*q^40+614176*
q^41+634235*q^42+649704*q^43+660188*q^44+665515*q^45+665515*q^46+660188*q^47+
649704*q^48+634235*q^49+614176*q^50+589937*q^51+562083*q^52+531126*q^53+497798*
q^54+462636*q^55+426394*q^56+389626*q^57+353012*q^58+316993*q^59+282200*q^60+
248907*q^61+217576*q^62+188383*q^63+161585*q^64+137209*q^65+115385*q^66+96012*q
^67+79058*q^68+64389*q^69+51843*q^70+41260*q^71+32437*q^72+25188*q^73+19283*q^
74+14582*q^75+10845*q^76+7954*q^77+5729*q^78+4064*q^79+2812*q^80+1921*q^81+1269
*q^82+828*q^83+520*q^84+319*q^85+185*q^86+104*q^87+56*q^88+24*q^89+13*q^90+q^91
, 1+14*q+26*q^2+61*q^3+114*q^4+204*q^5+354*q^6+581*q^7+932*q^8+1440*q^9+2199*q^
10+3251*q^11+4750*q^12+6780*q^13+9547*q^14+13197*q^15+18022*q^16+24224*q^17+
32180*q^18+42203*q^19+54704*q^20+70121*q^21+88910*q^22+111585*q^23+138612*q^24+
170575*q^25+207877*q^26+251119*q^27+300602*q^28+356828*q^29+419929*q^30+490254*
q^31+567658*q^32+652217*q^33+743527*q^34+841254*q^35+944699*q^36+1053112*q^37+
1165470*q^38+1280608*q^39+1397247*q^40+1513828*q^41+1628898*q^42+1740662*q^43+
1847584*q^44+1947779*q^45+2039815*q^46+2121898*q^47+2192799*q^48+2251078*q^49+
2295817*q^50+2326072*q^51+2341370*q^52+2341370*q^53+2326072*q^54+2295817*q^55+
2251078*q^56+2192799*q^57+2121898*q^58+2039815*q^59+1947779*q^60+1847584*q^61+
1740662*q^62+1628898*q^63+1513828*q^64+1397247*q^65+1280608*q^66+1165470*q^67+
1053112*q^68+944699*q^69+841254*q^70+743527*q^71+652217*q^72+567658*q^73+490254
*q^74+419929*q^75+356828*q^76+300602*q^77+251119*q^78+207877*q^79+170575*q^80+
138612*q^81+111585*q^82+88910*q^83+70121*q^84+54704*q^85+42203*q^86+32180*q^87+
24224*q^88+18022*q^89+13197*q^90+9547*q^91+6780*q^92+4750*q^93+3251*q^94+2199*q
^95+1440*q^96+932*q^97+581*q^98+354*q^99+204*q^100+114*q^101+61*q^102+26*q^103+
14*q^104+q^105]


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

[1, 2, 6, 22, 86, 340, 1340, 5254, 20518, 79932, 311028, 1209916, 4707964,

    18330728, 71429176]

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

     is given by

[1, 1, 3, 10, 42, 170, 670, 2630, 10262, 39966, 155514, 604948, 2353972,

    9165364, 35714588]

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

     is given by

[0, 1, 3, 12, 44, 170, 670, 2624, 10256, 39966, 155514, 604968, 2353992,

    9165364, 35714588]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 42, 170, 670, 2630, 10262, 39966, 155514, 604948, 2353972,

    9165364, 35714588]

 ODD:, [0, 1, 3, 12, 44, 170, 670, 2624, 10256, 39966, 155514, 604968, 2353992,

    9165364, 35714588]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.500000000,

    10.50000000, 14.00000000, 18.00000000, 22.50000000, 27.50000000,

    33.00000000, 39.00000000, 45.50000000, 52.50000000]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.507556723, 2.162040380, 2.902838368,

    3.713127177, 4.582118802, 5.503235674, 6.472356231, 7.486741886,

    8.544424339, 9.643862646, 10.78375582, 11.96294407]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.27273, 4.67442, 8.42647, 13.7873, 20.9958,

    30.2856, 41.8914, 56.0513, 73.0072, 93.0041, 116.289, 143.112]

                                                      -5
Skewness: , [Float(undefined), 0., 0., 0.2918624564 10  , 0., 0., 0., 0., 0.,

                                    -6
    0., 0., 0., 0., -0.7974290370 10  , 0.]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.340792622,

    2.474535437, 2.573269196, 2.648845022, 2.706070983, 2.749424008,

    2.782614116, 2.808385443, 2.828734526, 2.845106980, 2.858524295,

    2.869663209]

                                end of this data



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

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

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

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

                                  here it is:

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

    [0, 2, 0, 0, 0], [2, 1, 0, 0, 0], [3, 0, 0, 0, 0], [2, 0, 1, 0, 0],

    [1, 1, 1, 0, 0], [1, 0, 2, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0],

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

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

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

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

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

    [0, 0, 3, 0, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0], [0, 0, 1, 2, 0],

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

    [0, 0, 1, 0, 0], [0, 2, 0, 0, 0], [2, 1, 0, 0, 0], [3, 0, 0, 0, 0],

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+5*q^4+3*q^5, 1+4*q+9*q^2+13*q^3+17*
q^4+17*q^5+13*q^6+9*q^7+3*q^8, 1+5*q+14*q^2+26*q^3+41*q^4+53*q^5+58*q^6+55*q^7+
42*q^8+26*q^9+14*q^10+6*q^11+q^12, 1+6*q+20*q^2+45*q^3+83*q^4+128*q^5+171*q^6+
199*q^7+203*q^8+181*q^9+137*q^10+90*q^11+54*q^12+30*q^13+13*q^14+4*q^15+q^16, 1
+7*q+27*q^2+71*q^3+150*q^4+266*q^5+411*q^6+558*q^7+677*q^8+734*q^9+706*q^10+612
*q^11+472*q^12+321*q^13+204*q^14+125*q^15+70*q^16+33*q^17+12*q^18+4*q^19+q^20,
1+8*q+35*q^2+105*q^3+250*q^4+499*q^5+868*q^6+1334*q^7+1843*q^8+2300*q^9+2597*q^
10+2672*q^11+2496*q^12+2119*q^13+1651*q^14+1179*q^15+779*q^16+495*q^17+304*q^18
+172*q^19+85*q^20+36*q^21+13*q^22+4*q^23+q^24, 1+9*q+44*q^2+148*q^3+392*q^4+868
*q^5+1672*q^6+2853*q^7+4386*q^8+6124*q^9+7808*q^10+9136*q^11+9824*q^12+9730*q^
13+8891*q^14+7504*q^15+5886*q^16+4330*q^17+2993*q^18+1959*q^19+1244*q^20+762*q^
21+434*q^22+223*q^23+101*q^24+41*q^25+14*q^26+4*q^27+q^28, 1+10*q+54*q^2+201*q^
3+586*q^4+1424*q^5+3003*q^6+5614*q^7+9463*q^8+14527*q^9+20458*q^10+26566*q^11+
31919*q^12+35583*q^13+36859*q^14+35549*q^15+31998*q^16+26939*q^17+21316*q^18+
15976*q^19+11397*q^20+7742*q^21+5042*q^22+3188*q^23+1945*q^24+1114*q^25+586*q^
26+280*q^27+120*q^28+46*q^29+15*q^30+4*q^31+q^32, 1+11*q+65*q^2+265*q^3+843*q^4
+2229*q^5+5102*q^6+10349*q^7+18928*q^8+31569*q^9+48416*q^10+68677*q^11+90497*q^
12+111147*q^13+127511*q^14+136938*q^15+137901*q^16+130431*q^17+116189*q^18+
97774*q^19+77990*q^20+59305*q^21+43223*q^22+30236*q^23+20303*q^24+13151*q^25+
8275*q^26+5022*q^27+2887*q^28+1547*q^29+763*q^30+345*q^31+140*q^32+51*q^33+16*q
^34+4*q^35+q^36, 1+12*q+77*q^2+341*q^3+1175*q^4+3357*q^5+8283*q^6+18095*q^7+
35628*q^8+63999*q^9+105838*q^10+162185*q^11+231447*q^12+308754*q^13+386104*q^14
+453685*q^15+501832*q^16+523393*q^17+515646*q^18+480759*q^19+425180*q^20+357825
*q^21+287484*q^22+221264*q^23+163856*q^24+117097*q^25+80737*q^26+53712*q^27+
34599*q^28+21644*q^29+13080*q^30+7535*q^31+4089*q^32+2070*q^33+968*q^34+416*q^
35+161*q^36+56*q^37+17*q^38+4*q^39+q^40, 1+13*q+90*q^2+430*q^3+1595*q^4+4895*q^
5+12946*q^6+30279*q^7+63783*q^8+122575*q^9+216972*q^10+356263*q^11+545600*q^12+
782602*q^13+1054821*q^14+1339455*q^15+1605808*q^16+1820729*q^17+1955601*q^18+
1992804*q^19+1929892*q^20+1779595*q^21+1565979*q^22+1318564*q^23+1065528*q^24+
828615*q^25+621800*q^26+451563*q^27+317778*q^28+216554*q^29+142906*q^30+91516*q
^31+56946*q^32+34275*q^33+19777*q^34+10834*q^35+5586*q^36+2692*q^37+1200*q^38+
493*q^39+183*q^40+61*q^41+18*q^42+4*q^43+q^44, 1+14*q+104*q^2+533*q^3+2117*q^4+
6944*q^5+19591*q^6+48817*q^7+109464*q^8+223851*q^9+421687*q^10+737267*q^11+
1203366*q^12+1841984*q^13+2653682*q^14+3608615*q^15+4642714*q^16+5662211*q^17+
6556927*q^18+7220398*q^19+7571708*q^20+7572396*q^21+7234039*q^22+6613602*q^23+
5798234*q^24+4886010*q^25+3967400*q^26+3111540*q^27+2361718*q^28+1738248*q^29+
1242764*q^30+863474*q^31+582561*q^32+381622*q^33+243020*q^34+150467*q^35+90280*
q^36+52135*q^37+28766*q^38+15051*q^39+7414*q^40+3418*q^41+1460*q^42+576*q^43+
206*q^44+66*q^45+19*q^46+4*q^47+q^48]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+5*q^2+5*q+3), q^2*(q^8+4*q^7+9*q^6
+13*q^5+17*q^4+17*q^3+13*q^2+9*q+3), q^3*(q^12+5*q^11+14*q^10+26*q^9+41*q^8+53*
q^7+58*q^6+55*q^5+42*q^4+26*q^3+14*q^2+6*q+1), q^5*(q^16+6*q^15+20*q^14+45*q^13
+83*q^12+128*q^11+171*q^10+199*q^9+203*q^8+181*q^7+137*q^6+90*q^5+54*q^4+30*q^3
+13*q^2+4*q+1), q^8*(q^20+7*q^19+27*q^18+71*q^17+150*q^16+266*q^15+411*q^14+558
*q^13+677*q^12+734*q^11+706*q^10+612*q^9+472*q^8+321*q^7+204*q^6+125*q^5+70*q^4
+33*q^3+12*q^2+4*q+1), q^12*(1+35*q^22+q^24+105*q^21+8*q^23+4*q+13*q^2+36*q^3+
85*q^4+172*q^5+304*q^6+495*q^7+779*q^8+1179*q^9+1651*q^10+2119*q^11+2496*q^12+
2672*q^13+2597*q^14+2300*q^15+1843*q^16+1334*q^17+868*q^18+499*q^19+250*q^20),
q^17*(1+148*q^25+1672*q^22+392*q^24+2853*q^21+868*q^23+4*q+14*q^2+44*q^26+41*q^
3+101*q^4+223*q^5+434*q^6+762*q^7+1244*q^8+1959*q^9+2993*q^10+4330*q^11+5886*q^
12+7504*q^13+8891*q^14+9730*q^15+9824*q^16+q^28+9136*q^17+7808*q^18+6124*q^19+
4386*q^20+9*q^27), q^23*(1+5614*q^25+20458*q^22+9463*q^24+26566*q^21+14527*q^23
+4*q+54*q^30+201*q^29+15*q^2+3003*q^26+46*q^3+120*q^4+280*q^5+586*q^6+q^32+1114
*q^7+1945*q^8+3188*q^9+5042*q^10+7742*q^11+11397*q^12+15976*q^13+21316*q^14+
26939*q^15+31998*q^16+586*q^28+35549*q^17+36859*q^18+35583*q^19+31919*q^20+1424
*q^27+10*q^31), q^30*(1+68677*q^25+127511*q^22+90497*q^24+136938*q^21+111147*q^
23+4*q+5102*q^30+10349*q^29+65*q^34+265*q^33+16*q^2+48416*q^26+51*q^3+140*q^4+
345*q^5+763*q^6+843*q^32+1547*q^7+2887*q^8+5022*q^9+8275*q^10+13151*q^11+20303*
q^12+30236*q^13+43223*q^14+59305*q^15+77990*q^16+18928*q^28+97774*q^17+116189*q
^18+130431*q^19+137901*q^20+31569*q^27+2229*q^31+11*q^35+q^36), q^38*(1+453685*
q^25+515646*q^22+501832*q^24+480759*q^21+523393*q^23+4*q+105838*q^30+341*q^37+
162185*q^29+12*q^39+8283*q^34+77*q^38+18095*q^33+17*q^2+386104*q^26+56*q^3+161*
q^4+416*q^5+968*q^6+35628*q^32+2070*q^7+4089*q^8+7535*q^9+13080*q^10+21644*q^11
+34599*q^12+53712*q^13+80737*q^14+117097*q^15+163856*q^16+231447*q^28+221264*q^
17+287484*q^18+357825*q^19+425180*q^20+308754*q^27+q^40+63999*q^31+3357*q^35+
1175*q^36), q^47*(1+1992804*q^25+90*q^42+1565979*q^22+1929892*q^24+1318564*q^21
+1779595*q^23+4*q+q^44+1054821*q^30+30279*q^37+1339455*q^29+4895*q^39+430*q^41+
216972*q^34+12946*q^38+356263*q^33+18*q^2+1955601*q^26+61*q^3+183*q^4+493*q^5+
1200*q^6+545600*q^32+2692*q^7+5586*q^8+10834*q^9+19777*q^10+34275*q^11+56946*q^
12+91516*q^13+142906*q^14+216554*q^15+317778*q^16+13*q^43+1605808*q^28+451563*q
^17+621800*q^18+828615*q^19+1065528*q^20+1820729*q^27+1595*q^40+782602*q^31+
122575*q^35+63783*q^36), q^57*(1+6613602*q^25+19591*q^42+3967400*q^22+5798234*q
^24+3111540*q^21+4886010*q^23+4*q+2117*q^44+6556927*q^30+533*q^45+737267*q^37+
7220398*q^29+104*q^46+223851*q^39+48817*q^41+2653682*q^34+421687*q^38+3608615*q
^33+19*q^2+7234039*q^26+66*q^3+206*q^4+576*q^5+1460*q^6+4642714*q^32+3418*q^7+
7414*q^8+15051*q^9+28766*q^10+52135*q^11+90280*q^12+150467*q^13+243020*q^14+
381622*q^15+582561*q^16+6944*q^43+7571708*q^28+863474*q^17+1242764*q^18+1738248
*q^19+2361718*q^20+7572396*q^27+109464*q^40+5662211*q^31+1841984*q^35+q^48+
1203366*q^36+14*q^47)]


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

[1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406,

    22369622, 89478486]

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

     is given by

[1, 1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203,

    11184811, 44739243]

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

     is given by

[0, 1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203,

    11184811, 44739243]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203,

    11184811, 44739243]

 ODD:, [0, 1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203,

    11184811, 44739243]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.863636364, 4.406976744, 6.026315789,

    7.672767204, 9.328268034, 10.98667948, 12.64600261, 14.30560531,

    15.96529223, 17.62500416, 19.28472341, 20.94444478]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.391384351, 1.819646701, 2.231331362,

    2.609354573, 2.952275572, 3.264494182, 3.551303965, 3.817326767,

    4.066228574, 4.300833388, 4.523314351, 4.735363053]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.93595, 3.31111, 4.97884, 6.80873, 8.71593,

    10.6569, 12.6118, 14.5720, 16.5342, 18.4972, 20.4604, 22.4237]

Skewness: , [Float(undefined), 0., 0., -0.1606593136, -0.1051062402,

    0.0001440216319, 0.07966728601, 0.1307524791, 0.1618627460, 0.1801202723,

    0.1903080891, 0.1954833689, 0.1975527281, 0.1976879619, 0.1966229565]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.147201057,

    2.399454676, 2.627532375, 2.785921729, 2.889290626, 2.956251000,

    2.999292724, 3.026647655, 3.043714614, 3.054131144, 3.060235095,

    3.063607031]

                                end of this data



For the equivalence class of patterns,

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

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

                                  here it is:

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+5*q^4+2*q^5, 1+4*q+9*q^2+15*q^3+20*
q^4+20*q^5+14*q^6+6*q^7+q^8, 1+5*q+14*q^2+29*q^3+49*q^4+68*q^5+77*q^6+69*q^7+47
*q^8+25*q^9+10*q^10+2*q^11, 1+6*q+20*q^2+49*q^3+98*q^4+165*q^5+237*q^6+290*q^7+
299*q^8+259*q^9+190*q^10+120*q^11+66*q^12+28*q^13+8*q^14+q^15, 1+7*q+27*q^2+76*
q^3+174*q^4+338*q^5+569*q^6+838*q^7+1082*q^8+1225*q^9+1217*q^10+1069*q^11+846*q
^12+609*q^13+393*q^14+224*q^15+109*q^16+44*q^17+14*q^18+2*q^19, 1+8*q+35*q^2+
111*q^3+285*q^4+622*q^5+1184*q^6+1994*q^7+2994*q^8+4026*q^9+4861*q^10+5286*q^11
+5216*q^12+4725*q^13+3971*q^14+3109*q^15+2258*q^16+1510*q^17+927*q^18+518*q^19+
264*q^20+116*q^21+42*q^22+10*q^23+q^24, 1+9*q+44*q^2+155*q^3+440*q^4+1061*q^5+
2237*q^6+4195*q^7+7071*q^8+10787*q^9+14963*q^10+18940*q^11+21968*q^12+23490*q^
13+23349*q^14+21773*q^15+19168*q^16+15961*q^17+12572*q^18+9357*q^19+6577*q^20+
4366*q^21+2725*q^22+1595*q^23+863*q^24+419*q^25+180*q^26+66*q^27+18*q^28+2*q^29
, 1+10*q+54*q^2+209*q^3+649*q^4+1709*q^5+3937*q^6+8087*q^7+14995*q^8+25309*q^9+
39116*q^10+55606*q^11+72992*q^12+88843*q^13+100784*q^14+107252*q^15+107839*q^16
+103079*q^17+94052*q^18+82095*q^19+68642*q^20+55035*q^21+42329*q^22+31240*q^23+
22131*q^24+15021*q^25+9728*q^26+5982*q^27+3476*q^28+1885*q^29+952*q^30+436*q^31
+178*q^32+58*q^33+12*q^34+q^35, 1+11*q+65*q^2+274*q^3+923*q^4+2631*q^5+6558*q^6
+14590*q^7+29367*q^8+53985*q^9+91255*q^10+142575*q^11+206753*q^12+279340*q^13+
353007*q^14+419125*q^15+469990*q^16+500604*q^17+509235*q^18+496889*q^19+466533*
q^20+422466*q^21+369610*q^22+312841*q^23+256452*q^24+203740*q^25+156939*q^26+
117196*q^27+84757*q^28+59277*q^29+40001*q^30+25987*q^31+16212*q^32+9677*q^33+
5499*q^34+2950*q^35+1469*q^36+668*q^37+272*q^38+92*q^39+22*q^40+2*q^41, 1+12*q+
77*q^2+351*q^3+1274*q^4+3904*q^5+10451*q^6+24975*q^7+54058*q^8+107068*q^9+
195503*q^10+330982*q^11+521882*q^12+769349*q^13+1064134*q^14+1385960*q^15+
1706415*q^16+1994740*q^17+2224259*q^18+2376774*q^19+2443864*q^20+2426122*q^21+
2331605*q^22+2173918*q^23+1969951*q^24+1737450*q^25+1493163*q^26+1251509*q^27+
1023673*q^28+817390*q^29+637151*q^30+484650*q^31+359485*q^32+259831*q^33+182837
*q^34+125121*q^35+83138*q^36+53508*q^37+33283*q^38+19922*q^39+11414*q^40+6202*q
^41+3183*q^42+1520*q^43+668*q^44+256*q^45+76*q^46+14*q^47+q^48, 1+13*q+90*q^2+
441*q^3+1715*q^4+5618*q^5+16057*q^6+40954*q^7+94650*q^8+200381*q^9+391728*q^10+
711469*q^11+1206324*q^12+1917068*q^13+2865413*q^14+4041346*q^15+5395973*q^16+
6844021*q^17+8276704*q^18+9580911*q^19+10657740*q^20+11434843*q^21+11871486*q^
22+11958550*q^23+11715391*q^24+11183905*q^25+10420966*q^26+9490780*q^27+8458014
*q^28+7382583*q^29+6315909*q^30+5298795*q^31+4360874*q^32+3520997*q^33+2788844*
q^34+2166587*q^35+1650467*q^36+1232375*q^37+901455*q^38+645565*q^39+452216*q^40
+309447*q^41+206522*q^42+134177*q^43+84665*q^44+51758*q^45+30533*q^46+17305*q^
47+9371*q^48+4801*q^49+2298*q^50+1008*q^51+390*q^52+122*q^53+26*q^54+2*q^55, 1+
14*q+104*q^2+545*q^3+2260*q^4+7877*q^5+23921*q^6+64784*q^7+158981*q^8+357572*q^
9+743355*q^10+1437650*q^11+2599852*q^12+4414566*q^13+7063238*q^14+10682245*q^15
+15316225*q^16+20880952*q^17+27150611*q^18+33777628*q^19+40341069*q^20+46408412
*q^21+51592327*q^22+55590971*q^23+58210294*q^24+59370892*q^25+59101586*q^26+
57522233*q^27+54819747*q^28+51222459*q^29+46976179*q^30+42323509*q^31+37488302*
q^32+32664724*q^33+28010838*q^34+23646789*q^35+19656328*q^36+16090065*q^37+
12970148*q^38+10295328*q^39+8046079*q^40+6189977*q^41+4686122*q^42+3489565*q^43
+2554600*q^44+1837301*q^45+1297109*q^46+898009*q^47+608997*q^48+404011*q^49+
261788*q^50+165342*q^51+101566*q^52+60469*q^53+34742*q^54+19134*q^55+10056*q^56
+4991*q^57+2318*q^58+978*q^59+352*q^60+96*q^61+16*q^62+q^63]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+6*q^2+5*q+2), q^2*(q^8+4*q^7+9*q^6
+15*q^5+20*q^4+20*q^3+14*q^2+6*q+1), q^4*(q^11+5*q^10+14*q^9+29*q^8+49*q^7+68*q
^6+77*q^5+69*q^4+47*q^3+25*q^2+10*q+2), q^6*(q^15+6*q^14+20*q^13+49*q^12+98*q^
11+165*q^10+237*q^9+290*q^8+299*q^7+259*q^6+190*q^5+120*q^4+66*q^3+28*q^2+8*q+1
), q^9*(q^19+7*q^18+27*q^17+76*q^16+174*q^15+338*q^14+569*q^13+838*q^12+1082*q^
11+1225*q^10+1217*q^9+1069*q^8+846*q^7+609*q^6+393*q^5+224*q^4+109*q^3+44*q^2+
14*q+2), q^12*(1+35*q^22+q^24+111*q^21+8*q^23+10*q+42*q^2+116*q^3+264*q^4+518*q
^5+927*q^6+1510*q^7+2258*q^8+3109*q^9+3971*q^10+4725*q^11+5216*q^12+5286*q^13+
4861*q^14+4026*q^15+2994*q^16+1994*q^17+1184*q^18+622*q^19+285*q^20), q^16*(2+
440*q^25+4195*q^22+1061*q^24+7071*q^21+2237*q^23+18*q+q^29+66*q^2+155*q^26+180*
q^3+419*q^4+863*q^5+1595*q^6+2725*q^7+4366*q^8+6577*q^9+9357*q^10+12572*q^11+
15961*q^12+19168*q^13+21773*q^14+23349*q^15+23490*q^16+9*q^28+21968*q^17+18940*
q^18+14963*q^19+10787*q^20+44*q^27), q^20*(1+39116*q^25+88843*q^22+55606*q^24+
100784*q^21+72992*q^23+12*q+1709*q^30+3937*q^29+10*q^34+54*q^33+58*q^2+25309*q^
26+178*q^3+436*q^4+952*q^5+1885*q^6+209*q^32+3476*q^7+5982*q^8+9728*q^9+15021*q
^10+22131*q^11+31240*q^12+42329*q^13+55035*q^14+68642*q^15+82095*q^16+8087*q^28
+94052*q^17+103079*q^18+107839*q^19+107252*q^20+14995*q^27+649*q^31+q^35), q^25
*(2+469990*q^25+496889*q^22+500604*q^24+466533*q^21+509235*q^23+22*q+142575*q^
30+923*q^37+206753*q^29+65*q^39+q^41+14590*q^34+274*q^38+29367*q^33+92*q^2+
419125*q^26+272*q^3+668*q^4+1469*q^5+2950*q^6+53985*q^32+5499*q^7+9677*q^8+
16212*q^9+25987*q^10+40001*q^11+59277*q^12+84757*q^13+117196*q^14+156939*q^15+
203740*q^16+279340*q^28+256452*q^17+312841*q^18+369610*q^19+422466*q^20+353007*
q^27+11*q^40+91255*q^31+6558*q^35+2631*q^36), q^30*(1+2173918*q^25+10451*q^42+
1493163*q^22+1969951*q^24+1251509*q^21+1737450*q^23+14*q+1274*q^44+2224259*q^30
+351*q^45+330982*q^37+2376774*q^29+77*q^46+107068*q^39+24975*q^41+1064134*q^34+
195503*q^38+1385960*q^33+76*q^2+2331605*q^26+256*q^3+668*q^4+1520*q^5+3183*q^6+
1706415*q^32+6202*q^7+11414*q^8+19922*q^9+33283*q^10+53508*q^11+83138*q^12+
125121*q^13+182837*q^14+259831*q^15+359485*q^16+3904*q^43+2443864*q^28+484650*q
^17+637151*q^18+817390*q^19+1023673*q^20+2426122*q^27+54058*q^40+1994740*q^31+
769349*q^35+q^48+521882*q^36+12*q^47), q^36*(2+6315909*q^25+1917068*q^42+
3520997*q^22+5298795*q^24+2788844*q^21+4360874*q^23+26*q+1715*q^51+711469*q^44+
90*q^53+11183905*q^30+441*q^52+q^55+13*q^54+391728*q^45+8276704*q^37+10420966*q
^29+200381*q^46+5395973*q^39+2865413*q^41+11434843*q^34+5618*q^50+6844021*q^38+
11871486*q^33+122*q^2+7382583*q^26+390*q^3+1008*q^4+2298*q^5+4801*q^6+11958550*
q^32+9371*q^7+17305*q^8+30533*q^9+51758*q^10+84665*q^11+134177*q^12+206522*q^13
+309447*q^14+452216*q^15+645565*q^16+1206324*q^43+16057*q^49+9490780*q^28+
901455*q^17+1232375*q^18+1650467*q^19+2166587*q^20+8458014*q^27+4041346*q^40+
11715391*q^31+10657740*q^35+40954*q^48+9580911*q^36+94650*q^47), q^42*(1+
12970148*q^25+46408412*q^42+6189977*q^22+10295328*q^24+4686122*q^21+8046079*q^
23+16*q+2599852*q^51+33777628*q^44+743355*q^53+23921*q^57+32664724*q^30+1437650
*q^52+7877*q^58+158981*q^55+545*q^60+357572*q^54+27150611*q^45+59101586*q^37+
28010838*q^29+20880952*q^46+58210294*q^39+104*q^61+51592327*q^41+51222459*q^34+
4414566*q^50+59370892*q^38+46976179*q^33+64784*q^56+96*q^2+16090065*q^26+352*q^
3+978*q^4+2318*q^5+4991*q^6+42323509*q^32+10056*q^7+19134*q^8+34742*q^9+14*q^62
+60469*q^10+101566*q^11+165342*q^12+261788*q^13+404011*q^14+608997*q^15+898009*
q^16+40341069*q^43+7063238*q^49+23646789*q^28+1297109*q^17+1837301*q^18+2554600
*q^19+3489565*q^20+19656328*q^27+55590971*q^40+q^63+37488302*q^31+54819747*q^35
+10682245*q^48+2260*q^59+57522233*q^36+15316225*q^47)]


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

[1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074,

    176351644, 959658200]

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

     is given by

[1, 1, 3, 11, 45, 198, 919, 4432, 22039, 112176, 581867, 3064920, 16351546,

    88175822, 479829111]

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

     is given by

[0, 1, 3, 11, 45, 198, 918, 4432, 22035, 112176, 581857, 3064920, 16351528,

    88175822, 479829089]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 198, 918, 4432, 22039, 112176, 581857, 3064920,

    16351546, 88175822, 479829089]

 ODD:, [0, 1, 3, 11, 45, 198, 919, 4432, 22035, 112176, 581867, 3064920,

    16351528, 88175822, 479829111]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.772727273, 4.233333333, 5.868686869,

    7.669025585, 9.620712996, 11.71055498, 13.92756918, 16.26277107,

    18.70857445, 21.25835412, 23.90620141, 26.64679275]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.311897245, 1.646882577, 2.007040459,

    2.400413632, 2.824532266, 3.276099124, 3.752571860, 4.252047566,

    4.773029191, 5.314259465, 5.874624192, 6.453108156]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.72107, 2.71222, 4.02821, 5.76199, 7.97798,

    10.7328, 14.0818, 18.0799, 22.7818, 28.2414, 34.5112, 41.6426]

Skewness: , [Float(undefined), 0., 0., -0.1796919473, -0.1816260113,

    -0.1064222035, -0.02115512380, 0.05452635204, 0.1181826521, 0.1711750719,

    0.2154388334, 0.2527723056, 0.2846639085, 0.3122738751, 0.3364788312]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.318571912,

    2.587608027, 2.726229684, 2.788597154, 2.827214270, 2.860703446,

    2.892962254, 2.924271472, 2.954333799, 2.982973396, 3.010303097,

    3.036313576]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+5*q^4+2*q^5, 1+4*q+9*q^2+15*q^3+20*
q^4+20*q^5+14*q^6+6*q^7+q^8, 1+5*q+14*q^2+29*q^3+49*q^4+68*q^5+77*q^6+69*q^7+47
*q^8+24*q^9+9*q^10+2*q^11, 1+6*q+20*q^2+49*q^3+98*q^4+165*q^5+237*q^6+290*q^7+
299*q^8+257*q^9+184*q^10+111*q^11+57*q^12+24*q^13+7*q^14+q^15, 1+7*q+27*q^2+76*
q^3+174*q^4+338*q^5+569*q^6+838*q^7+1082*q^8+1222*q^9+1204*q^10+1037*q^11+790*q
^12+541*q^13+334*q^14+183*q^15+87*q^16+35*q^17+11*q^18+2*q^19, 1+8*q+35*q^2+111
*q^3+285*q^4+622*q^5+1184*q^6+1994*q^7+2994*q^8+4022*q^9+4839*q^10+5217*q^11+
5057*q^12+4445*q^13+3586*q^14+2683*q^15+1868*q^16+1206*q^17+718*q^18+392*q^19+
194*q^20+85*q^21+31*q^22+8*q^23+q^24, 1+9*q+44*q^2+155*q^3+440*q^4+1061*q^5+
2237*q^6+4195*q^7+7071*q^8+10782*q^9+14930*q^10+18817*q^11+21631*q^12+22760*q^
13+22058*q^14+19869*q^15+16795*q^16+13417*q^17+10163*q^18+7305*q^19+4981*q^20+
3218*q^21+1964*q^22+1125*q^23+597*q^24+288*q^25+124*q^26+46*q^27+13*q^28+2*q^29
, 1+10*q+54*q^2+209*q^3+649*q^4+1709*q^5+3937*q^6+8087*q^7+14995*q^8+25303*q^9+
39070*q^10+55409*q^11+72375*q^12+87300*q^13+97574*q^14+101558*q^15+99108*q^16+
91388*q^17+80196*q^18+67328*q^19+54259*q^20+42059*q^21+31393*q^22+22568*q^23+
15615*q^24+10378*q^25+6601*q^26+3998*q^27+2291*q^28+1232*q^29+616*q^30+283*q^31
+116*q^32+39*q^33+9*q^34+q^35, 1+11*q+65*q^2+274*q^3+923*q^4+2631*q^5+6558*q^6+
14590*q^7+29367*q^8+53978*q^9+91194*q^10+142281*q^11+205723*q^12+276452*q^13+
346220*q^14+405382*q^15+445607*q^16+462291*q^17+455437*q^18+428741*q^19+387776*
q^20+338401*q^21+285830*q^22+234198*q^23+186424*q^24+144282*q^25+108589*q^26+
79435*q^27+56415*q^28+38831*q^29+25845*q^30+16590*q^31+10242*q^32+6060*q^33+
3417*q^34+1820*q^35+905*q^36+414*q^37+170*q^38+59*q^39+15*q^40+2*q^41, 1+12*q+
77*q^2+351*q^3+1274*q^4+3904*q^5+10451*q^6+24975*q^7+54058*q^8+107060*q^9+
195425*q^10+330565*q^11+520271*q^12+764371*q^13+1051203*q^14+1356874*q^15+
1648691*q^16+1892406*q^17+2060731*q^18+2139465*q^19+2128782*q^20+2040127*q^21+
1891296*q^22+1702053*q^23+1491121*q^24+1274411*q^25+1064227*q^26+869211*q^27+
694730*q^28+543454*q^29+415972*q^30+311368*q^31+227735*q^32+162594*q^33+113193*
q^34+76732*q^35+50558*q^36+32307*q^37+19966*q^38+11886*q^39+6777*q^40+3675*q^41
+1882*q^42+903*q^43+399*q^44+155*q^45+48*q^46+10*q^47+q^48, 1+13*q+90*q^2+441*q
^3+1715*q^4+5618*q^5+16057*q^6+40954*q^7+94650*q^8+200372*q^9+391631*q^10+
710900*q^11+1203925*q^12+1908993*q^13+2842545*q^14+3985136*q^15+5273620*q^16+
6604953*q^17+7853287*q^18+8896048*q^19+9639656*q^20+10035293*q^21+10080568*q^22
+9810124*q^23+9281792*q^24+8563522*q^25+7723267*q^26+6822214*q^27+5911136*q^28+
5029235*q^29+4204609*q^30+3455555*q^31+2792179*q^32+2218058*q^33+1731829*q^34+
1328578*q^35+1000973*q^36+740224*q^37+536920*q^38+381684*q^39+265640*q^40+
180751*q^41+120037*q^42+77649*q^43+48818*q^44+29748*q^45+17506*q^46+9903*q^47+
5355*q^48+2747*q^49+1321*q^50+584*q^51+229*q^52+74*q^53+17*q^54+2*q^55, 1+14*q+
104*q^2+545*q^3+2260*q^4+7877*q^5+23921*q^6+64784*q^7+158981*q^8+357562*q^9+
743237*q^10+1436897*q^11+2596415*q^12+4402071*q^13+7025032*q^14+10580750*q^15+
15076991*q^16+20373339*q^17+26170922*q^18+32044223*q^19+37511646*q^20+42124369*
q^21+45544049*q^22+47585952*q^23+48221535*q^24+47551115*q^25+45763205*q^26+
43093419*q^27+39789851*q^28+36087793*q^29+32194168*q^30+28280228*q^31+24480130*
q^32+20893036*q^33+17586988*q^34+14603522*q^35+11962398*q^36+9666028*q^37+
7703462*q^38+6053971*q^39+4690208*q^40+3580892*q^41+2693065*q^42+1993979*q^43+
1452538*q^44+1040241*q^45+731722*q^46+505009*q^47+341553*q^48+226058*q^49+
146181*q^50+92179*q^51+56542*q^52+33626*q^53+19303*q^54+10638*q^55+5595*q^56+
2789*q^57+1302*q^58+554*q^59+203*q^60+58*q^61+11*q^62+q^63]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+3*q^4+5*q^3+6*q^2+5*q+2), q^2*(q^8+4*q^7+9*q^6
+15*q^5+20*q^4+20*q^3+14*q^2+6*q+1), q^4*(q^11+5*q^10+14*q^9+29*q^8+49*q^7+68*q
^6+77*q^5+69*q^4+47*q^3+24*q^2+9*q+2), q^6*(q^15+6*q^14+20*q^13+49*q^12+98*q^11
+165*q^10+237*q^9+290*q^8+299*q^7+257*q^6+184*q^5+111*q^4+57*q^3+24*q^2+7*q+1),
q^9*(q^19+7*q^18+27*q^17+76*q^16+174*q^15+338*q^14+569*q^13+838*q^12+1082*q^11+
1222*q^10+1204*q^9+1037*q^8+790*q^7+541*q^6+334*q^5+183*q^4+87*q^3+35*q^2+11*q+
2), q^12*(1+35*q^22+q^24+111*q^21+8*q^23+8*q+31*q^2+85*q^3+194*q^4+392*q^5+718*
q^6+1206*q^7+1868*q^8+2683*q^9+3586*q^10+4445*q^11+5057*q^12+5217*q^13+4839*q^
14+4022*q^15+2994*q^16+1994*q^17+1184*q^18+622*q^19+285*q^20), q^16*(2+440*q^25
+4195*q^22+1061*q^24+7071*q^21+2237*q^23+13*q+q^29+46*q^2+155*q^26+124*q^3+288*
q^4+597*q^5+1125*q^6+1964*q^7+3218*q^8+4981*q^9+7305*q^10+10163*q^11+13417*q^12
+16795*q^13+19869*q^14+22058*q^15+22760*q^16+9*q^28+21631*q^17+18817*q^18+14930
*q^19+10782*q^20+44*q^27), q^20*(1+39070*q^25+87300*q^22+55409*q^24+97574*q^21+
72375*q^23+9*q+1709*q^30+3937*q^29+10*q^34+54*q^33+39*q^2+25303*q^26+116*q^3+
283*q^4+616*q^5+1232*q^6+209*q^32+2291*q^7+3998*q^8+6601*q^9+10378*q^10+15615*q
^11+22568*q^12+31393*q^13+42059*q^14+54259*q^15+67328*q^16+8087*q^28+80196*q^17
+91388*q^18+99108*q^19+101558*q^20+14995*q^27+649*q^31+q^35), q^25*(2+445607*q^
25+428741*q^22+462291*q^24+387776*q^21+455437*q^23+15*q+142281*q^30+923*q^37+
205723*q^29+65*q^39+q^41+14590*q^34+274*q^38+29367*q^33+59*q^2+405382*q^26+170*
q^3+414*q^4+905*q^5+1820*q^6+53978*q^32+3417*q^7+6060*q^8+10242*q^9+16590*q^10+
25845*q^11+38831*q^12+56415*q^13+79435*q^14+108589*q^15+144282*q^16+276452*q^28
+186424*q^17+234198*q^18+285830*q^19+338401*q^20+346220*q^27+11*q^40+91194*q^31
+6558*q^35+2631*q^36), q^30*(1+1702053*q^25+10451*q^42+1064227*q^22+1491121*q^
24+869211*q^21+1274411*q^23+10*q+1274*q^44+2060731*q^30+351*q^45+330565*q^37+
2139465*q^29+77*q^46+107060*q^39+24975*q^41+1051203*q^34+195425*q^38+1356874*q^
33+48*q^2+1891296*q^26+155*q^3+399*q^4+903*q^5+1882*q^6+1648691*q^32+3675*q^7+
6777*q^8+11886*q^9+19966*q^10+32307*q^11+50558*q^12+76732*q^13+113193*q^14+
162594*q^15+227735*q^16+3904*q^43+2128782*q^28+311368*q^17+415972*q^18+543454*q
^19+694730*q^20+2040127*q^27+54058*q^40+1892406*q^31+764371*q^35+q^48+520271*q^
36+12*q^47), q^36*(2+4204609*q^25+1908993*q^42+2218058*q^22+3455555*q^24+
1731829*q^21+2792179*q^23+17*q+1715*q^51+710900*q^44+90*q^53+8563522*q^30+441*q
^52+q^55+13*q^54+391631*q^45+7853287*q^37+7723267*q^29+200372*q^46+5273620*q^39
+2842545*q^41+10035293*q^34+5618*q^50+6604953*q^38+10080568*q^33+74*q^2+5029235
*q^26+229*q^3+584*q^4+1321*q^5+2747*q^6+9810124*q^32+5355*q^7+9903*q^8+17506*q^
9+29748*q^10+48818*q^11+77649*q^12+120037*q^13+180751*q^14+265640*q^15+381684*q
^16+1203925*q^43+16057*q^49+6822214*q^28+536920*q^17+740224*q^18+1000973*q^19+
1328578*q^20+5911136*q^27+3985136*q^40+9281792*q^31+9639656*q^35+40954*q^48+
8896048*q^36+94650*q^47), q^42*(1+7703462*q^25+42124369*q^42+3580892*q^22+
6053971*q^24+2693065*q^21+4690208*q^23+11*q+2596415*q^51+32044223*q^44+743237*q
^53+23921*q^57+20893036*q^30+1436897*q^52+7877*q^58+158981*q^55+545*q^60+357562
*q^54+26170922*q^45+45763205*q^37+17586988*q^29+20373339*q^46+48221535*q^39+104
*q^61+45544049*q^41+36087793*q^34+4402071*q^50+47551115*q^38+32194168*q^33+
64784*q^56+58*q^2+9666028*q^26+203*q^3+554*q^4+1302*q^5+2789*q^6+28280228*q^32+
5595*q^7+10638*q^8+19303*q^9+14*q^62+33626*q^10+56542*q^11+92179*q^12+146181*q^
13+226058*q^14+341553*q^15+505009*q^16+37511646*q^43+7025032*q^49+14603522*q^28
+731722*q^17+1040241*q^18+1452538*q^19+1993979*q^20+11962398*q^27+47585952*q^40
+q^63+24480130*q^31+39789851*q^35+10580750*q^48+2260*q^59+43093419*q^36+
15076991*q^47)]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.772727273, 4.233333333, 5.850253807,

    7.606312292, 9.489717224, 11.49148752, 13.60440664, 15.82248067,

    18.14061483, 20.55440275, 23.05998101, 25.65392430]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.311897245, 1.646882577, 1.995022342,

    2.367073886, 2.766494757, 3.194053648, 3.649422541, 4.131819705,

    4.640296452, 5.173870324, 5.731586541, 6.312544591]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.72107, 2.71222, 3.98011, 5.60304, 7.65349,

    10.2020, 13.3183, 17.0719, 21.5324, 26.7689, 32.8511, 39.8482]

Skewness: , [Float(undefined), 0., 0., -0.1796919473, -0.1816260113,

    -0.1135458565, -0.02661577529, 0.05937656832, 0.1373237488, 0.2053261652,

    0.2636141537, 0.3132174948, 0.3553642719, 0.3912380045, 0.4218793528]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.318571912,

    2.587608027, 2.739892990, 2.828697779, 2.890949579, 2.941800872,

    2.986726897, 3.027561151, 3.064849830, 3.098926249, 3.129907678,

    3.158074302]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, 1+4*q+7*q^2+9*q^3+
12*q^4+13*q^5+14*q^6+13*q^7+9*q^8+4*q^9+q^10, 1+5*q+11*q^2+16*q^3+23*q^4+26*q^5
+33*q^6+36*q^7+38*q^8+41*q^9+40*q^10+36*q^11+26*q^12+14*q^13+5*q^14+q^15, 1+6*q
+16*q^2+27*q^3+41*q^4+52*q^5+66*q^6+77*q^7+89*q^8+102*q^9+108*q^10+114*q^11+119
*q^12+124*q^13+124*q^14+118*q^15+102*q^16+76*q^17+45*q^18+20*q^19+6*q^20+q^21,
1+7*q+22*q^2+43*q^3+70*q^4+98*q^5+126*q^6+158*q^7+191*q^8+226*q^9+249*q^10+281*
q^11+305*q^12+334*q^13+347*q^14+363*q^15+376*q^16+383*q^17+392*q^18+392*q^19+
381*q^20+349*q^21+297*q^22+223*q^23+141*q^24+71*q^25+27*q^26+7*q^27+q^28, 1+8*q
+29*q^2+65*q^3+115*q^4+175*q^5+237*q^6+307*q^7+381*q^8+467*q^9+552*q^10+641*q^
11+721*q^12+802*q^13+866*q^14+946*q^15+1018*q^16+1073*q^17+1123*q^18+1166*q^19+
1202*q^20+1231*q^21+1253*q^22+1271*q^23+1282*q^24+1272*q^25+1240*q^26+1168*q^27
+1047*q^28+875*q^29+662*q^30+435*q^31+239*q^32+105*q^33+35*q^34+8*q^35+q^36, 1+
9*q+37*q^2+94*q^3+182*q^4+299*q^5+432*q^6+580*q^7+747*q^8+928*q^9+1130*q^10+
1346*q^11+1585*q^12+1829*q^13+2062*q^14+2287*q^15+2513*q^16+2717*q^17+2939*q^18
+3146*q^19+3347*q^20+3506*q^21+3653*q^22+3780*q^23+3922*q^24+4017*q^25+4094*q^
26+4157*q^27+4217*q^28+4261*q^29+4283*q^30+4283*q^31+4228*q^32+4113*q^33+3907*q
^34+3598*q^35+3161*q^36+2611*q^37+1979*q^38+1337*q^39+779*q^40+379*q^41+148*q^
42+44*q^43+9*q^44+q^45, 1+10*q+46*q^2+131*q^3+278*q^4+492*q^5+760*q^6+1068*q^7+
1426*q^8+1816*q^9+2264*q^10+2742*q^11+3286*q^12+3886*q^13+4520*q^14+5170*q^15+
5842*q^16+6502*q^17+7179*q^18+7850*q^19+8533*q^20+9159*q^21+9796*q^22+10406*q^
23+11000*q^24+11525*q^25+12013*q^26+12448*q^27+12873*q^28+13247*q^29+13571*q^30
+13841*q^31+14056*q^32+14248*q^33+14411*q^34+14515*q^35+14600*q^36+14654*q^37+
14640*q^38+14532*q^39+14286*q^40+13863*q^41+13215*q^42+12310*q^43+11110*q^44+
9610*q^45+7856*q^46+5962*q^47+4103*q^48+2496*q^49+1306*q^50+571*q^51+201*q^52+
54*q^53+10*q^54+q^55, 1+11*q+56*q^2+177*q^3+411*q^4+783*q^5+1292*q^6+1913*q^7+
2653*q^8+3490*q^9+4444*q^10+5497*q^11+6682*q^12+8034*q^13+9498*q^14+11103*q^15+
12806*q^16+14609*q^17+16448*q^18+18371*q^19+20338*q^20+22334*q^21+24364*q^22+
26447*q^23+28533*q^24+30558*q^25+32465*q^26+34327*q^27+36128*q^28+37920*q^29+
39635*q^30+41256*q^31+42672*q^32+44006*q^33+45195*q^34+46273*q^35+47227*q^36+
48105*q^37+48868*q^38+49549*q^39+50034*q^40+50388*q^41+50637*q^42+50855*q^43+
51007*q^44+51035*q^45+50895*q^46+50550*q^47+49918*q^48+48860*q^49+47319*q^50+
45162*q^51+42331*q^52+38748*q^53+34411*q^54+29364*q^55+23808*q^56+18069*q^57+
12605*q^58+7914*q^59+4374*q^60+2078*q^61+826*q^62+265*q^63+65*q^64+11*q^65+q^66
, 1+12*q+67*q^2+233*q^3+590*q^4+1209*q^5+2128*q^6+3330*q^7+4814*q^8+6562*q^9+
8571*q^10+10850*q^11+13392*q^12+16310*q^13+19556*q^14+23221*q^15+27205*q^16+
31581*q^17+36186*q^18+41114*q^19+46272*q^20+51636*q^21+57185*q^22+62968*q^23+
68996*q^24+75237*q^25+81530*q^26+87940*q^27+94306*q^28+100671*q^29+106846*q^30+
112956*q^31+118852*q^32+124708*q^33+130433*q^34+136020*q^35+141289*q^36+146255*
q^37+150829*q^38+155141*q^39+159063*q^40+162684*q^41+165913*q^42+168812*q^43+
171362*q^44+173666*q^45+175663*q^46+177334*q^47+178643*q^48+179611*q^49+180245*
q^50+180582*q^51+180643*q^52+180519*q^53+180173*q^54+179475*q^55+178162*q^56+
176070*q^57+173029*q^58+168787*q^59+163093*q^60+155716*q^61+146472*q^62+135225*
q^63+121966*q^64+106818*q^65+90132*q^66+72557*q^67+55054*q^68+38789*q^69+24947*
q^70+14376*q^71+7279*q^72+3169*q^73+1156*q^74+341*q^75+77*q^76+12*q^77+q^78, 1+
13*q+79*q^2+300*q^3+825*q^4+1816*q^5+3405*q^6+5636*q^7+8521*q^8+12059*q^9+16226
*q^10+21046*q^11+26496*q^12+32724*q^13+39748*q^14+47690*q^15+56586*q^16+66535*q
^17+77431*q^18+89280*q^19+102007*q^20+115462*q^21+129692*q^22+144581*q^23+
160309*q^24+176769*q^25+194009*q^26+211904*q^27+230520*q^28+249653*q^29+269232*
q^30+289111*q^31+309170*q^32+329205*q^33+349188*q^34+369027*q^35+388731*q^36+
408341*q^37+427822*q^38+446957*q^39+465637*q^40+483468*q^41+500405*q^42+516394*
q^43+531736*q^44+546448*q^45+560674*q^46+573925*q^47+585978*q^48+596588*q^49+
605982*q^50+614250*q^51+621761*q^52+628474*q^53+634483*q^54+639599*q^55+643795*
q^56+646956*q^57+649108*q^58+650099*q^59+650181*q^60+649514*q^61+648259*q^62+
646339*q^63+643607*q^64+639747*q^65+634459*q^66+627224*q^67+617366*q^68+604361*
q^69+587632*q^70+566682*q^71+540879*q^72+509856*q^73+473282*q^74+431197*q^75+
383917*q^76+332286*q^77+277675*q^78+222181*q^79+168489*q^80+119617*q^81+78377*q
^82+46667*q^83+24835*q^84+11605*q^85+4666*q^86+1574*q^87+430*q^88+90*q^89+13*q^
90+q^91, 1+14*q+92*q^2+379*q^3+1127*q^4+2660*q^5+5306*q^6+9287*q^7+14716*q^8+
21660*q^9+30114*q^10+40106*q^11+51642*q^12+64845*q^13+79917*q^14+96970*q^15+
116276*q^16+137979*q^17+162325*q^18+189314*q^19+219128*q^20+251368*q^21+286138*
q^22+323033*q^23+362409*q^24+403966*q^25+447926*q^26+494087*q^27+542761*q^28+
593728*q^29+647194*q^30+702905*q^31+760648*q^32+819895*q^33+880531*q^34+942195*
q^35+1004951*q^36+1068538*q^37+1133143*q^38+1198180*q^39+1263735*q^40+1329022*q
^41+1394031*q^42+1458144*q^43+1521655*q^44+1584291*q^45+1646055*q^46+1706094*q^
47+1764389*q^48+1820417*q^49+1874375*q^50+1925915*q^51+1975355*q^52+2022293*q^
53+2066845*q^54+2108376*q^55+2146823*q^56+2181945*q^57+2213945*q^58+2242669*q^
59+2268295*q^60+2290373*q^61+2309218*q^62+2325178*q^63+2338970*q^64+2350615*q^
65+2360155*q^66+2366951*q^67+2370625*q^68+2370945*q^69+2368101*q^70+2362455*q^
71+2354535*q^72+2344631*q^73+2332990*q^74+2319093*q^75+2302006*q^76+2280428*q^
77+2253145*q^78+2218850*q^79+2176190*q^80+2123469*q^81+2059146*q^82+1981947*q^
83+1890666*q^84+1784329*q^85+1662394*q^86+1524983*q^87+1373075*q^88+1208791*q^
89+1035532*q^90+858171*q^91+683050*q^92+517642*q^93+369664*q^94+245818*q^95+
150220*q^96+83184*q^97+41118*q^98+17846*q^99+6670*q^100+2094*q^101+533*q^102+
104*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+5*q^3+5*q^2+3*q+1, q^10+4*q^9+7*q^8+9
*q^7+12*q^6+13*q^5+14*q^4+13*q^3+9*q^2+4*q+1, q^15+5*q^14+11*q^13+16*q^12+23*q^
11+26*q^10+33*q^9+36*q^8+38*q^7+41*q^6+40*q^5+36*q^4+26*q^3+14*q^2+5*q+1, q^21+
6*q^20+16*q^19+27*q^18+41*q^17+52*q^16+66*q^15+77*q^14+89*q^13+102*q^12+108*q^
11+114*q^10+119*q^9+124*q^8+124*q^7+118*q^6+102*q^5+76*q^4+45*q^3+20*q^2+6*q+1,
1+43*q^25+126*q^22+70*q^24+158*q^21+98*q^23+7*q+27*q^2+22*q^26+71*q^3+141*q^4+
223*q^5+297*q^6+349*q^7+381*q^8+392*q^9+392*q^10+383*q^11+376*q^12+363*q^13+347
*q^14+334*q^15+305*q^16+q^28+281*q^17+249*q^18+226*q^19+191*q^20+7*q^27, 1+641*
q^25+866*q^22+721*q^24+946*q^21+802*q^23+8*q+237*q^30+307*q^29+29*q^34+65*q^33+
35*q^2+552*q^26+105*q^3+239*q^4+435*q^5+662*q^6+115*q^32+875*q^7+1047*q^8+1168*
q^9+1240*q^10+1272*q^11+1282*q^12+1271*q^13+1253*q^14+1231*q^15+1202*q^16+381*q
^28+1166*q^17+1123*q^18+1073*q^19+1018*q^20+467*q^27+175*q^31+8*q^35+q^36, 1+
3347*q^25+94*q^42+3780*q^22+3506*q^24+3922*q^21+3653*q^23+9*q+9*q^44+2287*q^30+
q^45+747*q^37+2513*q^29+432*q^39+182*q^41+1346*q^34+580*q^38+1585*q^33+44*q^2+
3146*q^26+148*q^3+379*q^4+779*q^5+1337*q^6+1829*q^32+1979*q^7+2611*q^8+3161*q^9
+3598*q^10+3907*q^11+4113*q^12+4228*q^13+4283*q^14+4283*q^15+4261*q^16+37*q^43+
2717*q^28+4217*q^17+4157*q^18+4094*q^19+4017*q^20+2939*q^27+299*q^40+2062*q^31+
1130*q^35+928*q^36, 1+13571*q^25+3886*q^42+14248*q^22+13841*q^24+14411*q^21+
14056*q^23+10*q+278*q^51+2742*q^44+46*q^53+11525*q^30+131*q^52+q^55+10*q^54+
2264*q^45+7179*q^37+12013*q^29+1816*q^46+5842*q^39+4520*q^41+9159*q^34+492*q^50
+6502*q^38+9796*q^33+54*q^2+13247*q^26+201*q^3+571*q^4+1306*q^5+2496*q^6+10406*
q^32+4103*q^7+5962*q^8+7856*q^9+9610*q^10+11110*q^11+12310*q^12+13215*q^13+
13863*q^14+14286*q^15+14532*q^16+3286*q^43+760*q^49+12448*q^28+14640*q^17+14654
*q^18+14600*q^19+14515*q^20+12873*q^27+5170*q^40+11000*q^31+8533*q^35+1068*q^48
+7850*q^36+1426*q^47, 1+50388*q^25+28533*q^42+51007*q^22+50637*q^24+51035*q^21+
50855*q^23+11*q+11103*q^51+24364*q^44+8034*q^53+3490*q^57+47227*q^30+9498*q^52+
2653*q^58+5497*q^55+1292*q^60+6682*q^54+22334*q^45+37920*q^37+48105*q^29+20338*
q^46+11*q^65+34327*q^39+783*q^61+30558*q^41+42672*q^34+12806*q^50+56*q^64+36128
*q^38+44006*q^33+4444*q^56+65*q^2+50034*q^26+265*q^3+826*q^4+2078*q^5+4374*q^6+
45195*q^32+7914*q^7+12605*q^8+18069*q^9+411*q^62+23808*q^10+29364*q^11+34411*q^
12+38748*q^13+42331*q^14+45162*q^15+47319*q^16+q^66+26447*q^43+14609*q^49+48868
*q^28+48860*q^17+49918*q^18+50550*q^19+50895*q^20+49549*q^27+32465*q^40+177*q^
63+46273*q^31+41256*q^35+16448*q^48+1913*q^59+39635*q^36+18371*q^47, 1+180519*q
^25+141289*q^42+178162*q^22+180173*q^24+176070*q^21+179475*q^23+12*q+87940*q^51
+130433*q^44+75237*q^53+51636*q^57+178643*q^30+81530*q^52+46272*q^58+62968*q^55
+36186*q^60+68996*q^54+124708*q^45+3330*q^71+590*q^74+162684*q^37+10850*q^67+
179611*q^29+118852*q^46+16310*q^65+6562*q^69+155141*q^39+31581*q^61+146255*q^41
+171362*q^34+94306*q^50+19556*q^64+159063*q^38+173666*q^33+57185*q^56+77*q^2+
8571*q^68+180643*q^26+341*q^3+1156*q^4+3169*q^5+7279*q^6+175663*q^32+14376*q^7+
24947*q^8+38789*q^9+27205*q^62+55054*q^10+72557*q^11+90132*q^12+67*q^76+106818*
q^13+121966*q^14+135225*q^15+146472*q^16+13392*q^66+4814*q^70+12*q^77+q^78+
136020*q^43+100671*q^49+180245*q^28+155716*q^17+163093*q^18+168787*q^19+173029*
q^20+180582*q^27+2128*q^72+150829*q^40+1209*q^73+23221*q^63+177334*q^31+233*q^
75+168812*q^35+106846*q^48+41114*q^59+165913*q^36+112956*q^47, 1+634459*q^25+
596588*q^42+604361*q^22+13*q^90+627224*q^24+587632*q^21+617366*q^23+13*q+465637
*q^51+573925*q^44+427822*q^53+349188*q^57+649514*q^30+446957*q^52+329205*q^58+
388731*q^55+289111*q^60+q^91+408341*q^54+560674*q^45+102007*q^71+66535*q^74+
634483*q^37+160309*q^67+648259*q^29+546448*q^46+194009*q^65+129692*q^69+621761*
q^39+269232*q^61+605982*q^41+646956*q^34+26496*q^79+483468*q^50+211904*q^64+
628474*q^38+649108*q^33+369027*q^56+90*q^2+144581*q^68+639747*q^26+430*q^3+1574
*q^4+4666*q^5+11605*q^6+650099*q^32+24835*q^7+46667*q^8+78377*q^9+249653*q^62+
119617*q^10+168489*q^11+222181*q^12+47690*q^76+277675*q^13+332286*q^14+383917*q
^15+431197*q^16+176769*q^66+115462*q^70+39748*q^77+32724*q^78+585978*q^43+
500405*q^49+646339*q^28+473282*q^17+509856*q^18+540879*q^19+566682*q^20+643607*
q^27+89280*q^72+614250*q^40+77431*q^73+230520*q^63+650181*q^31+56586*q^75+
643795*q^35+21046*q^80+516394*q^48+8521*q^83+16226*q^81+3405*q^85+1816*q^86+825
*q^87+300*q^88+79*q^89+5636*q^84+12059*q^82+309170*q^59+639599*q^36+531736*q^47
, 1+2176190*q^25+2325178*q^42+1981947*q^22+96970*q^90+2123469*q^24+1890666*q^21
+2059146*q^23+14*q+14716*q^97+2066845*q^51+2290373*q^44+1975355*q^53+1764389*q^
57+2319093*q^30+40106*q^94+2022293*q^52+1706094*q^58+1874375*q^55+1584291*q^60+
79917*q^91+1925915*q^54+2268295*q^45+880531*q^71+702905*q^74+21660*q^96+9287*q^
98+2370625*q^37+1133143*q^67+2302006*q^29+2242669*q^46+51642*q^93+5306*q^99+
1263735*q^65+1004951*q^69+2360155*q^39+1521655*q^61+2338970*q^41+2362455*q^34+
447926*q^79+2108376*q^50+1329022*q^64+2660*q^100+2366951*q^38+2354535*q^33+
1820417*q^56+104*q^2+1068538*q^68+2218850*q^26+533*q^3+2094*q^4+6670*q^5+17846*
q^6+2344631*q^32+41118*q^7+83184*q^8+150220*q^9+30114*q^95+1458144*q^62+1127*q^
101+245818*q^10+369664*q^11+517642*q^12+593728*q^76+683050*q^13+858171*q^14+
1035532*q^15+1208791*q^16+1198180*q^66+942195*q^70+542761*q^77+494087*q^78+
2309218*q^43+379*q^102+2146823*q^49+2280428*q^28+1373075*q^17+1524983*q^18+
1662394*q^19+1784329*q^20+2253145*q^27+819895*q^72+2350615*q^40+760648*q^73+92*
q^103+1394031*q^63+2332990*q^31+647194*q^75+2368101*q^35+14*q^104+403966*q^80+
2181945*q^48+q^105+286138*q^83+362409*q^81+219128*q^85+189314*q^86+162325*q^87+
137979*q^88+116276*q^89+251368*q^84+323033*q^82+1646055*q^59+2370945*q^36+64845
*q^92+2213945*q^47]


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

[1, 2, 6, 22, 87, 352, 1434, 5861, 24019, 98677, 406291, 1676009, 6924618,

    28646875, 118638038]

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

     is given by

[1, 1, 3, 11, 44, 177, 717, 2926, 12004, 49355, 203182, 837947, 3462121,

    14323624, 59319865]

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

     is given by

[0, 1, 3, 11, 43, 175, 717, 2935, 12015, 49322, 203109, 838062, 3462497,

    14323251, 59318173]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 44, 175, 717, 2926, 12004, 49322, 203109, 837947, 3462121,

    14323251, 59318173]

 ODD:, [0, 1, 3, 11, 43, 177, 717, 2935, 12015, 49355, 203182, 838062, 3462497,

    14323624, 59319865]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.045454545, 5.183908046, 7.940340909,

    11.32705718, 15.35249957, 20.02443899, 25.35060855, 31.33840769,

    37.99450600, 45.32466239, 53.33375459, 62.02591165]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.521879004, 2.231069718, 3.105048656,

    4.145842820, 5.347468229, 6.701291590, 8.198472031, 9.831081763,

    11.59243483, 13.47698680, 15.48008992, 17.59775941]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.31612, 4.97767, 9.64133, 17.1880, 28.5954,

    44.9073, 67.2149, 96.6502, 134.385, 181.629, 239.633, 309.681]

Skewness: , [Float(undefined), 0., 0., -0.07672933926, -0.1567687948,

    -0.2182163527, -0.2605744247, -0.2891435964, -0.3083935724, -0.3213919091,

    -0.3301619311, -0.3360062468, -0.3397579230, -0.3419400664, -0.3429167195]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.266194381,

    2.317400803, 2.328152671, 2.334238905, 2.342762945, 2.354088857,

    2.367612269, 2.382590490, 2.398273041, 2.414103912, 2.429577555,

    2.444414230]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+2*q^5+q^6, 1+4*q+9*q^2+15*q^3
+18*q^4+16*q^5+12*q^6+8*q^7+4*q^8+2*q^9+q^10, 1+5*q+14*q^2+29*q^3+46*q^4+58*q^5
+60*q^6+54*q^7+43*q^8+32*q^9+23*q^10+14*q^11+8*q^12+4*q^13+2*q^14+q^15, 1+6*q+
20*q^2+49*q^3+94*q^4+147*q^5+193*q^6+218*q^7+218*q^8+200*q^9+172*q^10+142*q^11+
113*q^12+83*q^13+57*q^14+39*q^15+25*q^16+14*q^17+8*q^18+4*q^19+2*q^20+q^21, 1+7
*q+27*q^2+76*q^3+169*q^4+310*q^5+483*q^6+653*q^7+780*q^8+844*q^9+845*q^10+798*q
^11+725*q^12+639*q^13+543*q^14+445*q^15+353*q^16+266*q^17+196*q^18+142*q^19+97*
q^20+64*q^21+41*q^22+25*q^23+14*q^24+8*q^25+4*q^26+2*q^27+q^28, 1+8*q+35*q^2+
111*q^3+279*q^4+582*q^5+1038*q^6+1616*q^7+2232*q^8+2781*q^9+3180*q^10+3392*q^11
+3429*q^12+3333*q^13+3140*q^14+2886*q^15+2595*q^16+2269*q^17+1930*q^18+1601*q^
19+1296*q^20+1029*q^21+801*q^22+608*q^23+448*q^24+320*q^25+225*q^26+156*q^27+
104*q^28+66*q^29+41*q^30+25*q^31+14*q^32+8*q^33+4*q^34+2*q^35+q^36, 1+9*q+44*q^
2+155*q^3+433*q^4+1007*q^5+2010*q^6+3516*q^7+5475*q^8+7696*q^9+9896*q^10+11796*
q^11+13206*q^12+14060*q^13+14389*q^14+14282*q^15+13837*q^16+13138*q^17+12246*q^
18+11193*q^19+10027*q^20+8814*q^21+7613*q^22+6468*q^23+5415*q^24+4457*q^25+3603
*q^26+2869*q^27+2241*q^28+1717*q^29+1297*q^30+965*q^31+703*q^32+502*q^33+349*q^
34+239*q^35+163*q^36+106*q^37+66*q^38+41*q^39+25*q^40+14*q^41+8*q^42+4*q^43+2*q
^44+q^45, 1+10*q+54*q^2+209*q^3+641*q^4+1639*q^5+3605*q^6+6967*q^7+12016*q^8+
18734*q^9+26708*q^10+35198*q^11+43346*q^12+50409*q^13+55907*q^14+59668*q^15+
61751*q^16+62343*q^17+61683*q^18+59996*q^19+57445*q^20+54172*q^21+50334*q^22+
46077*q^23+41601*q^24+37084*q^25+32635*q^26+28381*q^27+24400*q^28+20709*q^29+
17348*q^30+14351*q^31+11719*q^32+9452*q^33+7535*q^34+5931*q^35+4611*q^36+3540*q
^37+2672*q^38+1986*q^39+1461*q^40+1060*q^41+757*q^42+531*q^43+363*q^44+246*q^45
+165*q^46+106*q^47+66*q^48+41*q^49+25*q^50+14*q^51+8*q^52+4*q^53+2*q^54+q^55, 1
+11*q+65*q^2+274*q^3+914*q^4+2543*q^5+6094*q^6+12853*q^7+24236*q^8+41368*q^9+
64596*q^10+93162*q^11+125236*q^12+158325*q^13+189858*q^14+217725*q^15+240568*q^
16+257781*q^17+269356*q^18+275675*q^19+277259*q^20+274685*q^21+268516*q^22+
259203*q^23+247215*q^24+233062*q^25+217258*q^26+200376*q^27+182978*q^28+165492*
q^29+148280*q^30+131626*q^31+115731*q^32+100776*q^33+86898*q^34+74212*q^35+
62794*q^36+52651*q^37+43737*q^38+35992*q^39+29335*q^40+23677*q^41+18921*q^42+
14965*q^43+11712*q^44+9078*q^45+6970*q^46+5292*q^47+3972*q^48+2941*q^49+2150*q^
50+1556*q^51+1114*q^52+786*q^53+545*q^54+370*q^55+248*q^56+165*q^57+106*q^58+66
*q^59+41*q^60+25*q^61+14*q^62+8*q^63+4*q^64+2*q^65+q^66, 1+12*q+77*q^2+351*q^3+
1264*q^4+3796*q^5+9825*q^6+22405*q^7+45736*q^8+84607*q^9+143280*q^10+224094*q^
11+326336*q^12+445911*q^13+576001*q^14+708462*q^15+835402*q^16+950385*q^17+
1049007*q^18+1128900*q^19+1189320*q^20+1230660*q^21+1254005*q^22+1260812*q^23+
1252679*q^24+1231152*q^25+1197727*q^26+1153943*q^27+1101485*q^28+1042164*q^29+
977844*q^30+910228*q^31+840792*q^32+770861*q^33+701489*q^34+633581*q^35+567984*
q^36+505362*q^37+446283*q^38+391230*q^39+340475*q^40+294166*q^41+252364*q^42+
214950*q^43+181726*q^44+152502*q^45+127013*q^46+104982*q^47+86134*q^48+70127*q^
49+56648*q^50+45419*q^51+36132*q^52+28509*q^53+22307*q^54+17294*q^55+13286*q^56
+10127*q^57+7652*q^58+5724*q^59+4241*q^60+3105*q^61+2245*q^62+1610*q^63+1143*q^
64+800*q^65+552*q^66+372*q^67+248*q^68+165*q^69+106*q^70+66*q^71+41*q^72+25*q^
73+14*q^74+8*q^75+4*q^76+2*q^77+q^78, 1+13*q+90*q^2+441*q^3+1704*q^4+5488*q^5+
15236*q^6+37291*q^7+81773*q^8+162640*q^9+296322*q^10+498742*q^11+781306*q^12+
1147147*q^13+1589062*q^14+2090024*q^15+2626083*q^16+3170550*q^17+3698084*q^18+
4187606*q^19+4623618*q^20+4996229*q^21+5300333*q^22+5534511*q^23+5700039*q^24+
5800074*q^25+5838805*q^26+5820875*q^27+5751153*q^28+5634597*q^29+5476633*q^30+
5283183*q^31+5060259*q^32+4813929*q^33+4549994*q^34+4273575*q^35+3989394*q^36+
3701705*q^37+3414154*q^38+3130108*q^39+2852597*q^40+2584188*q^41+2327210*q^42+
2083659*q^43+1854884*q^44+1641803*q^45+1444949*q^46+1264416*q^47+1100087*q^48+
951593*q^49+818285*q^50+699503*q^51+594494*q^52+502299*q^53+421910*q^54+352305*
q^55+292414*q^56+241227*q^57+197795*q^58+161151*q^59+130456*q^60+104938*q^61+
83843*q^62+66543*q^63+52478*q^64+41104*q^65+31964*q^66+24667*q^67+18878*q^68+
14336*q^69+10809*q^70+8084*q^71+5993*q^72+4405*q^73+3200*q^74+2299*q^75+1639*q^
76+1157*q^77+807*q^78+554*q^79+372*q^80+248*q^81+165*q^82+106*q^83+66*q^84+41*q
^85+25*q^86+14*q^87+8*q^88+4*q^89+2*q^90+q^91, 1+14*q+104*q^2+545*q^3+2248*q^4+
7723*q^5+22869*q^6+59720*q^7+139800*q^8+297019*q^9+578399*q^10+1040874*q^11+
1743385*q^12+2735351*q^13+4044519*q^14+5668099*q^15+7570242*q^16+9686608*q^17+
11934312*q^18+14223977*q^19+16470534*q^20+18600752*q^21+20556954*q^22+22297501*
q^23+23795400*q^24+25036187*q^25+26015371*q^26+26736218*q^27+27207643*q^28+
27441937*q^29+27453601*q^30+27258815*q^31+26875050*q^32+26321286*q^33+25617885*
q^34+24785595*q^35+23845113*q^36+22816520*q^37+21718451*q^38+20568237*q^39+
19381884*q^40+18173951*q^41+16957922*q^42+15746303*q^43+14550550*q^44+13381148*
q^45+12247380*q^46+11157063*q^47+10116597*q^48+9130847*q^49+8203147*q^50+
7335662*q^51+6529544*q^52+5785028*q^53+5101564*q^54+4477852*q^55+3911987*q^56+
3401673*q^57+2944178*q^58+2536328*q^59+2174738*q^60+1855892*q^61+1576223*q^62+
1332232*q^63+1120508*q^64+937796*q^65+781018*q^66+647208*q^67+533587*q^68+
437662*q^69+357137*q^70+289902*q^71+234076*q^72+187957*q^73+150065*q^74+119122*
q^75+94000*q^76+73742*q^77+57519*q^78+44590*q^79+34334*q^80+26252*q^81+19928*q^
82+15018*q^83+11241*q^84+8353*q^85+6157*q^86+4500*q^87+3254*q^88+2328*q^89+1653
*q^90+1164*q^91+809*q^92+554*q^93+372*q^94+248*q^95+165*q^96+106*q^97+66*q^98+
41*q^99+25*q^100+14*q^101+8*q^102+4*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+4*q^2+2*q+1, q^10+4*q^9+9*q^8+
15*q^7+18*q^6+16*q^5+12*q^4+8*q^3+4*q^2+2*q+1, q^15+5*q^14+14*q^13+29*q^12+46*q
^11+58*q^10+60*q^9+54*q^8+43*q^7+32*q^6+23*q^5+14*q^4+8*q^3+4*q^2+2*q+1, q^21+6
*q^20+20*q^19+49*q^18+94*q^17+147*q^16+193*q^15+218*q^14+218*q^13+200*q^12+172*
q^11+142*q^10+113*q^9+83*q^8+57*q^7+39*q^6+25*q^5+14*q^4+8*q^3+4*q^2+2*q+1, 1+
76*q^25+483*q^22+169*q^24+653*q^21+310*q^23+2*q+4*q^2+27*q^26+8*q^3+14*q^4+25*q
^5+41*q^6+64*q^7+97*q^8+142*q^9+196*q^10+266*q^11+353*q^12+445*q^13+543*q^14+
639*q^15+725*q^16+q^28+798*q^17+845*q^18+844*q^19+780*q^20+7*q^27, 1+3392*q^25+
3140*q^22+3429*q^24+2886*q^21+3333*q^23+2*q+1038*q^30+1616*q^29+35*q^34+111*q^
33+4*q^2+3180*q^26+8*q^3+14*q^4+25*q^5+41*q^6+279*q^32+66*q^7+104*q^8+156*q^9+
225*q^10+320*q^11+448*q^12+608*q^13+801*q^14+1029*q^15+1296*q^16+2232*q^28+1601
*q^17+1930*q^18+2269*q^19+2595*q^20+2781*q^27+582*q^31+8*q^35+q^36, 1+10027*q^
25+155*q^42+6468*q^22+8814*q^24+5415*q^21+7613*q^23+2*q+9*q^44+14282*q^30+q^45+
5475*q^37+13837*q^29+2010*q^39+433*q^41+11796*q^34+3516*q^38+13206*q^33+4*q^2+
11193*q^26+8*q^3+14*q^4+25*q^5+41*q^6+14060*q^32+66*q^7+106*q^8+163*q^9+239*q^
10+349*q^11+502*q^12+703*q^13+965*q^14+1297*q^15+1717*q^16+44*q^43+13138*q^28+
2241*q^17+2869*q^18+3603*q^19+4457*q^20+12246*q^27+1007*q^40+14389*q^31+9896*q^
35+7696*q^36, 1+17348*q^25+50409*q^42+9452*q^22+14351*q^24+7535*q^21+11719*q^23
+2*q+641*q^51+35198*q^44+54*q^53+37084*q^30+209*q^52+q^55+10*q^54+26708*q^45+
61683*q^37+32635*q^29+18734*q^46+61751*q^39+55907*q^41+54172*q^34+1639*q^50+
62343*q^38+50334*q^33+4*q^2+20709*q^26+8*q^3+14*q^4+25*q^5+41*q^6+46077*q^32+66
*q^7+106*q^8+165*q^9+246*q^10+363*q^11+531*q^12+757*q^13+1060*q^14+1461*q^15+
1986*q^16+43346*q^43+3605*q^49+28381*q^28+2672*q^17+3540*q^18+4611*q^19+5931*q^
20+24400*q^27+59668*q^40+41601*q^31+57445*q^35+6967*q^48+59996*q^36+12016*q^47,
1+23677*q^25+247215*q^42+11712*q^22+18921*q^24+9078*q^21+14965*q^23+2*q+217725*
q^51+268516*q^44+158325*q^53+41368*q^57+62794*q^30+189858*q^52+24236*q^58+93162
*q^55+6094*q^60+125236*q^54+274685*q^45+165492*q^37+52651*q^29+277259*q^46+11*q
^65+200376*q^39+2543*q^61+233062*q^41+115731*q^34+240568*q^50+65*q^64+182978*q^
38+100776*q^33+64596*q^56+4*q^2+29335*q^26+8*q^3+14*q^4+25*q^5+41*q^6+86898*q^
32+66*q^7+106*q^8+165*q^9+914*q^62+248*q^10+370*q^11+545*q^12+786*q^13+1114*q^
14+1556*q^15+2150*q^16+q^66+259203*q^43+257781*q^49+43737*q^28+2941*q^17+3972*q
^18+5292*q^19+6970*q^20+35992*q^27+217258*q^40+274*q^63+74212*q^31+131626*q^35+
269356*q^48+12853*q^59+148280*q^36+275675*q^47, 1+28509*q^25+567984*q^42+13286*
q^22+22307*q^24+10127*q^21+17294*q^23+2*q+1153943*q^51+701489*q^44+1231152*q^53
+1230660*q^57+86134*q^30+1197727*q^52+1189320*q^58+1260812*q^55+1049007*q^60+
1252679*q^54+770861*q^45+22405*q^71+1264*q^74+294166*q^37+224094*q^67+70127*q^
29+840792*q^46+445911*q^65+84607*q^69+391230*q^39+950385*q^61+505362*q^41+
181726*q^34+1101485*q^50+576001*q^64+340475*q^38+152502*q^33+1254005*q^56+4*q^2
+143280*q^68+36132*q^26+8*q^3+14*q^4+25*q^5+41*q^6+127013*q^32+66*q^7+106*q^8+
165*q^9+835402*q^62+248*q^10+372*q^11+552*q^12+77*q^76+800*q^13+1143*q^14+1610*
q^15+2245*q^16+326336*q^66+45736*q^70+12*q^77+q^78+633581*q^43+1042164*q^49+
56648*q^28+3105*q^17+4241*q^18+5724*q^19+7652*q^20+45419*q^27+9825*q^72+446283*
q^40+3796*q^73+708462*q^63+104982*q^31+351*q^75+214950*q^35+977844*q^48+1128900
*q^59+252364*q^36+910228*q^47, 1+31964*q^25+951593*q^42+14336*q^22+13*q^90+
24667*q^24+10809*q^21+18878*q^23+2*q+2852597*q^51+1264416*q^44+3414154*q^53+
4549994*q^57+104938*q^30+3130108*q^52+4813929*q^58+3989394*q^55+5283183*q^60+q^
91+3701705*q^54+1444949*q^45+4623618*q^71+3170550*q^74+421910*q^37+5700039*q^67
+83843*q^29+1641803*q^46+5838805*q^65+5300333*q^69+594494*q^39+5476633*q^61+
818285*q^41+241227*q^34+781306*q^79+2584188*q^50+5820875*q^64+502299*q^38+
197795*q^33+4273575*q^56+4*q^2+5534511*q^68+41104*q^26+8*q^3+14*q^4+25*q^5+41*q
^6+161151*q^32+66*q^7+106*q^8+165*q^9+5634597*q^62+248*q^10+372*q^11+554*q^12+
2090024*q^76+807*q^13+1157*q^14+1639*q^15+2299*q^16+5800074*q^66+4996229*q^70+
1589062*q^77+1147147*q^78+1100087*q^43+2327210*q^49+66543*q^28+3200*q^17+4405*q
^18+5993*q^19+8084*q^20+52478*q^27+4187606*q^72+699503*q^40+3698084*q^73+
5751153*q^63+130456*q^31+2626083*q^75+292414*q^35+498742*q^80+2083659*q^48+
81773*q^83+296322*q^81+15236*q^85+5488*q^86+1704*q^87+441*q^88+90*q^89+37291*q^
84+162640*q^82+5060259*q^59+352305*q^36+1854884*q^47, 1+34334*q^25+1332232*q^42
+15018*q^22+5668099*q^90+26252*q^24+11241*q^21+19928*q^23+2*q+139800*q^97+
5101564*q^51+1855892*q^44+6529544*q^53+10116597*q^57+119122*q^30+1040874*q^94+
5785028*q^52+11157063*q^58+8203147*q^55+13381148*q^60+4044519*q^91+7335662*q^54
+2174738*q^45+25617885*q^71+27258815*q^74+297019*q^96+59720*q^98+533587*q^37+
21718451*q^67+94000*q^29+2536328*q^46+1743385*q^93+22869*q^99+19381884*q^65+
23845113*q^69+781018*q^39+14550550*q^61+1120508*q^41+289902*q^34+26015371*q^79+
4477852*q^50+18173951*q^64+7723*q^100+647208*q^38+234076*q^33+9130847*q^56+4*q^
2+22816520*q^68+44590*q^26+8*q^3+14*q^4+25*q^5+41*q^6+187957*q^32+66*q^7+106*q^
8+165*q^9+578399*q^95+15746303*q^62+2248*q^101+248*q^10+372*q^11+554*q^12+
27441937*q^76+809*q^13+1164*q^14+1653*q^15+2328*q^16+20568237*q^66+24785595*q^
70+27207643*q^77+26736218*q^78+1576223*q^43+545*q^102+3911987*q^49+73742*q^28+
3254*q^17+4500*q^18+6157*q^19+8353*q^20+57519*q^27+26321286*q^72+937796*q^40+
26875050*q^73+104*q^103+16957922*q^63+150065*q^31+27453601*q^75+357137*q^35+14*
q^104+25036187*q^80+3401673*q^48+q^105+20556954*q^83+23795400*q^81+16470534*q^
85+14223977*q^86+11934312*q^87+9686608*q^88+7570242*q^89+18600752*q^84+22297501
*q^82+12247380*q^59+437662*q^36+2735351*q^92+2944178*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.863636364, 4.522222222, 6.444162437,

    8.610741971, 11.00853003, 13.62686962, 16.45684577, 19.49076628,

    22.72185850, 26.14407329, 29.75194882, 33.54051108]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.455255509, 2.012338483, 2.634827060,

    3.322242202, 4.072123457, 4.881688628, 5.748330464, 6.669732044,

    7.643863452, 8.668946037, 9.743412200, 10.86586936]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.11777, 4.04951, 6.94231, 11.0373, 16.5822,

    23.8309, 33.0433, 44.4853, 58.4286, 75.1506, 94.9341, 118.067]

Skewness: , [Float(undefined), 0., 0., 0.1491970460, 0.2825172660, 0.3762673426,

    0.4378078836, 0.4775786994, 0.5033486337, 0.5201786029, 0.5312550474,

    0.5385808304, 0.5434302880, 0.5466256783, 0.5487084109]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.547029639,

    2.797768857, 2.937151936, 3.019175734, 3.069672962, 3.101890146,

    3.123011206, 3.137095611, 3.146576032, 3.152978260, 3.157243006,

    3.160041600]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+6*q^3+5*q^4+2*q^5+q^6, 1+2*q+5*q^2+10*q^3
+16*q^4+18*q^5+16*q^6+10*q^7+5*q^8+2*q^9+q^10, 1+2*q+5*q^2+10*q^3+20*q^4+31*q^5
+44*q^6+52*q^7+52*q^8+44*q^9+31*q^10+20*q^11+10*q^12+5*q^13+2*q^14+q^15, 1+2*q+
5*q^2+10*q^3+20*q^4+35*q^5+56*q^6+81*q^7+110*q^8+135*q^9+148*q^10+148*q^11+135*
q^12+110*q^13+81*q^14+56*q^15+35*q^16+20*q^17+10*q^18+5*q^19+2*q^20+q^21, 1+2*q
+5*q^2+10*q^3+20*q^4+35*q^5+60*q^6+93*q^7+136*q^8+190*q^9+251*q^10+311*q^11+365
*q^12+401*q^13+414*q^14+401*q^15+365*q^16+311*q^17+251*q^18+190*q^19+136*q^20+
93*q^21+60*q^22+35*q^23+20*q^24+10*q^25+5*q^26+2*q^27+q^28, 1+2*q+5*q^2+10*q^3+
20*q^4+35*q^5+60*q^6+97*q^7+148*q^8+216*q^9+302*q^10+406*q^11+530*q^12+661*q^13
+794*q^14+915*q^15+1019*q^16+1087*q^17+1110*q^18+1087*q^19+1019*q^20+915*q^21+
794*q^22+661*q^23+530*q^24+406*q^25+302*q^26+216*q^27+148*q^28+97*q^29+60*q^30+
35*q^31+20*q^32+10*q^33+5*q^34+2*q^35+q^36, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+60
*q^6+97*q^7+152*q^8+228*q^9+328*q^10+457*q^11+621*q^12+817*q^13+1049*q^14+1307*
q^15+1586*q^16+1873*q^17+2162*q^18+2423*q^19+2647*q^20+2804*q^21+2883*q^22+2883
*q^23+2804*q^24+2647*q^25+2423*q^26+2162*q^27+1873*q^28+1586*q^29+1307*q^30+
1049*q^31+817*q^32+621*q^33+457*q^34+328*q^35+228*q^36+152*q^37+97*q^38+60*q^39
+35*q^40+20*q^41+10*q^42+5*q^43+2*q^44+q^45, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+
60*q^6+97*q^7+152*q^8+232*q^9+340*q^10+483*q^11+672*q^12+908*q^13+1201*q^14+
1554*q^15+1975*q^16+2447*q^17+2981*q^18+3557*q^19+4168*q^20+4800*q^21+5419*q^22
+5996*q^23+6512*q^24+6928*q^25+7219*q^26+7377*q^27+7377*q^28+7219*q^29+6928*q^
30+6512*q^31+5996*q^32+5419*q^33+4800*q^34+4168*q^35+3557*q^36+2981*q^37+2447*q
^38+1975*q^39+1554*q^40+1201*q^41+908*q^42+672*q^43+483*q^44+340*q^45+232*q^46+
152*q^47+97*q^48+60*q^49+35*q^50+20*q^51+10*q^52+5*q^53+2*q^54+q^55, 1+2*q+5*q^
2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+495*q^11+698*q^12
+959*q^13+1292*q^14+1706*q^15+2218*q^16+2828*q^17+3555*q^18+4390*q^19+5339*q^20
+6395*q^21+7557*q^22+8800*q^23+10113*q^24+11461*q^25+12800*q^26+14098*q^27+
15305*q^28+16381*q^29+17275*q^30+17955*q^31+18371*q^32+18516*q^33+18371*q^34+
17955*q^35+17275*q^36+16381*q^37+15305*q^38+14098*q^39+12800*q^40+11461*q^41+
10113*q^42+8800*q^43+7557*q^44+6395*q^45+5339*q^46+4390*q^47+3555*q^48+2828*q^
49+2218*q^50+1706*q^51+1292*q^52+959*q^53+698*q^54+495*q^55+344*q^56+232*q^57+
152*q^58+97*q^59+60*q^60+35*q^61+20*q^62+10*q^63+5*q^64+2*q^65+q^66, 1+2*q+5*q^
2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+499*q^11+710*q^12
+985*q^13+1343*q^14+1797*q^15+2370*q^16+3071*q^17+3932*q^18+4956*q^19+6172*q^20
+7584*q^21+9201*q^22+11027*q^23+13072*q^24+15320*q^25+17760*q^26+20378*q^27+
23116*q^28+25950*q^29+28827*q^30+31681*q^31+34457*q^32+37083*q^33+39470*q^34+
41556*q^35+43266*q^36+44521*q^37+45292*q^38+45562*q^39+45292*q^40+44521*q^41+
43266*q^42+41556*q^43+39470*q^44+37083*q^45+34457*q^46+31681*q^47+28827*q^48+
25950*q^49+23116*q^50+20378*q^51+17760*q^52+15320*q^53+13072*q^54+11027*q^55+
9201*q^56+7584*q^57+6172*q^58+4956*q^59+3932*q^60+3071*q^61+2370*q^62+1797*q^63
+1343*q^64+985*q^65+710*q^66+499*q^67+344*q^68+232*q^69+152*q^70+97*q^71+60*q^
72+35*q^73+20*q^74+10*q^75+5*q^76+2*q^77+q^78, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5
+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+499*q^11+714*q^12+997*q^13+1369*q^14+
1848*q^15+2461*q^16+3223*q^17+4175*q^18+5333*q^19+6734*q^20+8409*q^21+10390*q^
22+12689*q^23+15352*q^24+18381*q^25+21795*q^26+25625*q^27+29848*q^28+34461*q^29
+39454*q^30+44786*q^31+50398*q^32+56270*q^33+62292*q^34+68401*q^35+74524*q^36+
80536*q^37+86292*q^38+91724*q^39+96660*q^40+100988*q^41+104633*q^42+107457*q^43
+109371*q^44+110360*q^45+110360*q^46+109371*q^47+107457*q^48+104633*q^49+100988
*q^50+96660*q^51+91724*q^52+86292*q^53+80536*q^54+74524*q^55+68401*q^56+62292*q
^57+56270*q^58+50398*q^59+44786*q^60+39454*q^61+34461*q^62+29848*q^63+25625*q^
64+21795*q^65+18381*q^66+15352*q^67+12689*q^68+10390*q^69+8409*q^70+6734*q^71+
5333*q^72+4175*q^73+3223*q^74+2461*q^75+1848*q^76+1369*q^77+997*q^78+714*q^79+
499*q^80+344*q^81+232*q^82+152*q^83+97*q^84+60*q^85+35*q^86+20*q^87+10*q^88+5*q
^89+2*q^90+q^91, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9
+344*q^10+499*q^11+714*q^12+1001*q^13+1381*q^14+1874*q^15+2512*q^16+3314*q^17+
4327*q^18+5576*q^19+7111*q^20+8971*q^21+11211*q^22+13870*q^23+17014*q^24+20679*
q^25+24909*q^26+29766*q^27+35283*q^28+41499*q^29+48448*q^30+56151*q^31+64599*q^
32+73813*q^33+83751*q^34+94373*q^35+105634*q^36+117462*q^37+129762*q^38+142444*
q^39+155364*q^40+168366*q^41+181296*q^42+193968*q^43+206174*q^44+217731*q^45+
228418*q^46+238042*q^47+246421*q^48+253364*q^49+258709*q^50+262351*q^51+264185*
q^52+264185*q^53+262351*q^54+258709*q^55+253364*q^56+246421*q^57+238042*q^58+
228418*q^59+217731*q^60+206174*q^61+193968*q^62+181296*q^63+168366*q^64+155364*
q^65+142444*q^66+129762*q^67+117462*q^68+105634*q^69+94373*q^70+83751*q^71+
73813*q^72+64599*q^73+56151*q^74+48448*q^75+41499*q^76+35283*q^77+29766*q^78+
24909*q^79+20679*q^80+17014*q^81+13870*q^82+11211*q^83+8971*q^84+7111*q^85+5576
*q^86+4327*q^87+3314*q^88+2512*q^89+1874*q^90+1381*q^91+1001*q^92+714*q^93+499*
q^94+344*q^95+232*q^96+152*q^97+97*q^98+60*q^99+35*q^100+20*q^101+10*q^102+5*q^
103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+6*q^3+5*q^4+2*q^5+q^6, 1+2*q+5*q^2+10*q^3
+16*q^4+18*q^5+16*q^6+10*q^7+5*q^8+2*q^9+q^10, 1+2*q+5*q^2+10*q^3+20*q^4+31*q^5
+44*q^6+52*q^7+52*q^8+44*q^9+31*q^10+20*q^11+10*q^12+5*q^13+2*q^14+q^15, 1+2*q+
5*q^2+10*q^3+20*q^4+35*q^5+56*q^6+81*q^7+110*q^8+135*q^9+148*q^10+148*q^11+135*
q^12+110*q^13+81*q^14+56*q^15+35*q^16+20*q^17+10*q^18+5*q^19+2*q^20+q^21, 1+2*q
+5*q^2+10*q^3+20*q^4+35*q^5+60*q^6+93*q^7+136*q^8+190*q^9+251*q^10+311*q^11+365
*q^12+401*q^13+414*q^14+401*q^15+365*q^16+311*q^17+251*q^18+190*q^19+136*q^20+
93*q^21+60*q^22+35*q^23+20*q^24+10*q^25+5*q^26+2*q^27+q^28, 1+2*q+5*q^2+10*q^3+
20*q^4+35*q^5+60*q^6+97*q^7+148*q^8+216*q^9+302*q^10+406*q^11+530*q^12+661*q^13
+794*q^14+915*q^15+1019*q^16+1087*q^17+1110*q^18+1087*q^19+1019*q^20+915*q^21+
794*q^22+661*q^23+530*q^24+406*q^25+302*q^26+216*q^27+148*q^28+97*q^29+60*q^30+
35*q^31+20*q^32+10*q^33+5*q^34+2*q^35+q^36, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+60
*q^6+97*q^7+152*q^8+228*q^9+328*q^10+457*q^11+621*q^12+817*q^13+1049*q^14+1307*
q^15+1586*q^16+1873*q^17+2162*q^18+2423*q^19+2647*q^20+2804*q^21+2883*q^22+2883
*q^23+2804*q^24+2647*q^25+2423*q^26+2162*q^27+1873*q^28+1586*q^29+1307*q^30+
1049*q^31+817*q^32+621*q^33+457*q^34+328*q^35+228*q^36+152*q^37+97*q^38+60*q^39
+35*q^40+20*q^41+10*q^42+5*q^43+2*q^44+q^45, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+
60*q^6+97*q^7+152*q^8+232*q^9+340*q^10+483*q^11+672*q^12+908*q^13+1201*q^14+
1554*q^15+1975*q^16+2447*q^17+2981*q^18+3557*q^19+4168*q^20+4800*q^21+5419*q^22
+5996*q^23+6512*q^24+6928*q^25+7219*q^26+7377*q^27+7377*q^28+7219*q^29+6928*q^
30+6512*q^31+5996*q^32+5419*q^33+4800*q^34+4168*q^35+3557*q^36+2981*q^37+2447*q
^38+1975*q^39+1554*q^40+1201*q^41+908*q^42+672*q^43+483*q^44+340*q^45+232*q^46+
152*q^47+97*q^48+60*q^49+35*q^50+20*q^51+10*q^52+5*q^53+2*q^54+q^55, 1+2*q+5*q^
2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+495*q^11+698*q^12
+959*q^13+1292*q^14+1706*q^15+2218*q^16+2828*q^17+3555*q^18+4390*q^19+5339*q^20
+6395*q^21+7557*q^22+8800*q^23+10113*q^24+11461*q^25+12800*q^26+14098*q^27+
15305*q^28+16381*q^29+17275*q^30+17955*q^31+18371*q^32+18516*q^33+18371*q^34+
17955*q^35+17275*q^36+16381*q^37+15305*q^38+14098*q^39+12800*q^40+11461*q^41+
10113*q^42+8800*q^43+7557*q^44+6395*q^45+5339*q^46+4390*q^47+3555*q^48+2828*q^
49+2218*q^50+1706*q^51+1292*q^52+959*q^53+698*q^54+495*q^55+344*q^56+232*q^57+
152*q^58+97*q^59+60*q^60+35*q^61+20*q^62+10*q^63+5*q^64+2*q^65+q^66, 1+2*q+5*q^
2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+499*q^11+710*q^12
+985*q^13+1343*q^14+1797*q^15+2370*q^16+3071*q^17+3932*q^18+4956*q^19+6172*q^20
+7584*q^21+9201*q^22+11027*q^23+13072*q^24+15320*q^25+17760*q^26+20378*q^27+
23116*q^28+25950*q^29+28827*q^30+31681*q^31+34457*q^32+37083*q^33+39470*q^34+
41556*q^35+43266*q^36+44521*q^37+45292*q^38+45562*q^39+45292*q^40+44521*q^41+
43266*q^42+41556*q^43+39470*q^44+37083*q^45+34457*q^46+31681*q^47+28827*q^48+
25950*q^49+23116*q^50+20378*q^51+17760*q^52+15320*q^53+13072*q^54+11027*q^55+
9201*q^56+7584*q^57+6172*q^58+4956*q^59+3932*q^60+3071*q^61+2370*q^62+1797*q^63
+1343*q^64+985*q^65+710*q^66+499*q^67+344*q^68+232*q^69+152*q^70+97*q^71+60*q^
72+35*q^73+20*q^74+10*q^75+5*q^76+2*q^77+q^78, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5
+60*q^6+97*q^7+152*q^8+232*q^9+344*q^10+499*q^11+714*q^12+997*q^13+1369*q^14+
1848*q^15+2461*q^16+3223*q^17+4175*q^18+5333*q^19+6734*q^20+8409*q^21+10390*q^
22+12689*q^23+15352*q^24+18381*q^25+21795*q^26+25625*q^27+29848*q^28+34461*q^29
+39454*q^30+44786*q^31+50398*q^32+56270*q^33+62292*q^34+68401*q^35+74524*q^36+
80536*q^37+86292*q^38+91724*q^39+96660*q^40+100988*q^41+104633*q^42+107457*q^43
+109371*q^44+110360*q^45+110360*q^46+109371*q^47+107457*q^48+104633*q^49+100988
*q^50+96660*q^51+91724*q^52+86292*q^53+80536*q^54+74524*q^55+68401*q^56+62292*q
^57+56270*q^58+50398*q^59+44786*q^60+39454*q^61+34461*q^62+29848*q^63+25625*q^
64+21795*q^65+18381*q^66+15352*q^67+12689*q^68+10390*q^69+8409*q^70+6734*q^71+
5333*q^72+4175*q^73+3223*q^74+2461*q^75+1848*q^76+1369*q^77+997*q^78+714*q^79+
499*q^80+344*q^81+232*q^82+152*q^83+97*q^84+60*q^85+35*q^86+20*q^87+10*q^88+5*q
^89+2*q^90+q^91, 1+2*q+5*q^2+10*q^3+20*q^4+35*q^5+60*q^6+97*q^7+152*q^8+232*q^9
+344*q^10+499*q^11+714*q^12+1001*q^13+1381*q^14+1874*q^15+2512*q^16+3314*q^17+
4327*q^18+5576*q^19+7111*q^20+8971*q^21+11211*q^22+13870*q^23+17014*q^24+20679*
q^25+24909*q^26+29766*q^27+35283*q^28+41499*q^29+48448*q^30+56151*q^31+64599*q^
32+73813*q^33+83751*q^34+94373*q^35+105634*q^36+117462*q^37+129762*q^38+142444*
q^39+155364*q^40+168366*q^41+181296*q^42+193968*q^43+206174*q^44+217731*q^45+
228418*q^46+238042*q^47+246421*q^48+253364*q^49+258709*q^50+262351*q^51+264185*
q^52+264185*q^53+262351*q^54+258709*q^55+253364*q^56+246421*q^57+238042*q^58+
228418*q^59+217731*q^60+206174*q^61+193968*q^62+181296*q^63+168366*q^64+155364*
q^65+142444*q^66+129762*q^67+117462*q^68+105634*q^69+94373*q^70+83751*q^71+
73813*q^72+64599*q^73+56151*q^74+48448*q^75+41499*q^76+35283*q^77+29766*q^78+
24909*q^79+20679*q^80+17014*q^81+13870*q^82+11211*q^83+8971*q^84+7111*q^85+5576
*q^86+4327*q^87+3314*q^88+2512*q^89+1874*q^90+1381*q^91+1001*q^92+714*q^93+499*
q^94+344*q^95+232*q^96+152*q^97+97*q^98+60*q^99+35*q^100+20*q^101+10*q^102+5*q^
103+2*q^104+q^105]


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

[1, 2, 6, 22, 86, 330, 1206, 4174, 13726, 43134, 130302, 380414, 1078270,

    2978814, 8046590]

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

     is given by

[1, 1, 3, 12, 44, 165, 603, 2090, 6868, 21567, 65151, 190210, 539140, 1489407,

    4023295]

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

     is given by

[0, 1, 3, 10, 42, 165, 603, 2084, 6858, 21567, 65151, 190204, 539130, 1489407,

    4023295]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 44, 165, 603, 2090, 6868, 21567, 65151, 190210, 539140,

    1489407, 4023295]

 ODD:, [0, 1, 3, 10, 42, 165, 603, 2084, 6858, 21567, 65151, 190204, 539130,

    1489407, 4023295]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.500000000,

    10.50000000, 14.00000000, 18.00000000, 22.50000000, 27.50000000,

    33.00000000, 39.00000000, 45.50000000, 52.50000000]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.414213562, 1.916877305, 2.516912490,

    3.225301612, 4.037319913, 4.945705503, 5.943848431, 7.026161377,

    8.187931832, 9.425141989, 10.73432744, 12.11247196]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.00000, 3.67442, 6.33485, 10.4026, 16.3000,

    24.4600, 35.3293, 49.3669, 67.0422, 88.8333, 115.226, 146.712]

                                                              -5
Skewness: , [Float(undefined), 0., 0., 0., 0., 0.6271847594 10  ,

                   -5                 -5                                     -6
    0.2980486445 10  , 0.1519562590 10  , 0., 0., 0., 0., 0., 0.8084892893 10  ,

    0.]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.681825000,

    2.959197672, 2.997558615, 2.973392924, 2.948646919, 2.932717090,

    2.923729044, 2.919074641, 2.916997053, 2.916457243, 2.916766540,

    2.917585070]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+5*q^2+6*q^3+5*q^4+3*q^5, 1+2*q+5*q^2+10*q^3+16*
q^4+20*q^5+18*q^6+11*q^7+3*q^8, 1+2*q+5*q^2+10*q^3+20*q^4+33*q^5+48*q^6+59*q^7+
60*q^8+46*q^9+26*q^10+10*q^11+q^12, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+60*q^6+90*
q^7+124*q^8+150*q^9+162*q^10+154*q^11+124*q^12+84*q^13+44*q^14+15*q^15+3*q^16,
1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+102*q^7+152*q^8+210*q^9+273*q^10+328*q^
11+367*q^12+383*q^13+373*q^14+329*q^15+261*q^16+179*q^17+104*q^18+48*q^19+15*q^
20+3*q^21, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+106*q^7+164*q^8+238*q^9+329*
q^10+430*q^11+532*q^12+632*q^13+718*q^14+781*q^15+819*q^16+823*q^17+781*q^18+
695*q^19+573*q^20+435*q^21+296*q^22+175*q^23+86*q^24+34*q^25+10*q^26+q^27, 1+2*
q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+106*q^7+168*q^8+250*q^9+357*q^10+486*q^11+
630*q^12+788*q^13+952*q^14+1110*q^15+1257*q^16+1395*q^17+1500*q^18+1589*q^19+
1630*q^20+1624*q^21+1555*q^22+1437*q^23+1265*q^24+1050*q^25+818*q^26+585*q^27+
382*q^28+225*q^29+114*q^30+48*q^31+15*q^32+3*q^33, 1+2*q+5*q^2+10*q^3+20*q^4+37
*q^5+64*q^6+106*q^7+168*q^8+254*q^9+369*q^10+514*q^11+686*q^12+886*q^13+1104*q^
14+1336*q^15+1575*q^16+1820*q^17+2048*q^18+2273*q^19+2470*q^20+2663*q^21+2813*q
^22+2934*q^23+2994*q^24+3004*q^25+2943*q^26+2820*q^27+2618*q^28+2347*q^29+2009*
q^30+1646*q^31+1279*q^32+941*q^33+638*q^34+400*q^35+229*q^36+114*q^37+48*q^38+
15*q^39+3*q^40, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+106*q^7+168*q^8+254*q^9
+373*q^10+526*q^11+714*q^12+942*q^13+1202*q^14+1488*q^15+1797*q^16+2130*q^17+
2464*q^18+2816*q^19+3146*q^20+3486*q^21+3797*q^22+4116*q^23+4384*q^24+4658*q^25
+4864*q^26+5054*q^27+5153*q^28+5236*q^29+5223*q^30+5155*q^31+4960*q^32+4695*q^
33+4310*q^34+3860*q^35+3348*q^36+2805*q^37+2256*q^38+1733*q^39+1264*q^40+876*q^
41+570*q^42+344*q^43+185*q^44+86*q^45+34*q^46+10*q^47+q^48, 1+2*q+5*q^2+10*q^3+
20*q^4+37*q^5+64*q^6+106*q^7+168*q^8+254*q^9+373*q^10+530*q^11+726*q^12+970*q^
13+1258*q^14+1586*q^15+1949*q^16+2352*q^17+2770*q^18+3224*q^19+3680*q^20+4159*q
^21+4621*q^22+5103*q^23+5546*q^24+6014*q^25+6440*q^26+6878*q^27+7246*q^28+7630*
q^29+7920*q^30+8210*q^31+8414*q^32+8605*q^33+8690*q^34+8730*q^35+8635*q^36+8443
*q^37+8115*q^38+7702*q^39+7150*q^40+6526*q^41+5788*q^42+5005*q^43+4204*q^44+
3425*q^45+2692*q^46+2048*q^47+1487*q^48+1032*q^49+678*q^50+414*q^51+231*q^52+
114*q^53+48*q^54+15*q^55+3*q^56, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+106*q^
7+168*q^8+254*q^9+373*q^10+530*q^11+730*q^12+982*q^13+1286*q^14+1642*q^15+2047*
q^16+2504*q^17+2992*q^18+3530*q^19+4084*q^20+4685*q^21+5285*q^22+5924*q^23+6536
*q^24+7187*q^25+7794*q^26+8450*q^27+9038*q^28+9680*q^29+10236*q^30+10838*q^31+
11336*q^32+11873*q^33+12284*q^34+12733*q^35+13073*q^36+13440*q^37+13652*q^38+
13874*q^39+13902*q^40+13916*q^41+13731*q^42+13498*q^43+13060*q^44+12550*q^45+
11825*q^46+11016*q^47+10033*q^48+8993*q^49+7885*q^50+6790*q^51+5691*q^52+4653*q
^53+3696*q^54+2851*q^55+2131*q^56+1538*q^57+1056*q^58+688*q^59+418*q^60+231*q^
61+114*q^62+48*q^63+15*q^64+3*q^65, 1+2*q+5*q^2+10*q^3+20*q^4+37*q^5+64*q^6+106
*q^7+168*q^8+254*q^9+373*q^10+530*q^11+730*q^12+986*q^13+1298*q^14+1670*q^15+
2103*q^16+2602*q^17+3144*q^18+3752*q^19+4390*q^20+5089*q^21+5807*q^22+6580*q^23
+7348*q^24+8174*q^25+8970*q^26+9817*q^27+10616*q^28+11484*q^29+12278*q^30+13140
*q^31+13918*q^32+14767*q^33+15508*q^34+16301*q^35+16977*q^36+17706*q^37+18302*q
^38+18974*q^39+19490*q^40+20067*q^41+20477*q^42+20901*q^43+21147*q^44+21391*q^
45+21429*q^46+21464*q^47+21304*q^48+21069*q^49+20583*q^50+19993*q^51+19152*q^52
+18195*q^53+17035*q^54+15770*q^55+14365*q^56+12912*q^57+11380*q^58+9872*q^59+
8396*q^60+7005*q^61+5713*q^62+4571*q^63+3565*q^64+2701*q^65+1972*q^66+1389*q^67
+936*q^68+594*q^69+350*q^70+185*q^71+86*q^72+34*q^73+10*q^74+q^75]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q*(q^5+2*q^4+5*q^3+6*q^2+5*q+3), q^2*(q^8+2*q^7+5*q^6
+10*q^5+16*q^4+20*q^3+18*q^2+11*q+3), q^3*(q^12+2*q^11+5*q^10+10*q^9+20*q^8+33*
q^7+48*q^6+59*q^5+60*q^4+46*q^3+26*q^2+10*q+1), q^5*(q^16+2*q^15+5*q^14+10*q^13
+20*q^12+37*q^11+60*q^10+90*q^9+124*q^8+150*q^7+162*q^6+154*q^5+124*q^4+84*q^3+
44*q^2+15*q+3), q^7*(q^21+2*q^20+5*q^19+10*q^18+20*q^17+37*q^16+64*q^15+102*q^
14+152*q^13+210*q^12+273*q^11+328*q^10+367*q^9+383*q^8+373*q^7+329*q^6+261*q^5+
179*q^4+104*q^3+48*q^2+15*q+3), q^9*(1+5*q^25+37*q^22+10*q^24+64*q^21+20*q^23+
10*q+34*q^2+2*q^26+86*q^3+175*q^4+296*q^5+435*q^6+573*q^7+695*q^8+781*q^9+823*q
^10+819*q^11+781*q^12+718*q^13+632*q^14+532*q^15+430*q^16+329*q^17+238*q^18+164
*q^19+106*q^20+q^27), q^12*(3+168*q^25+486*q^22+250*q^24+630*q^21+357*q^23+15*q
+10*q^30+20*q^29+q^33+48*q^2+106*q^26+114*q^3+225*q^4+382*q^5+585*q^6+2*q^32+
818*q^7+1050*q^8+1265*q^9+1437*q^10+1555*q^11+1624*q^12+1630*q^13+1589*q^14+
1500*q^15+1395*q^16+37*q^28+1257*q^17+1110*q^18+952*q^19+788*q^20+64*q^27+5*q^
31), q^15*(3+1336*q^25+2048*q^22+1575*q^24+2273*q^21+1820*q^23+15*q+369*q^30+10
*q^37+514*q^29+2*q^39+64*q^34+5*q^38+106*q^33+48*q^2+1104*q^26+114*q^3+229*q^4+
400*q^5+638*q^6+168*q^32+941*q^7+1279*q^8+1646*q^9+2009*q^10+2347*q^11+2618*q^
12+2820*q^13+2943*q^14+3004*q^15+2994*q^16+686*q^28+2934*q^17+2813*q^18+2663*q^
19+2470*q^20+886*q^27+q^40+254*q^31+37*q^35+20*q^36), q^18*(1+4116*q^25+64*q^42
+4864*q^22+4384*q^24+5054*q^21+4658*q^23+10*q+20*q^44+2464*q^30+10*q^45+526*q^
37+2816*q^29+5*q^46+254*q^39+106*q^41+1202*q^34+373*q^38+1488*q^33+34*q^2+3797*
q^26+86*q^3+185*q^4+344*q^5+570*q^6+1797*q^32+876*q^7+1264*q^8+1733*q^9+2256*q^
10+2805*q^11+3348*q^12+3860*q^13+4310*q^14+4695*q^15+4960*q^16+37*q^43+3146*q^
28+5155*q^17+5223*q^18+5236*q^19+5153*q^20+3486*q^27+168*q^40+2130*q^31+942*q^
35+q^48+714*q^36+2*q^47), q^22*(3+8210*q^25+1258*q^42+8690*q^22+8414*q^24+8730*
q^21+8605*q^23+15*q+37*q^51+726*q^44+10*q^53+6440*q^30+20*q^52+2*q^55+5*q^54+
530*q^45+3224*q^37+6878*q^29+373*q^46+2352*q^39+1586*q^41+4621*q^34+64*q^50+
2770*q^38+5103*q^33+q^56+48*q^2+7920*q^26+114*q^3+231*q^4+414*q^5+678*q^6+5546*
q^32+1032*q^7+1487*q^8+2048*q^9+2692*q^10+3425*q^11+4204*q^12+5005*q^13+5788*q^
14+6526*q^15+7150*q^16+970*q^43+106*q^49+7246*q^28+7702*q^17+8115*q^18+8443*q^
19+8635*q^20+7630*q^27+1949*q^40+6014*q^31+4159*q^35+168*q^48+3680*q^36+254*q^
47), q^26*(3+13902*q^25+5924*q^42+13498*q^22+13916*q^24+13060*q^21+13731*q^23+
15*q+1286*q^51+4685*q^44+730*q^53+168*q^57+12733*q^30+982*q^52+106*q^58+373*q^
55+37*q^60+530*q^54+4084*q^45+9038*q^37+13073*q^29+3530*q^46+q^65+7794*q^39+20*
q^61+6536*q^41+10838*q^34+1642*q^50+2*q^64+8450*q^38+11336*q^33+254*q^56+48*q^2
+13874*q^26+114*q^3+231*q^4+418*q^5+688*q^6+11873*q^32+1056*q^7+1538*q^8+2131*q
^9+10*q^62+2851*q^10+3696*q^11+4653*q^12+5691*q^13+6790*q^14+7885*q^15+8993*q^
16+5285*q^43+2047*q^49+13440*q^28+10033*q^17+11016*q^18+11825*q^19+12550*q^20+
13652*q^27+7187*q^40+5*q^63+12284*q^31+10236*q^35+2504*q^48+64*q^59+9680*q^36+
2992*q^47), q^30*(1+20583*q^25+14767*q^42+18195*q^22+19993*q^24+17035*q^21+
19152*q^23+10*q+7348*q^51+13140*q^44+5807*q^53+3144*q^57+21391*q^30+6580*q^52+
2602*q^58+4390*q^55+1670*q^60+5089*q^54+12278*q^45+20*q^71+2*q^74+18302*q^37+
168*q^67+21429*q^29+11484*q^46+373*q^65+64*q^69+16977*q^39+1298*q^61+15508*q^41
+20067*q^34+8174*q^50+530*q^64+17706*q^38+20477*q^33+3752*q^56+34*q^2+106*q^68+
21069*q^26+86*q^3+185*q^4+350*q^5+594*q^6+20901*q^32+936*q^7+1389*q^8+1972*q^9+
986*q^62+2701*q^10+3565*q^11+4571*q^12+5713*q^13+7005*q^14+8396*q^15+9872*q^16+
254*q^66+37*q^70+13918*q^43+8970*q^49+21464*q^28+11380*q^17+12912*q^18+14365*q^
19+15770*q^20+21304*q^27+10*q^72+16301*q^40+5*q^73+730*q^63+21147*q^31+q^75+
19490*q^35+9817*q^48+2103*q^59+18974*q^36+10616*q^47)]


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

[1, 2, 6, 22, 86, 321, 1085, 3266, 8797, 21478, 48206, 100728, 198046, 369617,

    659505]

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

     is given by

[1, 1, 3, 11, 43, 161, 543, 1635, 4398, 10733, 24084, 50303, 98922, 184561,

    329410]

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

     is given by

[0, 1, 3, 11, 43, 160, 542, 1631, 4399, 10745, 24122, 50425, 99124, 185056,

    330095]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 43, 160, 542, 1635, 4398, 10745, 24122, 50303, 98922,

    185056, 330095]

 ODD:, [0, 1, 3, 11, 43, 161, 543, 1631, 4399, 10733, 24084, 50425, 99124,

    184561, 329410]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.954545455, 4.825581395, 7.052959502,

    9.605529954, 12.47856705, 15.68318745, 19.23312226, 23.13977513,

    27.41165316, 32.05494683, 37.07418219, 42.47271969]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.330661924, 1.685725712, 2.109243629,

    2.639153407, 3.293324898, 4.078016649, 4.993476314, 6.037963324,

    7.209443266, 8.506092232, 9.926384331, 11.46906349]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.77066, 2.84167, 4.44891, 6.96513, 10.8460,

    16.6302, 24.9348, 36.4570, 51.9761, 72.3536, 98.5331, 131.539]

Skewness: , [Float(undefined), 0., 0., -0.2640282852, -0.4094315924,

    -0.4063263482, -0.3530891028, -0.3016090393, -0.2671023723, -0.2484320322,

    -0.2406757668, -0.2395929678, -0.2423054329, -0.2470479353, -0.2527689502]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.436083907,

    2.869318438, 3.021371379, 2.977323461, 2.876252311, 2.781064224,

    2.706820364, 2.652167574, 2.612519300, 2.583837207, 2.563063642,

    2.548087172]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+2*q^5+q^6, 1+4*q+9*q^2+15*q^3
+18*q^4+16*q^5+12*q^6+8*q^7+4*q^8+2*q^9+q^10, 1+5*q+14*q^2+29*q^3+46*q^4+58*q^5
+60*q^6+54*q^7+44*q^8+32*q^9+22*q^10+14*q^11+8*q^12+4*q^13+2*q^14+q^15, 1+6*q+
20*q^2+49*q^3+94*q^4+147*q^5+193*q^6+218*q^7+220*q^8+204*q^9+175*q^10+142*q^11+
110*q^12+80*q^13+56*q^14+38*q^15+24*q^16+14*q^17+8*q^18+4*q^19+2*q^20+q^21, 1+7
*q+27*q^2+76*q^3+169*q^4+310*q^5+483*q^6+653*q^7+783*q^8+854*q^9+864*q^10+820*q
^11+741*q^12+642*q^13+534*q^14+431*q^15+338*q^16+256*q^17+188*q^18+134*q^19+92*
q^20+62*q^21+40*q^22+24*q^23+14*q^24+8*q^25+4*q^26+2*q^27+q^28, 1+8*q+35*q^2+
111*q^3+279*q^4+582*q^5+1038*q^6+1616*q^7+2236*q^8+2799*q^9+3227*q^10+3478*q^11
+3544*q^12+3450*q^13+3232*q^14+2928*q^15+2580*q^16+2218*q^17+1862*q^18+1528*q^
19+1228*q^20+967*q^21+746*q^22+564*q^23+416*q^24+300*q^25+212*q^26+146*q^27+98*
q^28+64*q^29+40*q^30+24*q^31+14*q^32+8*q^33+4*q^34+2*q^35+q^36, 1+9*q+44*q^2+
155*q^3+433*q^4+1007*q^5+2010*q^6+3516*q^7+5480*q^8+7724*q^9+9986*q^10+12004*q^
11+13577*q^12+14589*q^13+15020*q^14+14917*q^15+14364*q^16+13475*q^17+12358*q^18
+11102*q^19+9789*q^20+8486*q^21+7238*q^22+6080*q^23+5032*q^24+4100*q^25+3292*q^
26+2606*q^27+2031*q^28+1560*q^29+1180*q^30+876*q^31+640*q^32+460*q^33+324*q^34+
224*q^35+152*q^36+100*q^37+64*q^38+40*q^39+24*q^40+14*q^41+8*q^42+4*q^43+2*q^44
+q^45, 1+10*q+54*q^2+209*q^3+641*q^4+1639*q^5+3605*q^6+6967*q^7+12022*q^8+18774
*q^9+26859*q^10+35610*q^11+44228*q^12+51956*q^13+58204*q^14+62627*q^15+65111*q^
16+65728*q^17+64700*q^18+62312*q^19+58863*q^20+54659*q^21+49978*q^22+45055*q^23
+40090*q^24+35234*q^25+30596*q^26+26268*q^27+22304*q^28+18730*q^29+15562*q^30+
12790*q^31+10394*q^32+8356*q^33+6644*q^34+5222*q^35+4059*q^36+3118*q^37+2364*q^
38+1770*q^39+1308*q^40+952*q^41+684*q^42+484*q^43+336*q^44+230*q^45+154*q^46+
100*q^47+64*q^48+40*q^49+24*q^50+14*q^51+8*q^52+4*q^53+2*q^54+q^55, 1+11*q+65*q
^2+274*q^3+914*q^4+2543*q^5+6094*q^6+12853*q^7+24243*q^8+41422*q^9+64829*q^10+
93888*q^11+127022*q^12+161960*q^13+196176*q^14+227315*q^15+253522*q^16+273586*q
^17+286949*q^18+293644*q^19+294124*q^20+289110*q^21+279494*q^22+266224*q^23+
250222*q^24+232344*q^25+213342*q^26+193852*q^27+174417*q^28+155472*q^29+137344*
q^30+120280*q^31+104444*q^32+89932*q^33+76798*q^34+65048*q^35+54646*q^36+45534*
q^37+37632*q^38+30842*q^39+25064*q^40+20196*q^41+16132*q^42+12772*q^43+10022*q^
44+7791*q^45+5998*q^46+4572*q^47+3448*q^48+2572*q^49+1898*q^50+1384*q^51+996*q^
52+708*q^53+496*q^54+342*q^55+232*q^56+154*q^57+100*q^58+64*q^59+40*q^60+24*q^
61+14*q^62+8*q^63+4*q^64+2*q^65+q^66, 1+12*q+77*q^2+351*q^3+1264*q^4+3796*q^5+
9825*q^6+22405*q^7+45744*q^8+84677*q^9+143619*q^10+225276*q^11+329597*q^12+
453388*q^13+590707*q^14+733830*q^15+874454*q^16+1004846*q^17+1118666*q^18+
1211408*q^19+1280530*q^20+1325249*q^21+1346205*q^22+1345143*q^23+1324538*q^24+
1287256*q^25+1236320*q^26+1174701*q^27+1105180*q^28+1030318*q^29+952381*q^30+
873298*q^31+794704*q^32+717924*q^33+643996*q^34+573742*q^35+507755*q^36+446418*
q^37+389969*q^38+338492*q^39+291944*q^40+250208*q^41+213088*q^42+180324*q^43+
151632*q^44+126696*q^45+105176*q^46+86744*q^47+71070*q^48+57832*q^49+46740*q^50
+37514*q^51+29892*q^52+23646*q^53+18566*q^54+14463*q^55+11178*q^56+8568*q^57+
6508*q^58+4900*q^59+3656*q^60+2700*q^61+1974*q^62+1428*q^63+1020*q^64+720*q^65+
502*q^66+344*q^67+232*q^68+154*q^69+100*q^70+64*q^71+40*q^72+24*q^73+14*q^74+8*
q^75+4*q^76+2*q^77+q^78, 1+13*q+90*q^2+441*q^3+1704*q^4+5488*q^5+15236*q^6+
37291*q^7+81782*q^8+162728*q^9+296794*q^10+500558*q^11+786836*q^12+1161172*q^13
+1619650*q^14+2148677*q^15+2726644*q^16+3326873*q^17+3921076*q^18+4482511*q^19+
4988371*q^20+5421262*q^21+5769660*q^22+6027670*q^23+6194420*q^24+6273104*q^25+
6269941*q^26+6193237*q^27+6052497*q^28+5857728*q^29+5618984*q^30+5345925*q^31+
5047518*q^32+4731930*q^33+4406383*q^34+4077146*q^35+3749618*q^36+3428296*q^37+
3116836*q^38+2818168*q^39+2534506*q^40+2267440*q^41+2018055*q^42+1786952*q^43+
1574326*q^44+1380065*q^45+1203756*q^46+1044751*q^47+902254*q^48+775316*q^49+
662902*q^50+563950*q^51+477348*q^52+401984*q^53+336784*q^54+280694*q^55+232710*
q^56+191902*q^57+157390*q^58+128364*q^59+104104*q^60+83944*q^61+67286*q^62+
53612*q^63+42452*q^64+33398*q^65+26105*q^66+20266*q^67+15620*q^68+11952*q^69+
9076*q^70+6836*q^71+5108*q^72+3784*q^73+2776*q^74+2018*q^75+1452*q^76+1032*q^77
+726*q^78+504*q^79+344*q^80+232*q^81+154*q^82+100*q^83+64*q^84+40*q^85+24*q^86+
14*q^87+8*q^88+4*q^89+2*q^90+q^91, 1+14*q+104*q^2+545*q^3+2248*q^4+7723*q^5+
22869*q^6+59720*q^7+139810*q^8+297127*q^9+579034*q^10+1043542*q^11+1752251*q^12
+2759910*q^13+4103101*q^14+5791144*q^15+7801615*q^16+10081342*q^17+12552248*q^
18+15120390*q^19+17686061*q^20+20153157*q^21+22436630*q^22+24467299*q^23+
26194194*q^24+27584947*q^25+28624512*q^26+29312862*q^27+29662244*q^28+29694234*
q^29+29436993*q^30+28922940*q^31+28186660*q^32+27263200*q^33+26186866*q^34+
24990254*q^35+23703643*q^36+22354710*q^37+20968264*q^38+19566160*q^39+18167396*
q^40+16788160*q^41+15441980*q^42+14139976*q^43+12891024*q^44+11701950*q^45+
10577784*q^46+9521920*q^47+8536290*q^48+7621604*q^49+6777500*q^50+6002714*q^51+
5295276*q^52+4652616*q^53+4071686*q^54+3549111*q^55+3081276*q^56+2664407*q^57+
2294678*q^58+1968268*q^59+1681408*q^60+1430458*q^61+1211926*q^62+1022480*q^63+
858996*q^64+718560*q^65+598468*q^66+496246*q^67+409640*q^68+336602*q^69+275296*
q^70+224088*q^71+181520*q^72+146308*q^73+117330*q^74+93600*q^75+74268*q^76+
58606*q^77+45983*q^78+35866*q^79+27806*q^80+21420*q^81+16392*q^82+12460*q^83+
9404*q^84+7044*q^85+5236*q^86+3860*q^87+2820*q^88+2042*q^89+1464*q^90+1038*q^91
+728*q^92+504*q^93+344*q^94+232*q^95+154*q^96+100*q^97+64*q^98+40*q^99+24*q^100
+14*q^101+8*q^102+4*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+4*q^2+2*q+1, q^10+4*q^9+9*q^8+
15*q^7+18*q^6+16*q^5+12*q^4+8*q^3+4*q^2+2*q+1, q^15+5*q^14+14*q^13+29*q^12+46*q
^11+58*q^10+60*q^9+54*q^8+44*q^7+32*q^6+22*q^5+14*q^4+8*q^3+4*q^2+2*q+1, q^21+6
*q^20+20*q^19+49*q^18+94*q^17+147*q^16+193*q^15+218*q^14+220*q^13+204*q^12+175*
q^11+142*q^10+110*q^9+80*q^8+56*q^7+38*q^6+24*q^5+14*q^4+8*q^3+4*q^2+2*q+1, 1+
76*q^25+483*q^22+169*q^24+653*q^21+310*q^23+2*q+4*q^2+27*q^26+8*q^3+14*q^4+24*q
^5+40*q^6+62*q^7+92*q^8+134*q^9+188*q^10+256*q^11+338*q^12+431*q^13+534*q^14+
642*q^15+741*q^16+q^28+820*q^17+864*q^18+854*q^19+783*q^20+7*q^27, 1+3478*q^25+
3232*q^22+3544*q^24+2928*q^21+3450*q^23+2*q+1038*q^30+1616*q^29+35*q^34+111*q^
33+4*q^2+3227*q^26+8*q^3+14*q^4+24*q^5+40*q^6+279*q^32+64*q^7+98*q^8+146*q^9+
212*q^10+300*q^11+416*q^12+564*q^13+746*q^14+967*q^15+1228*q^16+2236*q^28+1528*
q^17+1862*q^18+2218*q^19+2580*q^20+2799*q^27+582*q^31+8*q^35+q^36, 1+9789*q^25+
155*q^42+6080*q^22+8486*q^24+5032*q^21+7238*q^23+2*q+9*q^44+14917*q^30+q^45+
5480*q^37+14364*q^29+2010*q^39+433*q^41+12004*q^34+3516*q^38+13577*q^33+4*q^2+
11102*q^26+8*q^3+14*q^4+24*q^5+40*q^6+14589*q^32+64*q^7+100*q^8+152*q^9+224*q^
10+324*q^11+460*q^12+640*q^13+876*q^14+1180*q^15+1560*q^16+44*q^43+13475*q^28+
2031*q^17+2606*q^18+3292*q^19+4100*q^20+12358*q^27+1007*q^40+15020*q^31+9986*q^
35+7724*q^36, 1+15562*q^25+51956*q^42+8356*q^22+12790*q^24+6644*q^21+10394*q^23
+2*q+641*q^51+35610*q^44+54*q^53+35234*q^30+209*q^52+q^55+10*q^54+26859*q^45+
64700*q^37+30596*q^29+18774*q^46+65111*q^39+58204*q^41+54659*q^34+1639*q^50+
65728*q^38+49978*q^33+4*q^2+18730*q^26+8*q^3+14*q^4+24*q^5+40*q^6+45055*q^32+64
*q^7+100*q^8+154*q^9+230*q^10+336*q^11+484*q^12+684*q^13+952*q^14+1308*q^15+
1770*q^16+44228*q^43+3605*q^49+26268*q^28+2364*q^17+3118*q^18+4059*q^19+5222*q^
20+22304*q^27+62627*q^40+40090*q^31+58863*q^35+6967*q^48+62312*q^36+12022*q^47,
1+20196*q^25+250222*q^42+10022*q^22+16132*q^24+7791*q^21+12772*q^23+2*q+227315*
q^51+279494*q^44+161960*q^53+41422*q^57+54646*q^30+196176*q^52+24243*q^58+93888
*q^55+6094*q^60+127022*q^54+289110*q^45+155472*q^37+45534*q^29+294124*q^46+11*q
^65+193852*q^39+2543*q^61+232344*q^41+104444*q^34+253522*q^50+65*q^64+174417*q^
38+89932*q^33+64829*q^56+4*q^2+25064*q^26+8*q^3+14*q^4+24*q^5+40*q^6+76798*q^32
+64*q^7+100*q^8+154*q^9+914*q^62+232*q^10+342*q^11+496*q^12+708*q^13+996*q^14+
1384*q^15+1898*q^16+q^66+266224*q^43+273586*q^49+37632*q^28+2572*q^17+3448*q^18
+4572*q^19+5998*q^20+30842*q^27+213342*q^40+274*q^63+65048*q^31+120280*q^35+
286949*q^48+12853*q^59+137344*q^36+293644*q^47, 1+23646*q^25+507755*q^42+11178*
q^22+18566*q^24+8568*q^21+14463*q^23+2*q+1174701*q^51+643996*q^44+1287256*q^53+
1325249*q^57+71070*q^30+1236320*q^52+1280530*q^58+1345143*q^55+1118666*q^60+
1324538*q^54+717924*q^45+22405*q^71+1264*q^74+250208*q^37+225276*q^67+57832*q^
29+794704*q^46+453388*q^65+84677*q^69+338492*q^39+1004846*q^61+446418*q^41+
151632*q^34+1105180*q^50+590707*q^64+291944*q^38+126696*q^33+1346205*q^56+4*q^2
+143619*q^68+29892*q^26+8*q^3+14*q^4+24*q^5+40*q^6+105176*q^32+64*q^7+100*q^8+
154*q^9+874454*q^62+232*q^10+344*q^11+502*q^12+77*q^76+720*q^13+1020*q^14+1428*
q^15+1974*q^16+329597*q^66+45744*q^70+12*q^77+q^78+573742*q^43+1030318*q^49+
46740*q^28+2700*q^17+3656*q^18+4900*q^19+6508*q^20+37514*q^27+9825*q^72+389969*
q^40+3796*q^73+733830*q^63+86744*q^31+351*q^75+180324*q^35+952381*q^48+1211408*
q^59+213088*q^36+873298*q^47, 1+26105*q^25+775316*q^42+11952*q^22+13*q^90+20266
*q^24+9076*q^21+15620*q^23+2*q+2534506*q^51+1044751*q^44+3116836*q^53+4406383*q
^57+83944*q^30+2818168*q^52+4731930*q^58+3749618*q^55+5345925*q^60+q^91+3428296
*q^54+1203756*q^45+4988371*q^71+3326873*q^74+336784*q^37+6194420*q^67+67286*q^
29+1380065*q^46+6269941*q^65+5769660*q^69+477348*q^39+5618984*q^61+662902*q^41+
191902*q^34+786836*q^79+2267440*q^50+6193237*q^64+401984*q^38+157390*q^33+
4077146*q^56+4*q^2+6027670*q^68+33398*q^26+8*q^3+14*q^4+24*q^5+40*q^6+128364*q^
32+64*q^7+100*q^8+154*q^9+5857728*q^62+232*q^10+344*q^11+504*q^12+2148677*q^76+
726*q^13+1032*q^14+1452*q^15+2018*q^16+6273104*q^66+5421262*q^70+1619650*q^77+
1161172*q^78+902254*q^43+2018055*q^49+53612*q^28+2776*q^17+3784*q^18+5108*q^19+
6836*q^20+42452*q^27+4482511*q^72+563950*q^40+3921076*q^73+6052497*q^63+104104*
q^31+2726644*q^75+232710*q^35+500558*q^80+1786952*q^48+81782*q^83+296794*q^81+
15236*q^85+5488*q^86+1704*q^87+441*q^88+90*q^89+37291*q^84+162728*q^82+5047518*
q^59+280694*q^36+1574326*q^47, 1+27806*q^25+1022480*q^42+12460*q^22+5791144*q^
90+21420*q^24+9404*q^21+16392*q^23+2*q+139810*q^97+4071686*q^51+1430458*q^44+
5295276*q^53+8536290*q^57+93600*q^30+1043542*q^94+4652616*q^52+9521920*q^58+
6777500*q^55+11701950*q^60+4103101*q^91+6002714*q^54+1681408*q^45+26186866*q^71
+28922940*q^74+297127*q^96+59720*q^98+409640*q^37+20968264*q^67+74268*q^29+
1968268*q^46+1752251*q^93+22869*q^99+18167396*q^65+23703643*q^69+598468*q^39+
12891024*q^61+858996*q^41+224088*q^34+28624512*q^79+3549111*q^50+16788160*q^64+
7723*q^100+496246*q^38+181520*q^33+7621604*q^56+4*q^2+22354710*q^68+35866*q^26+
8*q^3+14*q^4+24*q^5+40*q^6+146308*q^32+64*q^7+100*q^8+154*q^9+579034*q^95+
14139976*q^62+2248*q^101+232*q^10+344*q^11+504*q^12+29694234*q^76+728*q^13+1038
*q^14+1464*q^15+2042*q^16+19566160*q^66+24990254*q^70+29662244*q^77+29312862*q^
78+1211926*q^43+545*q^102+3081276*q^49+58606*q^28+2820*q^17+3860*q^18+5236*q^19
+7044*q^20+45983*q^27+27263200*q^72+718560*q^40+28186660*q^73+104*q^103+
15441980*q^63+117330*q^31+29436993*q^75+275296*q^35+14*q^104+27584947*q^80+
2664407*q^48+q^105+22436630*q^83+26194194*q^81+17686061*q^85+15120390*q^86+
12552248*q^87+10081342*q^88+7801615*q^89+20153157*q^84+24467299*q^82+10577784*q
^59+336602*q^36+2759910*q^92+2294678*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.863636364, 4.522222222, 6.439086294,

    8.589700997, 10.95571395, 13.52260376, 16.27847432, 19.21333638,

    22.31864215, 25.58696662, 29.01178033, 32.58728202]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.455255509, 2.012338483, 2.629893587,

    3.304663676, 4.032965000, 4.811473059, 5.637309898, 6.507994624,

    7.421376085, 8.375573109, 9.368925986, 10.39995810]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.11777, 4.04951, 6.91634, 10.9208, 16.2648,

    23.1503, 31.7793, 42.3540, 55.0768, 70.1502, 87.7768, 108.159]

Skewness: , [Float(undefined), 0., 0., 0.1491970460, 0.2825172660, 0.3784325686,

    0.4457770208, 0.4938825271, 0.5291492000, 0.5556811268, 0.5761218138,

    0.5922021498, 0.6050926652, 0.6155906127, 0.6242690131]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.547029639,

    2.797768857, 2.954003510, 3.062603889, 3.142342608, 3.202898761,

    3.250056736, 3.287567410, 3.317901092, 3.342843118, 3.363593122,

    3.381103729]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+5*q^4+2*q^5+q^6, 1+4*q+7*q^2+11*q^3
+15*q^4+17*q^5+15*q^6+10*q^7+5*q^8+2*q^9+q^10, 1+5*q+11*q^2+19*q^3+29*q^4+39*q^
5+49*q^6+52*q^7+50*q^8+41*q^9+30*q^10+19*q^11+10*q^12+5*q^13+2*q^14+q^15, 1+6*q
+16*q^2+31*q^3+52*q^4+76*q^5+105*q^6+134*q^7+155*q^8+167*q^9+167*q^10+154*q^11+
135*q^12+107*q^13+77*q^14+53*q^15+34*q^16+19*q^17+10*q^18+5*q^19+2*q^20+q^21, 1
+7*q+22*q^2+48*q^3+88*q^4+140*q^5+205*q^6+282*q^7+363*q^8+441*q^9+502*q^10+544*
q^11+558*q^12+549*q^13+516*q^14+465*q^15+400*q^16+325*q^17+251*q^18+186*q^19+
131*q^20+89*q^21+57*q^22+34*q^23+19*q^24+10*q^25+5*q^26+2*q^27+q^28, 1+8*q+29*q
^2+71*q^3+142*q^4+245*q^5+382*q^6+556*q^7+759*q^8+988*q^9+1225*q^10+1449*q^11+
1640*q^12+1781*q^13+1868*q^14+1897*q^15+1866*q^16+1787*q^17+1657*q^18+1497*q^19
+1307*q^20+1104*q^21+904*q^22+720*q^23+554*q^24+412*q^25+298*q^26+210*q^27+143*
q^28+93*q^29+57*q^30+34*q^31+19*q^32+10*q^33+5*q^34+2*q^35+q^36, 1+9*q+37*q^2+
101*q^3+220*q^4+410*q^5+682*q^6+1049*q^7+1509*q^8+2060*q^9+2693*q^10+3382*q^11+
4091*q^12+4769*q^13+5373*q^14+5869*q^15+6242*q^16+6464*q^17+6544*q^18+6486*q^19
+6295*q^20+5986*q^21+5579*q^22+5090*q^23+4552*q^24+3983*q^25+3412*q^26+2860*q^
27+2351*q^28+1885*q^29+1485*q^30+1145*q^31+863*q^32+637*q^33+459*q^34+322*q^35+
222*q^36+147*q^37+93*q^38+57*q^39+34*q^40+19*q^41+10*q^42+5*q^43+2*q^44+q^45, 1
+10*q+46*q^2+139*q^3+329*q^4+660*q^5+1171*q^6+1903*q^7+2881*q^8+4117*q^9+5617*q
^10+7363*q^11+9327*q^12+11437*q^13+13603*q^14+15716*q^15+17676*q^16+19398*q^17+
20808*q^18+21868*q^19+22561*q^20+22876*q^21+22824*q^22+22421*q^23+21700*q^24+
20696*q^25+19456*q^26+18008*q^27+16436*q^28+14773*q^29+13098*q^30+11442*q^31+
9845*q^32+8338*q^33+6962*q^34+5715*q^35+4634*q^36+3701*q^37+2915*q^38+2255*q^39
+1716*q^40+1284*q^41+946*q^42+684*q^43+483*q^44+334*q^45+226*q^46+147*q^47+93*q
^48+57*q^49+34*q^50+19*q^51+10*q^52+5*q^53+2*q^54+q^55, 1+11*q+56*q^2+186*q^3+
477*q^4+1027*q^5+1941*q^6+3332*q^7+5303*q^8+7931*q^9+11279*q^10+15369*q^11+
20203*q^12+25736*q^13+31861*q^14+38415*q^15+45175*q^16+51888*q^17+58297*q^18+
64176*q^19+69329*q^20+73630*q^21+76971*q^22+79326*q^23+80647*q^24+80971*q^25+
80336*q^26+78788*q^27+76413*q^28+73295*q^29+69534*q^30+65255*q^31+60587*q^32+
55651*q^33+50588*q^34+45512*q^35+40509*q^36+35675*q^37+31061*q^38+26751*q^39+
22781*q^40+19207*q^41+16013*q^42+13211*q^43+10775*q^44+8697*q^45+6933*q^46+5471
*q^47+4257*q^48+3275*q^49+2482*q^50+1855*q^51+1367*q^52+993*q^53+708*q^54+495*q
^55+338*q^56+226*q^57+147*q^58+93*q^59+57*q^60+34*q^61+19*q^62+10*q^63+5*q^64+2
*q^65+q^66, 1+12*q+67*q^2+243*q^3+673*q^4+1551*q^5+3117*q^6+5649*q^7+9443*q^8+
14773*q^9+21891*q^10+30989*q^11+42199*q^12+55589*q^13+71125*q^14+88678*q^15+
107972*q^16+128596*q^17+149990*q^18+171543*q^19+192602*q^20+212554*q^21+230864*
q^22+247091*q^23+260903*q^24+272082*q^25+280479*q^26+286039*q^27+288783*q^28+
288772*q^29+286103*q^30+280973*q^31+273538*q^32+264060*q^33+252760*q^34+239964*
q^35+225952*q^36+211099*q^37+195639*q^38+179909*q^39+164147*q^40+148588*q^41+
133421*q^42+118846*q^43+104980*q^44+92001*q^45+79968*q^46+68966*q^47+58995*q^48
+50072*q^49+42124*q^50+35162*q^51+29094*q^52+23882*q^53+19432*q^54+15678*q^55+
12522*q^56+9915*q^57+7762*q^58+6017*q^59+4613*q^60+3502*q^61+2621*q^62+1938*q^
63+1414*q^64+1017*q^65+720*q^66+499*q^67+338*q^68+226*q^69+147*q^70+93*q^71+57*
q^72+34*q^73+19*q^74+10*q^75+5*q^76+2*q^77+q^78, 1+13*q+79*q^2+311*q^3+927*q^4+
2281*q^5+4865*q^6+9300*q^7+16315*q^8+26682*q^9+41181*q^10+60539*q^11+85377*q^12
+116201*q^13+153332*q^14+196921*q^15+246886*q^16+302864*q^17+364145*q^18+429669
*q^19+498084*q^20+567821*q^21+637208*q^22+704589*q^23+768432*q^24+827386*q^25+
880380*q^26+926509*q^27+965193*q^28+996020*q^29+1018771*q^30+1033360*q^31+
1039905*q^32+1038591*q^33+1029761*q^34+1013842*q^35+991326*q^36+962832*q^37+
929002*q^38+890568*q^39+848344*q^40+803101*q^41+755624*q^42+706661*q^43+656884*
q^44+606888*q^45+557323*q^46+508655*q^47+461443*q^48+416055*q^49+372845*q^50+
332056*q^51+293949*q^52+258602*q^53+226132*q^54+196501*q^55+169670*q^56+145581*
q^57+124149*q^58+105190*q^59+88581*q^60+74102*q^61+61577*q^62+50800*q^63+41631*
q^64+33856*q^65+27348*q^66+21915*q^67+17425*q^68+13732*q^69+10734*q^70+8304*q^
71+6373*q^72+4840*q^73+3641*q^74+2704*q^75+1985*q^76+1438*q^77+1029*q^78+724*q^
79+499*q^80+338*q^81+226*q^82+147*q^83+93*q^84+57*q^85+34*q^86+19*q^87+10*q^88+
5*q^89+2*q^90+q^91, 1+14*q+92*q^2+391*q^3+1250*q^4+3276*q^5+7401*q^6+14906*q^7+
27420*q^8+46845*q^9+75270*q^10+114862*q^11+167719*q^12+235794*q^13+320749*q^14+
423896*q^15+546145*q^16+687903*q^17+848973*q^18+1028394*q^19+1224380*q^20+
1434271*q^21+1654683*q^22+1881576*q^23+2110590*q^24+2337223*q^25+2557146*q^26+
2766340*q^27+2961271*q^28+3139011*q^29+3297197*q^30+3434030*q^31+3548241*q^32+
3638947*q^33+3705763*q^34+3748663*q^35+3767942*q^36+3764199*q^37+3738361*q^38+
3691469*q^39+3624985*q^40+3540393*q^41+3439515*q^42+3324197*q^43+3196537*q^44+
3058394*q^45+2911996*q^46+2759162*q^47+2601904*q^48+2441865*q^49+2280822*q^50+
2120190*q^51+1961548*q^52+1806117*q^53+1655183*q^54+1509676*q^55+1370533*q^56+
1238298*q^57+1113558*q^58+996549*q^59+887619*q^60+786725*q^61+693969*q^62+
609129*q^63+532088*q^64+462513*q^65+400103*q^66+344398*q^67+294995*q^68+251361*
q^69+213073*q^70+179637*q^71+150653*q^72+125632*q^73+104207*q^74+85922*q^75+
70440*q^76+57372*q^77+46444*q^78+37338*q^79+29831*q^80+23654*q^81+18625*q^82+
14547*q^83+11276*q^84+8660*q^85+6600*q^86+4979*q^87+3724*q^88+2751*q^89+2009*q^
90+1450*q^91+1033*q^92+724*q^93+499*q^94+338*q^95+226*q^96+147*q^97+93*q^98+57*
q^99+34*q^100+19*q^101+10*q^102+5*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+6*q^3+5*q^2+2*q+1, q^10+4*q^9+7*q^8+
11*q^7+15*q^6+17*q^5+15*q^4+10*q^3+5*q^2+2*q+1, q^15+5*q^14+11*q^13+19*q^12+29*
q^11+39*q^10+49*q^9+52*q^8+50*q^7+41*q^6+30*q^5+19*q^4+10*q^3+5*q^2+2*q+1, q^21
+6*q^20+16*q^19+31*q^18+52*q^17+76*q^16+105*q^15+134*q^14+155*q^13+167*q^12+167
*q^11+154*q^10+135*q^9+107*q^8+77*q^7+53*q^6+34*q^5+19*q^4+10*q^3+5*q^2+2*q+1,
1+48*q^25+205*q^22+88*q^24+282*q^21+140*q^23+2*q+5*q^2+22*q^26+10*q^3+19*q^4+34
*q^5+57*q^6+89*q^7+131*q^8+186*q^9+251*q^10+325*q^11+400*q^12+465*q^13+516*q^14
+549*q^15+558*q^16+q^28+544*q^17+502*q^18+441*q^19+363*q^20+7*q^27, 1+1449*q^25
+1868*q^22+1640*q^24+1897*q^21+1781*q^23+2*q+382*q^30+556*q^29+29*q^34+71*q^33+
5*q^2+1225*q^26+10*q^3+19*q^4+34*q^5+57*q^6+142*q^32+93*q^7+143*q^8+210*q^9+298
*q^10+412*q^11+554*q^12+720*q^13+904*q^14+1104*q^15+1307*q^16+759*q^28+1497*q^
17+1657*q^18+1787*q^19+1866*q^20+988*q^27+245*q^31+8*q^35+q^36, 1+6295*q^25+101
*q^42+5090*q^22+5986*q^24+4552*q^21+5579*q^23+2*q+9*q^44+5869*q^30+q^45+1509*q^
37+6242*q^29+682*q^39+220*q^41+3382*q^34+1049*q^38+4091*q^33+5*q^2+6486*q^26+10
*q^3+19*q^4+34*q^5+57*q^6+4769*q^32+93*q^7+147*q^8+222*q^9+322*q^10+459*q^11+
637*q^12+863*q^13+1145*q^14+1485*q^15+1885*q^16+37*q^43+6464*q^28+2351*q^17+
2860*q^18+3412*q^19+3983*q^20+6544*q^27+410*q^40+5373*q^31+2693*q^35+2060*q^36,
1+13098*q^25+11437*q^42+8338*q^22+11442*q^24+6962*q^21+9845*q^23+2*q+329*q^51+
7363*q^44+46*q^53+20696*q^30+139*q^52+q^55+10*q^54+5617*q^45+20808*q^37+19456*q
^29+4117*q^46+17676*q^39+13603*q^41+22876*q^34+660*q^50+19398*q^38+22824*q^33+5
*q^2+14773*q^26+10*q^3+19*q^4+34*q^5+57*q^6+22421*q^32+93*q^7+147*q^8+226*q^9+
334*q^10+483*q^11+684*q^12+946*q^13+1284*q^14+1716*q^15+2255*q^16+9327*q^43+
1171*q^49+18008*q^28+2915*q^17+3701*q^18+4634*q^19+5715*q^20+16436*q^27+15716*q
^40+21700*q^31+22561*q^35+1903*q^48+21868*q^36+2881*q^47, 1+19207*q^25+80647*q^
42+10775*q^22+16013*q^24+8697*q^21+13211*q^23+2*q+38415*q^51+76971*q^44+25736*q
^53+7931*q^57+40509*q^30+31861*q^52+5303*q^58+15369*q^55+1941*q^60+20203*q^54+
73630*q^45+73295*q^37+35675*q^29+69329*q^46+11*q^65+78788*q^39+1027*q^61+80971*
q^41+60587*q^34+45175*q^50+56*q^64+76413*q^38+55651*q^33+11279*q^56+5*q^2+22781
*q^26+10*q^3+19*q^4+34*q^5+57*q^6+50588*q^32+93*q^7+147*q^8+226*q^9+477*q^62+
338*q^10+495*q^11+708*q^12+993*q^13+1367*q^14+1855*q^15+2482*q^16+q^66+79326*q^
43+51888*q^49+31061*q^28+3275*q^17+4257*q^18+5471*q^19+6933*q^20+26751*q^27+
80336*q^40+186*q^63+45512*q^31+65255*q^35+58297*q^48+3332*q^59+69534*q^36+64176
*q^47, 1+23882*q^25+225952*q^42+12522*q^22+19432*q^24+9915*q^21+15678*q^23+2*q+
286039*q^51+252760*q^44+272082*q^53+212554*q^57+58995*q^30+280479*q^52+192602*q
^58+247091*q^55+149990*q^60+260903*q^54+264060*q^45+5649*q^71+673*q^74+148588*q
^37+30989*q^67+50072*q^29+273538*q^46+55589*q^65+14773*q^69+179909*q^39+128596*
q^61+211099*q^41+104980*q^34+288783*q^50+71125*q^64+164147*q^38+92001*q^33+
230864*q^56+5*q^2+21891*q^68+29094*q^26+10*q^3+19*q^4+34*q^5+57*q^6+79968*q^32+
93*q^7+147*q^8+226*q^9+107972*q^62+338*q^10+499*q^11+720*q^12+67*q^76+1017*q^13
+1414*q^14+1938*q^15+2621*q^16+42199*q^66+9443*q^70+12*q^77+q^78+239964*q^43+
288772*q^49+42124*q^28+3502*q^17+4613*q^18+6017*q^19+7762*q^20+35162*q^27+3117*
q^72+195639*q^40+1551*q^73+88678*q^63+68966*q^31+243*q^75+118846*q^35+286103*q^
48+171543*q^59+133421*q^36+280973*q^47, 1+27348*q^25+416055*q^42+13732*q^22+13*
q^90+21915*q^24+10734*q^21+17425*q^23+2*q+848344*q^51+508655*q^44+929002*q^53+
1029761*q^57+74102*q^30+890568*q^52+1038591*q^58+991326*q^55+1033360*q^60+q^91+
962832*q^54+557323*q^45+498084*q^71+302864*q^74+226132*q^37+768432*q^67+61577*q
^29+606888*q^46+880380*q^65+637208*q^69+293949*q^39+1018771*q^61+372845*q^41+
145581*q^34+85377*q^79+803101*q^50+926509*q^64+258602*q^38+124149*q^33+1013842*
q^56+5*q^2+704589*q^68+33856*q^26+10*q^3+19*q^4+34*q^5+57*q^6+105190*q^32+93*q^
7+147*q^8+226*q^9+996020*q^62+338*q^10+499*q^11+724*q^12+196921*q^76+1029*q^13+
1438*q^14+1985*q^15+2704*q^16+827386*q^66+567821*q^70+153332*q^77+116201*q^78+
461443*q^43+755624*q^49+50800*q^28+3641*q^17+4840*q^18+6373*q^19+8304*q^20+
41631*q^27+429669*q^72+332056*q^40+364145*q^73+965193*q^63+88581*q^31+246886*q^
75+169670*q^35+60539*q^80+706661*q^48+16315*q^83+41181*q^81+4865*q^85+2281*q^86
+927*q^87+311*q^88+79*q^89+9300*q^84+26682*q^82+1039905*q^59+196501*q^36+656884
*q^47, 1+29831*q^25+609129*q^42+14547*q^22+423896*q^90+23654*q^24+11276*q^21+
18625*q^23+2*q+27420*q^97+1655183*q^51+786725*q^44+1961548*q^53+2601904*q^57+
85922*q^30+114862*q^94+1806117*q^52+2759162*q^58+2280822*q^55+3058394*q^60+
320749*q^91+2120190*q^54+887619*q^45+3705763*q^71+3434030*q^74+46845*q^96+14906
*q^98+294995*q^37+3738361*q^67+70440*q^29+996549*q^46+167719*q^93+7401*q^99+
3624985*q^65+3767942*q^69+400103*q^39+3196537*q^61+532088*q^41+179637*q^34+
2557146*q^79+1509676*q^50+3540393*q^64+3276*q^100+344398*q^38+150653*q^33+
2441865*q^56+5*q^2+3764199*q^68+37338*q^26+10*q^3+19*q^4+34*q^5+57*q^6+125632*q
^32+93*q^7+147*q^8+226*q^9+75270*q^95+3324197*q^62+1250*q^101+338*q^10+499*q^11
+724*q^12+3139011*q^76+1033*q^13+1450*q^14+2009*q^15+2751*q^16+3691469*q^66+
3748663*q^70+2961271*q^77+2766340*q^78+693969*q^43+391*q^102+1370533*q^49+57372
*q^28+3724*q^17+4979*q^18+6600*q^19+8660*q^20+46444*q^27+3638947*q^72+462513*q^
40+3548241*q^73+92*q^103+3439515*q^63+104207*q^31+3297197*q^75+213073*q^35+14*q
^104+2337223*q^80+1238298*q^48+q^105+1654683*q^83+2110590*q^81+1224380*q^85+
1028394*q^86+848973*q^87+687903*q^88+546145*q^89+1434271*q^84+1881576*q^82+
2911996*q^59+251361*q^36+235794*q^92+1113558*q^47]


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

[1, 2, 6, 22, 88, 363, 1507, 6241, 25721, 105485, 430767, 1752945, 7113095,

    28797292, 116368938]

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

     is given by

[1, 1, 3, 11, 44, 182, 754, 3119, 12857, 52749, 215400, 876451, 3556481,

    14398721, 58184728]

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

     is given by

[0, 1, 3, 11, 44, 181, 753, 3122, 12864, 52736, 215367, 876494, 3556614,

    14398571, 58184210]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 44, 181, 753, 3119, 12857, 52736, 215367, 876451, 3556481,

    14398571, 58184210]

 ODD:, [0, 1, 3, 11, 44, 182, 754, 3122, 12864, 52749, 215400, 876494, 3556614,

    14398721, 58184728]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.954545455, 4.818181818, 7.049586777,

    9.609157266, 12.46274635, 15.58329769, 18.95001185, 22.54678051,

    26.36079398, 30.38153771, 34.60012716, 39.00887725]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.460923516, 2.036850591, 2.703485883,

    3.463602923, 4.310969238, 5.237049340, 6.234054437, 7.295691601,

    8.417033153, 9.594187729, 10.82400616, 12.10386897]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.13430, 4.14876, 7.30884, 11.9965, 18.5845,

    27.4267, 38.8634, 53.2271, 70.8464, 92.0484, 117.159, 146.504]

Skewness: , [Float(undefined), 0., 0., -0.008672084295, 0.01300529549,

    0.05287622984, 0.09972796299, 0.1461382191, 0.1885090440, 0.2257714342,

    0.2580753495, 0.2860210264, 0.3102897327, 0.3315040910, 0.3501818225]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.497740792,

    2.679464243, 2.741227098, 2.764973778, 2.783513427, 2.804776711,

    2.828570064, 2.853344754, 2.877961495, 2.901742845, 2.924422529,

    2.945839668]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+5*q^4+q^5+q^6, 1+4*q+9*q^2+15*q^3+
20*q^4+18*q^5+12*q^6+7*q^7+2*q^8+q^9+q^10, 1+5*q+14*q^2+29*q^3+49*q^4+65*q^5+70
*q^6+61*q^7+43*q^8+27*q^9+14*q^10+9*q^11+3*q^12+2*q^13+q^14+q^15, 1+6*q+20*q^2+
49*q^3+98*q^4+161*q^5+223*q^6+263*q^7+262*q^8+226*q^9+177*q^10+122*q^11+82*q^12
+50*q^13+26*q^14+17*q^15+11*q^16+5*q^17+3*q^18+2*q^19+q^20+q^21, 1+7*q+27*q^2+
76*q^3+174*q^4+333*q^5+546*q^6+779*q^7+971*q^8+1063*q^9+1042*q^10+929*q^11+762*
q^12+593*q^13+429*q^14+290*q^15+198*q^16+125*q^17+80*q^18+48*q^19+30*q^20+21*q^
21+15*q^22+7*q^23+5*q^24+3*q^25+2*q^26+q^27+q^28, 1+8*q+35*q^2+111*q^3+285*q^4+
616*q^5+1150*q^6+1887*q^7+2747*q^8+3570*q^9+4180*q^10+4451*q^11+4368*q^12+4008*
q^13+3466*q^14+2854*q^15+2232*q^16+1689*q^17+1231*q^18+869*q^19+595*q^20+401*q^
21+272*q^22+185*q^23+124*q^24+80*q^25+56*q^26+38*q^27+28*q^28+19*q^29+11*q^30+7
*q^31+5*q^32+3*q^33+2*q^34+q^35+q^36, 1+9*q+44*q^2+155*q^3+440*q^4+1054*q^5+
2190*q^6+4021*q^7+6601*q^8+9765*q^9+13117*q^10+16108*q^11+18228*q^12+19214*q^13
+19045*q^14+17905*q^15+16101*q^16+13881*q^17+11554*q^18+9327*q^19+7286*q^20+
5549*q^21+4124*q^22+3008*q^23+2164*q^24+1542*q^25+1072*q^26+756*q^27+533*q^28+
379*q^29+273*q^30+195*q^31+133*q^32+97*q^33+70*q^34+50*q^35+36*q^36+26*q^37+15*
q^38+11*q^39+7*q^40+5*q^41+3*q^42+2*q^43+q^44+q^45, 1+10*q+54*q^2+209*q^3+649*q
^4+1701*q^5+3875*q^6+7824*q^7+14186*q^8+23306*q^9+34955*q^10+48166*q^11+61348*q
^12+72738*q^13+80911*q^14+85103*q^15+85281*q^16+81954*q^17+75863*q^18+67974*q^
19+59168*q^20+50089*q^21+41411*q^22+33429*q^23+26449*q^24+20562*q^25+15705*q^26
+11800*q^27+8774*q^28+6459*q^29+4737*q^30+3477*q^31+2532*q^32+1846*q^33+1356*q^
34+995*q^35+740*q^36+555*q^37+411*q^38+305*q^39+222*q^40+163*q^41+121*q^42+91*q
^43+64*q^44+48*q^45+34*q^46+22*q^47+15*q^48+11*q^49+7*q^50+5*q^51+3*q^52+2*q^53
+q^54+q^55, 1+11*q+65*q^2+274*q^3+923*q^4+2622*q^5+6479*q^6+14213*q^7+28070*q^8
+50402*q^9+82919*q^10+125779*q^11+176899*q^12+231966*q^13+285269*q^14+331079*q^
15+365000*q^16+384630*q^17+389584*q^18+381123*q^19+361523*q^20+333708*q^21+
300465*q^22+264484*q^23+227999*q^24+192785*q^25+160176*q^26+130864*q^27+105249*
q^28+83419*q^29+65314*q^30+50559*q^31+38810*q^32+29523*q^33+22334*q^34+16823*q^
35+12661*q^36+9526*q^37+7200*q^38+5446*q^39+4131*q^40+3135*q^41+2385*q^42+1820*
q^43+1397*q^44+1072*q^45+827*q^46+630*q^47+479*q^48+357*q^49+273*q^50+204*q^51+
155*q^52+115*q^53+85*q^54+62*q^55+46*q^56+30*q^57+22*q^58+15*q^59+11*q^60+7*q^
61+5*q^62+3*q^63+2*q^64+q^65+q^66, 1+12*q+77*q^2+351*q^3+1274*q^4+3894*q^5+
10353*q^6+24456*q^7+52087*q^8+101076*q^9+180148*q^10+296806*q^11+454505*q^12+
650121*q^13+872900*q^14+1105655*q^15+1328141*q^16+1521187*q^17+1669934*q^18+
1765746*q^19+1806165*q^20+1793912*q^21+1735846*q^22+1640649*q^23+1517982*q^24+
1377530*q^25+1227783*q^26+1076404*q^27+929163*q^28+790359*q^29+663023*q^30+
549159*q^31+449383*q^32+363727*q^33+291400*q^34+231251*q^35+182059*q^36+142379*
q^37+110722*q^38+85790*q^39+66318*q^40+51168*q^41+39474*q^42+30484*q^43+23554*q
^44+18272*q^45+14233*q^46+11096*q^47+8688*q^48+6804*q^49+5317*q^50+4174*q^51+
3275*q^52+2567*q^53+2021*q^54+1585*q^55+1240*q^56+966*q^57+747*q^58+570*q^59+
441*q^60+339*q^61+259*q^62+196*q^63+149*q^64+109*q^65+83*q^66+60*q^67+42*q^68+
30*q^69+22*q^70+15*q^71+11*q^72+7*q^73+5*q^74+3*q^75+2*q^76+q^77+q^78, 1+13*q+
90*q^2+441*q^3+1715*q^4+5607*q^5+15938*q^6+40262*q^7+91778*q^8+190870*q^9+
365191*q^10+647085*q^11+1067662*q^12+1648202*q^13+2391181*q^14+3274199*q^15+
4249887*q^16+5252602*q^17+6209622*q^18+7053270*q^19+7730286*q^20+8206680*q^21+
8468632*q^22+8520558*q^23+8380437*q^24+8075553*q^25+7638690*q^26+7103436*q^27+
6503284*q^28+5867896*q^29+5222715*q^30+4589170*q^31+3983836*q^32+3418899*q^33+
2902487*q^34+2438847*q^35+2029258*q^36+1673291*q^37+1368077*q^38+1109960*q^39+
894408*q^40+716343*q^41+570680*q^42+452776*q^43+357932*q^44+282314*q^45+222417*
q^46+175192*q^47+138062*q^48+108979*q^49+86109*q^50+68175*q^51+54101*q^52+43015
*q^53+34275*q^54+27402*q^55+21914*q^56+17577*q^57+14098*q^58+11297*q^59+9037*q^
60+7245*q^61+5789*q^62+4638*q^63+3705*q^64+2954*q^65+2348*q^66+1867*q^67+1467*q
^68+1156*q^69+898*q^70+703*q^71+546*q^72+427*q^73+325*q^74+251*q^75+190*q^76+
143*q^77+107*q^78+81*q^79+56*q^80+42*q^81+30*q^82+22*q^83+15*q^84+11*q^85+7*q^
86+5*q^87+3*q^88+2*q^89+q^90+q^91, 1+14*q+104*q^2+545*q^3+2260*q^4+7865*q^5+
23779*q^6+63885*q^7+154936*q^8+343095*q^9+699752*q^10+1323431*q^11+2333983*q^12
+3856414*q^13+5994964*q^14+8802534*q^15+12254609*q^16+16237437*q^17+20555984*q^
18+24960460*q^19+29183105*q^20+32974656*q^21+36132831*q^22+38518545*q^23+
40060567*q^24+40750702*q^25+40632183*q^26+39787413*q^27+38323366*q^28+36360291*
q^29+34021130*q^30+31422996*q^31+28674024*q^32+25869040*q^33+23088721*q^34+
20397716*q^35+17845957*q^36+15468650*q^37+13289717*q^38+11321540*q^39+9567798*q
^40+8024783*q^41+6682950*q^42+5528677*q^43+4545902*q^44+3717039*q^45+3024230*q^
46+2450095*q^47+1977904*q^48+1592220*q^49+1279046*q^50+1026037*q^51+822430*q^52
+659203*q^53+528636*q^54+424387*q^55+341245*q^56+274893*q^57+221885*q^58+179478
*q^59+145428*q^60+118014*q^61+95922*q^62+78073*q^63+63623*q^64+51916*q^65+42373
*q^66+34609*q^67+28264*q^68+23069*q^69+18803*q^70+15304*q^71+12438*q^72+10106*q
^73+8200*q^74+6644*q^75+5368*q^76+4330*q^77+3476*q^78+2788*q^79+2224*q^80+1767*
q^81+1395*q^82+1104*q^83+868*q^84+683*q^85+532*q^86+413*q^87+317*q^88+245*q^89+
184*q^90+141*q^91+105*q^92+77*q^93+56*q^94+42*q^95+30*q^96+22*q^97+15*q^98+11*q
^99+7*q^100+5*q^101+3*q^102+2*q^103+q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+5*q^2+q+1, q^10+4*q^9+9*q^8+15*
q^7+20*q^6+18*q^5+12*q^4+7*q^3+2*q^2+q+1, q^15+5*q^14+14*q^13+29*q^12+49*q^11+
65*q^10+70*q^9+61*q^8+43*q^7+27*q^6+14*q^5+9*q^4+3*q^3+2*q^2+q+1, q^21+6*q^20+
20*q^19+49*q^18+98*q^17+161*q^16+223*q^15+263*q^14+262*q^13+226*q^12+177*q^11+
122*q^10+82*q^9+50*q^8+26*q^7+17*q^6+11*q^5+5*q^4+3*q^3+2*q^2+q+1, 1+76*q^25+
546*q^22+174*q^24+779*q^21+333*q^23+q+2*q^2+27*q^26+3*q^3+5*q^4+7*q^5+15*q^6+21
*q^7+30*q^8+48*q^9+80*q^10+125*q^11+198*q^12+290*q^13+429*q^14+593*q^15+762*q^
16+q^28+929*q^17+1042*q^18+1063*q^19+971*q^20+7*q^27, 1+4451*q^25+3466*q^22+
4368*q^24+2854*q^21+4008*q^23+q+1150*q^30+1887*q^29+35*q^34+111*q^33+2*q^2+4180
*q^26+3*q^3+5*q^4+7*q^5+11*q^6+285*q^32+19*q^7+28*q^8+38*q^9+56*q^10+80*q^11+
124*q^12+185*q^13+272*q^14+401*q^15+595*q^16+2747*q^28+869*q^17+1231*q^18+1689*
q^19+2232*q^20+3570*q^27+616*q^31+8*q^35+q^36, 1+7286*q^25+155*q^42+3008*q^22+
5549*q^24+2164*q^21+4124*q^23+q+9*q^44+17905*q^30+q^45+6601*q^37+16101*q^29+
2190*q^39+440*q^41+16108*q^34+4021*q^38+18228*q^33+2*q^2+9327*q^26+3*q^3+5*q^4+
7*q^5+11*q^6+19214*q^32+15*q^7+26*q^8+36*q^9+50*q^10+70*q^11+97*q^12+133*q^13+
195*q^14+273*q^15+379*q^16+44*q^43+13881*q^28+533*q^17+756*q^18+1072*q^19+1542*
q^20+11554*q^27+1054*q^40+19045*q^31+13117*q^35+9765*q^36, 1+4737*q^25+72738*q^
42+1846*q^22+3477*q^24+1356*q^21+2532*q^23+q+649*q^51+48166*q^44+54*q^53+20562*
q^30+209*q^52+q^55+10*q^54+34955*q^45+75863*q^37+15705*q^29+23306*q^46+85281*q^
39+80911*q^41+50089*q^34+1701*q^50+81954*q^38+41411*q^33+2*q^2+6459*q^26+3*q^3+
5*q^4+7*q^5+11*q^6+33429*q^32+15*q^7+22*q^8+34*q^9+48*q^10+64*q^11+91*q^12+121*
q^13+163*q^14+222*q^15+305*q^16+61348*q^43+3875*q^49+11800*q^28+411*q^17+555*q^
18+740*q^19+995*q^20+8774*q^27+85103*q^40+26449*q^31+59168*q^35+7824*q^48+67974
*q^36+14186*q^47, 1+3135*q^25+227999*q^42+1397*q^22+2385*q^24+1072*q^21+1820*q^
23+q+331079*q^51+300465*q^44+231966*q^53+50402*q^57+12661*q^30+285269*q^52+
28070*q^58+125779*q^55+6479*q^60+176899*q^54+333708*q^45+83419*q^37+9526*q^29+
361523*q^46+11*q^65+130864*q^39+2622*q^61+192785*q^41+38810*q^34+365000*q^50+65
*q^64+105249*q^38+29523*q^33+82919*q^56+2*q^2+4131*q^26+3*q^3+5*q^4+7*q^5+11*q^
6+22334*q^32+15*q^7+22*q^8+30*q^9+923*q^62+46*q^10+62*q^11+85*q^12+115*q^13+155
*q^14+204*q^15+273*q^16+q^66+264484*q^43+384630*q^49+7200*q^28+357*q^17+479*q^
18+630*q^19+827*q^20+5446*q^27+160176*q^40+274*q^63+16823*q^31+50559*q^35+
389584*q^48+14213*q^59+65314*q^36+381123*q^47, 1+2567*q^25+182059*q^42+1240*q^
22+2021*q^24+966*q^21+1585*q^23+q+1076404*q^51+291400*q^44+1377530*q^53+1793912
*q^57+8688*q^30+1227783*q^52+1806165*q^58+1640649*q^55+1669934*q^60+1517982*q^
54+363727*q^45+24456*q^71+1274*q^74+51168*q^37+296806*q^67+6804*q^29+449383*q^
46+650121*q^65+101076*q^69+85790*q^39+1521187*q^61+142379*q^41+23554*q^34+
929163*q^50+872900*q^64+66318*q^38+18272*q^33+1735846*q^56+2*q^2+180148*q^68+
3275*q^26+3*q^3+5*q^4+7*q^5+11*q^6+14233*q^32+15*q^7+22*q^8+30*q^9+1328141*q^62
+42*q^10+60*q^11+83*q^12+77*q^76+109*q^13+149*q^14+196*q^15+259*q^16+454505*q^
66+52087*q^70+12*q^77+q^78+231251*q^43+790359*q^49+5317*q^28+339*q^17+441*q^18+
570*q^19+747*q^20+4174*q^27+10353*q^72+110722*q^40+3894*q^73+1105655*q^63+11096
*q^31+351*q^75+30484*q^35+663023*q^48+1765746*q^59+39474*q^36+549159*q^47, 1+
2348*q^25+108979*q^42+1156*q^22+13*q^90+1867*q^24+898*q^21+1467*q^23+q+894408*q
^51+175192*q^44+1368077*q^53+2902487*q^57+7245*q^30+1109960*q^52+3418899*q^58+
2029258*q^55+4589170*q^60+q^91+1673291*q^54+222417*q^45+7730286*q^71+5252602*q^
74+34275*q^37+8380437*q^67+5789*q^29+282314*q^46+7638690*q^65+8468632*q^69+
54101*q^39+5222715*q^61+86109*q^41+17577*q^34+1067662*q^79+716343*q^50+7103436*
q^64+43015*q^38+14098*q^33+2438847*q^56+2*q^2+8520558*q^68+2954*q^26+3*q^3+5*q^
4+7*q^5+11*q^6+11297*q^32+15*q^7+22*q^8+30*q^9+5867896*q^62+42*q^10+56*q^11+81*
q^12+3274199*q^76+107*q^13+143*q^14+190*q^15+251*q^16+8075553*q^66+8206680*q^70
+2391181*q^77+1648202*q^78+138062*q^43+570680*q^49+4638*q^28+325*q^17+427*q^18+
546*q^19+703*q^20+3705*q^27+7053270*q^72+68175*q^40+6209622*q^73+6503284*q^63+
9037*q^31+4249887*q^75+21914*q^35+647085*q^80+452776*q^48+91778*q^83+365191*q^
81+15938*q^85+5607*q^86+1715*q^87+441*q^88+90*q^89+40262*q^84+190870*q^82+
3983836*q^59+27402*q^36+357932*q^47, 1+2224*q^25+78073*q^42+1104*q^22+8802534*q
^90+1767*q^24+868*q^21+1395*q^23+q+154936*q^97+528636*q^51+118014*q^44+822430*q
^53+1977904*q^57+6644*q^30+1323431*q^94+659203*q^52+2450095*q^58+1279046*q^55+
3717039*q^60+5994964*q^91+1026037*q^54+145428*q^45+23088721*q^71+31422996*q^74+
343095*q^96+63885*q^98+28264*q^37+13289717*q^67+5368*q^29+179478*q^46+2333983*q
^93+23779*q^99+9567798*q^65+17845957*q^69+42373*q^39+4545902*q^61+63623*q^41+
15304*q^34+40632183*q^79+424387*q^50+8024783*q^64+7865*q^100+34609*q^38+12438*q
^33+1592220*q^56+2*q^2+15468650*q^68+2788*q^26+3*q^3+5*q^4+7*q^5+11*q^6+10106*q
^32+15*q^7+22*q^8+30*q^9+699752*q^95+5528677*q^62+2260*q^101+42*q^10+56*q^11+77
*q^12+36360291*q^76+105*q^13+141*q^14+184*q^15+245*q^16+11321540*q^66+20397716*
q^70+38323366*q^77+39787413*q^78+95922*q^43+545*q^102+341245*q^49+4330*q^28+317
*q^17+413*q^18+532*q^19+683*q^20+3476*q^27+25869040*q^72+51916*q^40+28674024*q^
73+104*q^103+6682950*q^63+8200*q^31+34021130*q^75+18803*q^35+14*q^104+40750702*
q^80+274893*q^48+q^105+36132831*q^83+40060567*q^81+29183105*q^85+24960460*q^86+
20555984*q^87+16237437*q^88+12254609*q^89+32974656*q^84+38518545*q^82+3024230*q
^59+23069*q^36+3856414*q^92+221885*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 12, 45, 195, 904, 4283, 20789, 103038, 518873, 2646759, 13648828,

    71039287, 372693631]

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

     is given by

[0, 1, 3, 10, 45, 199, 902, 4275, 20797, 103060, 518845, 2646687, 13648910,

    71039459, 372693407]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 45, 199, 902, 4283, 20789, 103060, 518845, 2646759,

    13648828, 71039459, 372693407]

 ODD:, [0, 1, 3, 10, 45, 195, 904, 4275, 20797, 103038, 518873, 2646687,

    13648910, 71039287, 372693631]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.818181818, 4.366666667, 6.104060914,

    8.005537099, 10.05386773, 12.23616121, 14.54231482, 16.96415115,

    19.49488726, 22.12878701, 24.86092236, 27.68700362]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.402477147, 1.864582169, 2.351548524,

    2.862862663, 3.396556120, 3.950840310, 4.524348068, 5.116078073,

    5.725298921, 6.351467813, 6.994170865, 7.653081446]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.96694, 3.47667, 5.52978, 8.19598, 11.5366,

    15.6091, 20.4697, 26.1743, 32.7790, 40.3411, 48.9184, 58.5697]

Skewness: , [Float(undefined), 0., 0., 0.1274601866, 0.2817367885, 0.4044896846,

    0.4912725444, 0.5498878078, 0.5883411329, 0.6127435545, 0.6274570731,

    0.6355430171, 0.6391256177, 0.6397168546, 0.6383605465]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.737684812,

    3.174961209, 3.460408938, 3.648966539, 3.770089138, 3.842528399,

    3.879633807, 3.891436906, 3.885615994, 3.867821789, 3.842324329,

    3.812255947]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3
+14*q^4+14*q^5+14*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+28*q^4+35*q^
5+41*q^6+43*q^7+43*q^8+41*q^9+35*q^10+28*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*
q+16*q^2+31*q^3+51*q^4+71*q^5+93*q^6+111*q^7+124*q^8+135*q^9+137*q^10+137*q^11+
135*q^12+124*q^13+111*q^14+93*q^15+71*q^16+51*q^17+31*q^18+16*q^19+6*q^20+q^21,
1+7*q+22*q^2+48*q^3+87*q^4+134*q^5+188*q^6+246*q^7+299*q^8+351*q^9+392*q^10+423
*q^11+444*q^12+456*q^13+456*q^14+456*q^15+444*q^16+423*q^17+392*q^18+351*q^19+
299*q^20+246*q^21+188*q^22+134*q^23+87*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+
29*q^2+71*q^3+141*q^4+238*q^5+359*q^6+502*q^7+654*q^8+814*q^9+972*q^10+1118*q^
11+1250*q^12+1356*q^13+1438*q^14+1499*q^15+1542*q^16+1567*q^17+1578*q^18+1567*q
^19+1542*q^20+1499*q^21+1438*q^22+1356*q^23+1250*q^24+1118*q^25+972*q^26+814*q^
27+654*q^28+502*q^29+359*q^30+238*q^31+141*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+
9*q+37*q^2+101*q^3+219*q^4+402*q^5+652*q^6+971*q^7+1344*q^8+1763*q^9+2216*q^10+
2683*q^11+3155*q^12+3605*q^13+4015*q^14+4380*q^15+4690*q^16+4945*q^17+5159*q^18
+5316*q^19+5433*q^20+5513*q^21+5546*q^22+5546*q^23+5513*q^24+5433*q^25+5316*q^
26+5159*q^27+4945*q^28+4690*q^29+4380*q^30+4015*q^31+3605*q^32+3155*q^33+2683*q
^34+2216*q^35+1763*q^36+1344*q^37+971*q^38+652*q^39+402*q^40+219*q^41+101*q^42+
37*q^43+9*q^44+q^45, 1+10*q+46*q^2+139*q^3+328*q^4+651*q^5+1133*q^6+1794*q^7+
2631*q^8+3631*q^9+4782*q^10+6049*q^11+7416*q^12+8840*q^13+10274*q^14+11678*q^15
+13013*q^16+14243*q^17+15357*q^18+16341*q^19+17189*q^20+17926*q^21+18538*q^22+
19034*q^23+19423*q^24+19704*q^25+19878*q^26+19971*q^27+19971*q^28+19878*q^29+
19704*q^30+19423*q^31+19034*q^32+18538*q^33+17926*q^34+17189*q^35+16341*q^36+
15357*q^37+14243*q^38+13013*q^39+11678*q^40+10274*q^41+8840*q^42+7416*q^43+6049
*q^44+4782*q^45+3631*q^46+2631*q^47+1794*q^48+1133*q^49+651*q^50+328*q^51+139*q
^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+186*q^3+476*q^4+1017*q^5+1894*q^6+3184*
q^7+4936*q^8+7164*q^9+9872*q^10+13024*q^11+16586*q^12+20509*q^13+24707*q^14+
29101*q^15+33584*q^16+38049*q^17+42402*q^18+46556*q^19+50432*q^20+54021*q^21+
57283*q^22+60231*q^23+62854*q^24+65160*q^25+67133*q^26+68816*q^27+70191*q^28+
71299*q^29+72148*q^30+72763*q^31+73132*q^32+73264*q^33+73132*q^34+72763*q^35+
72148*q^36+71299*q^37+70191*q^38+68816*q^39+67133*q^40+65160*q^41+62854*q^42+
60231*q^43+57283*q^44+54021*q^45+50432*q^46+46556*q^47+42402*q^48+38049*q^49+
33584*q^50+29101*q^51+24707*q^52+20509*q^53+16586*q^54+13024*q^55+9872*q^56+
7164*q^57+4936*q^58+3184*q^59+1894*q^60+1017*q^61+476*q^62+186*q^63+56*q^64+11*
q^65+q^66, 1+12*q+67*q^2+243*q^3+672*q^4+1540*q^5+3060*q^6+5453*q^7+8919*q^8+
13600*q^9+19599*q^10+26948*q^11+35634*q^12+45617*q^13+56788*q^14+69023*q^15+
82144*q^16+95930*q^17+110137*q^18+124507*q^19+138755*q^20+152676*q^21+166074*q^
22+178800*q^23+190792*q^24+201957*q^25+212271*q^26+221724*q^27+230298*q^28+
237986*q^29+244840*q^30+250823*q^31+256025*q^32+260469*q^33+264179*q^34+267201*
q^35+269563*q^36+271239*q^37+272255*q^38+272606*q^39+272255*q^40+271239*q^41+
269563*q^42+267201*q^43+264179*q^44+260469*q^45+256025*q^46+250823*q^47+244840*
q^48+237986*q^49+230298*q^50+221724*q^51+212271*q^52+201957*q^53+190792*q^54+
178800*q^55+166074*q^56+152676*q^57+138755*q^58+124507*q^59+110137*q^60+95930*q
^61+82144*q^62+69023*q^63+56788*q^64+45617*q^65+35634*q^66+26948*q^67+19599*q^
68+13600*q^69+8919*q^70+5453*q^71+3060*q^72+1540*q^73+672*q^74+243*q^75+67*q^76
+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+926*q^4+2269*q^5+4797*q^6+9046*q^7+15585*q
^8+24937*q^9+37558*q^10+53791*q^11+73847*q^12+97850*q^13+125773*q^14+157509*q^
15+192849*q^16+231464*q^17+272935*q^18+316732*q^19+362214*q^20+408720*q^21+
455592*q^22+502174*q^23+547948*q^24+592430*q^25+635249*q^26+676142*q^27+714931*
q^28+751449*q^29+785670*q^30+817472*q^31+846834*q^32+873735*q^33+898197*q^34+
920264*q^35+940100*q^36+957776*q^37+973393*q^38+987019*q^39+998655*q^40+1008279
*q^41+1015954*q^42+1021640*q^43+1025385*q^44+1027264*q^45+1027264*q^46+1025385*
q^47+1021640*q^48+1015954*q^49+1008279*q^50+998655*q^51+987019*q^52+973393*q^53
+957776*q^54+940100*q^55+920264*q^56+898197*q^57+873735*q^58+846834*q^59+817472
*q^60+785670*q^61+751449*q^62+714931*q^63+676142*q^64+635249*q^65+592430*q^66+
547948*q^67+502174*q^68+455592*q^69+408720*q^70+362214*q^71+316732*q^72+272935*
q^73+231464*q^74+192849*q^75+157509*q^76+125773*q^77+97850*q^78+73847*q^79+
53791*q^80+37558*q^81+24937*q^82+15585*q^83+9046*q^84+4797*q^85+2269*q^86+926*q
^87+311*q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1249*q^4+3263*q^5+7321
*q^6+14583*q^7+26425*q^8+44312*q^9+69695*q^10+103909*q^11+148062*q^12+203048*q^
13+269458*q^14+347600*q^15+437545*q^16+539064*q^17+651698*q^18+774700*q^19+
907008*q^20+1047287*q^21+1194022*q^22+1345462*q^23+1499905*q^24+1655603*q^25+
1810930*q^26+1964460*q^27+2114947*q^28+2261373*q^29+2402987*q^30+2539175*q^31+
2669430*q^32+2793413*q^33+2910734*q^34+3021197*q^35+3124659*q^36+3221132*q^37+
3310656*q^38+3393452*q^39+3469656*q^40+3539495*q^41+3603159*q^42+3660790*q^43+
3712461*q^44+3758265*q^45+3798207*q^46+3832310*q^47+3860654*q^48+3883252*q^49+
3900162*q^50+3911442*q^51+3917069*q^52+3917069*q^53+3911442*q^54+3900162*q^55+
3883252*q^56+3860654*q^57+3832310*q^58+3798207*q^59+3758265*q^60+3712461*q^61+
3660790*q^62+3603159*q^63+3539495*q^64+3469656*q^65+3393452*q^66+3310656*q^67+
3221132*q^68+3124659*q^69+3021197*q^70+2910734*q^71+2793413*q^72+2669430*q^73+
2539175*q^74+2402987*q^75+2261373*q^76+2114947*q^77+1964460*q^78+1810930*q^79+
1655603*q^80+1499905*q^81+1345462*q^82+1194022*q^83+1047287*q^84+907008*q^85+
774700*q^86+651698*q^87+539064*q^88+437545*q^89+347600*q^90+269458*q^91+203048*
q^92+148062*q^93+103909*q^94+69695*q^95+44312*q^96+26425*q^97+14583*q^98+7321*q
^99+3263*q^100+1249*q^101+391*q^102+92*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3
+14*q^4+14*q^5+14*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+28*q^4+35*q^
5+41*q^6+43*q^7+43*q^8+41*q^9+35*q^10+28*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*
q+16*q^2+31*q^3+51*q^4+71*q^5+93*q^6+111*q^7+124*q^8+135*q^9+137*q^10+137*q^11+
135*q^12+124*q^13+111*q^14+93*q^15+71*q^16+51*q^17+31*q^18+16*q^19+6*q^20+q^21,
1+7*q+22*q^2+48*q^3+87*q^4+134*q^5+188*q^6+246*q^7+299*q^8+351*q^9+392*q^10+423
*q^11+444*q^12+456*q^13+456*q^14+456*q^15+444*q^16+423*q^17+392*q^18+351*q^19+
299*q^20+246*q^21+188*q^22+134*q^23+87*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+
29*q^2+71*q^3+141*q^4+238*q^5+359*q^6+502*q^7+654*q^8+814*q^9+972*q^10+1118*q^
11+1250*q^12+1356*q^13+1438*q^14+1499*q^15+1542*q^16+1567*q^17+1578*q^18+1567*q
^19+1542*q^20+1499*q^21+1438*q^22+1356*q^23+1250*q^24+1118*q^25+972*q^26+814*q^
27+654*q^28+502*q^29+359*q^30+238*q^31+141*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+
9*q+37*q^2+101*q^3+219*q^4+402*q^5+652*q^6+971*q^7+1344*q^8+1763*q^9+2216*q^10+
2683*q^11+3155*q^12+3605*q^13+4015*q^14+4380*q^15+4690*q^16+4945*q^17+5159*q^18
+5316*q^19+5433*q^20+5513*q^21+5546*q^22+5546*q^23+5513*q^24+5433*q^25+5316*q^
26+5159*q^27+4945*q^28+4690*q^29+4380*q^30+4015*q^31+3605*q^32+3155*q^33+2683*q
^34+2216*q^35+1763*q^36+1344*q^37+971*q^38+652*q^39+402*q^40+219*q^41+101*q^42+
37*q^43+9*q^44+q^45, 1+10*q+46*q^2+139*q^3+328*q^4+651*q^5+1133*q^6+1794*q^7+
2631*q^8+3631*q^9+4782*q^10+6049*q^11+7416*q^12+8840*q^13+10274*q^14+11678*q^15
+13013*q^16+14243*q^17+15357*q^18+16341*q^19+17189*q^20+17926*q^21+18538*q^22+
19034*q^23+19423*q^24+19704*q^25+19878*q^26+19971*q^27+19971*q^28+19878*q^29+
19704*q^30+19423*q^31+19034*q^32+18538*q^33+17926*q^34+17189*q^35+16341*q^36+
15357*q^37+14243*q^38+13013*q^39+11678*q^40+10274*q^41+8840*q^42+7416*q^43+6049
*q^44+4782*q^45+3631*q^46+2631*q^47+1794*q^48+1133*q^49+651*q^50+328*q^51+139*q
^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+186*q^3+476*q^4+1017*q^5+1894*q^6+3184*
q^7+4936*q^8+7164*q^9+9872*q^10+13024*q^11+16586*q^12+20509*q^13+24707*q^14+
29101*q^15+33584*q^16+38049*q^17+42402*q^18+46556*q^19+50432*q^20+54021*q^21+
57283*q^22+60231*q^23+62854*q^24+65160*q^25+67133*q^26+68816*q^27+70191*q^28+
71299*q^29+72148*q^30+72763*q^31+73132*q^32+73264*q^33+73132*q^34+72763*q^35+
72148*q^36+71299*q^37+70191*q^38+68816*q^39+67133*q^40+65160*q^41+62854*q^42+
60231*q^43+57283*q^44+54021*q^45+50432*q^46+46556*q^47+42402*q^48+38049*q^49+
33584*q^50+29101*q^51+24707*q^52+20509*q^53+16586*q^54+13024*q^55+9872*q^56+
7164*q^57+4936*q^58+3184*q^59+1894*q^60+1017*q^61+476*q^62+186*q^63+56*q^64+11*
q^65+q^66, 1+12*q+67*q^2+243*q^3+672*q^4+1540*q^5+3060*q^6+5453*q^7+8919*q^8+
13600*q^9+19599*q^10+26948*q^11+35634*q^12+45617*q^13+56788*q^14+69023*q^15+
82144*q^16+95930*q^17+110137*q^18+124507*q^19+138755*q^20+152676*q^21+166074*q^
22+178800*q^23+190792*q^24+201957*q^25+212271*q^26+221724*q^27+230298*q^28+
237986*q^29+244840*q^30+250823*q^31+256025*q^32+260469*q^33+264179*q^34+267201*
q^35+269563*q^36+271239*q^37+272255*q^38+272606*q^39+272255*q^40+271239*q^41+
269563*q^42+267201*q^43+264179*q^44+260469*q^45+256025*q^46+250823*q^47+244840*
q^48+237986*q^49+230298*q^50+221724*q^51+212271*q^52+201957*q^53+190792*q^54+
178800*q^55+166074*q^56+152676*q^57+138755*q^58+124507*q^59+110137*q^60+95930*q
^61+82144*q^62+69023*q^63+56788*q^64+45617*q^65+35634*q^66+26948*q^67+19599*q^
68+13600*q^69+8919*q^70+5453*q^71+3060*q^72+1540*q^73+672*q^74+243*q^75+67*q^76
+12*q^77+q^78, 1+13*q+79*q^2+311*q^3+926*q^4+2269*q^5+4797*q^6+9046*q^7+15585*q
^8+24937*q^9+37558*q^10+53791*q^11+73847*q^12+97850*q^13+125773*q^14+157509*q^
15+192849*q^16+231464*q^17+272935*q^18+316732*q^19+362214*q^20+408720*q^21+
455592*q^22+502174*q^23+547948*q^24+592430*q^25+635249*q^26+676142*q^27+714931*
q^28+751449*q^29+785670*q^30+817472*q^31+846834*q^32+873735*q^33+898197*q^34+
920264*q^35+940100*q^36+957776*q^37+973393*q^38+987019*q^39+998655*q^40+1008279
*q^41+1015954*q^42+1021640*q^43+1025385*q^44+1027264*q^45+1027264*q^46+1025385*
q^47+1021640*q^48+1015954*q^49+1008279*q^50+998655*q^51+987019*q^52+973393*q^53
+957776*q^54+940100*q^55+920264*q^56+898197*q^57+873735*q^58+846834*q^59+817472
*q^60+785670*q^61+751449*q^62+714931*q^63+676142*q^64+635249*q^65+592430*q^66+
547948*q^67+502174*q^68+455592*q^69+408720*q^70+362214*q^71+316732*q^72+272935*
q^73+231464*q^74+192849*q^75+157509*q^76+125773*q^77+97850*q^78+73847*q^79+
53791*q^80+37558*q^81+24937*q^82+15585*q^83+9046*q^84+4797*q^85+2269*q^86+926*q
^87+311*q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1249*q^4+3263*q^5+7321
*q^6+14583*q^7+26425*q^8+44312*q^9+69695*q^10+103909*q^11+148062*q^12+203048*q^
13+269458*q^14+347600*q^15+437545*q^16+539064*q^17+651698*q^18+774700*q^19+
907008*q^20+1047287*q^21+1194022*q^22+1345462*q^23+1499905*q^24+1655603*q^25+
1810930*q^26+1964460*q^27+2114947*q^28+2261373*q^29+2402987*q^30+2539175*q^31+
2669430*q^32+2793413*q^33+2910734*q^34+3021197*q^35+3124659*q^36+3221132*q^37+
3310656*q^38+3393452*q^39+3469656*q^40+3539495*q^41+3603159*q^42+3660790*q^43+
3712461*q^44+3758265*q^45+3798207*q^46+3832310*q^47+3860654*q^48+3883252*q^49+
3900162*q^50+3911442*q^51+3917069*q^52+3917069*q^53+3911442*q^54+3900162*q^55+
3883252*q^56+3860654*q^57+3832310*q^58+3798207*q^59+3758265*q^60+3712461*q^61+
3660790*q^62+3603159*q^63+3539495*q^64+3469656*q^65+3393452*q^66+3310656*q^67+
3221132*q^68+3124659*q^69+3021197*q^70+2910734*q^71+2793413*q^72+2669430*q^73+
2539175*q^74+2402987*q^75+2261373*q^76+2114947*q^77+1964460*q^78+1810930*q^79+
1655603*q^80+1499905*q^81+1345462*q^82+1194022*q^83+1047287*q^84+907008*q^85+
774700*q^86+651698*q^87+539064*q^88+437545*q^89+347600*q^90+269458*q^91+203048*
q^92+148062*q^93+103909*q^94+69695*q^95+44312*q^96+26425*q^97+14583*q^98+7321*q
^99+3263*q^100+1249*q^101+391*q^102+92*q^103+14*q^104+q^105]


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

[1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248,

    44725516, 195354368]

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

     is given by

[1, 1, 3, 10, 44, 183, 776, 3322, 14350, 62155, 270020, 1175374, 5124146,

    22362758, 97677184]

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

     is given by

[0, 1, 3, 12, 44, 183, 776, 3330, 14346, 62155, 270020, 1175446, 5124102,

    22362758, 97677184]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 44, 183, 776, 3322, 14350, 62155, 270020, 1175374, 5124146,

    22362758, 97677184]

 ODD:, [0, 1, 3, 12, 44, 183, 776, 3330, 14346, 62155, 270020, 1175446, 5124102,

    22362758, 97677184]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.500000000,

    10.50000000, 14.00000000, 18.00000000, 22.50000000, 27.50000000,

    33.00000000, 39.00000000, 45.50000000, 52.50000000]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.507556723, 2.184657243, 3.002503508,

    3.966678217, 5.080382135, 6.345837558, 7.764493429, 9.337059455,

    11.06356819, 12.94346914, 14.97573261, 17.15894853]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.27273, 4.77273, 9.01503, 15.7345, 25.8103,

    40.2697, 60.2874, 87.1807, 122.403, 167.533, 224.273, 294.430]

                                                      -5
Skewness: , [Float(undefined), 0., 0., 0.2918624564 10  , 0., 0.,

                    -5                 -6                             -6
    -0.1602214213 10  , 0.7626239169 10  , 0., 0., 0., 0.7384343772 10  ,

                    -6                 -6                 -6
    -0.4611576870 10  , 0.2977381677 10  , 0.1979369672 10  ]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.340792622,

    2.400545066, 2.402093764, 2.388994115, 2.372286710, 2.355473112,

    2.340004072, 2.326512655, 2.315248653, 2.306299001, 2.299497586,

    2.294731083]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+3*q^2+6*q^3+5*q^4+3*q^5+q^6, 1+4*q+4*q^2+8*q^3+
12*q^4+14*q^5+16*q^6+15*q^7+9*q^8+4*q^9+q^10, 1+5*q+5*q^2+10*q^3+15*q^4+24*q^5+
31*q^6+37*q^7+47*q^8+50*q^9+51*q^10+43*q^11+29*q^12+14*q^13+5*q^14+q^15, 1+6*q+
6*q^2+12*q^3+18*q^4+29*q^5+45*q^6+58*q^7+75*q^8+97*q^9+125*q^10+143*q^11+159*q^
12+174*q^13+174*q^14+163*q^15+133*q^16+90*q^17+49*q^18+20*q^19+6*q^20+q^21, 1+7
*q+7*q^2+14*q^3+21*q^4+34*q^5+53*q^6+77*q^7+104*q^8+135*q^9+186*q^10+232*q^11+
298*q^12+359*q^13+422*q^14+487*q^15+559*q^16+603*q^17+626*q^18+633*q^19+599*q^
20+529*q^21+420*q^22+287*q^23+164*q^24+76*q^25+27*q^26+7*q^27+q^28, 1+8*q+8*q^2
+16*q^3+24*q^4+39*q^5+61*q^6+89*q^7+130*q^8+174*q^9+238*q^10+312*q^11+413*q^12+
529*q^13+670*q^14+820*q^15+993*q^16+1188*q^17+1399*q^18+1621*q^19+1827*q^20+
2027*q^21+2188*q^22+2334*q^23+2401*q^24+2371*q^25+2268*q^26+2052*q^27+1741*q^28
+1348*q^29+926*q^30+548*q^31+273*q^32+111*q^33+35*q^34+8*q^35+q^36, 1+9*q+9*q^2
+18*q^3+27*q^4+44*q^5+69*q^6+101*q^7+148*q^8+209*q^9+291*q^10+381*q^11+519*q^12
+678*q^13+887*q^14+1124*q^15+1428*q^16+1760*q^17+2163*q^18+2627*q^19+3142*q^20+
3738*q^21+4373*q^22+5038*q^23+5732*q^24+6463*q^25+7178*q^26+7864*q^27+8452*q^28
+8922*q^29+9225*q^30+9362*q^31+9224*q^32+8773*q^33+8044*q^34+7025*q^35+5777*q^
36+4391*q^37+3015*q^38+1830*q^39+960*q^40+426*q^41+155*q^42+44*q^43+9*q^44+q^45
, 1+10*q+10*q^2+20*q^3+30*q^4+49*q^5+77*q^6+113*q^7+166*q^8+235*q^9+339*q^10+
451*q^11+612*q^12+816*q^13+1081*q^14+1401*q^15+1813*q^16+2292*q^17+2894*q^18+
3607*q^19+4430*q^20+5421*q^21+6574*q^22+7893*q^23+9365*q^24+11063*q^25+12897*q^
26+14914*q^27+17108*q^28+19443*q^29+21882*q^30+24435*q^31+26900*q^32+29325*q^33
+31679*q^34+33754*q^35+35516*q^36+36743*q^37+37394*q^38+37300*q^39+36460*q^40+
34687*q^41+31934*q^42+28353*q^43+24051*q^44+19305*q^45+14449*q^46+9910*q^47+
6115*q^48+3335*q^49+1577*q^50+633*q^51+209*q^52+54*q^53+10*q^54+q^55, 1+11*q+11
*q^2+22*q^3+33*q^4+54*q^5+85*q^6+125*q^7+184*q^8+261*q^9+377*q^10+515*q^11+706*
q^12+939*q^13+1262*q^14+1651*q^15+2168*q^16+2778*q^17+3561*q^18+4502*q^19+5658*
q^20+7040*q^21+8687*q^22+10654*q^23+12956*q^24+15674*q^25+18789*q^26+22383*q^27
+26451*q^28+31087*q^29+36224*q^30+42003*q^31+48318*q^32+55250*q^33+62697*q^34+
70666*q^35+79087*q^36+87793*q^37+96798*q^38+105923*q^39+114964*q^40+123713*q^41
+131829*q^42+139078*q^43+145272*q^44+150002*q^45+152823*q^46+153367*q^47+151408
*q^48+146610*q^49+138996*q^50+128432*q^51+115044*q^52+99472*q^53+82412*q^54+
64861*q^55+47939*q^56+32814*q^57+20492*q^58+11499*q^59+5709*q^60+2464*q^61+905*
q^62+274*q^63+65*q^64+11*q^65+q^66, 1+12*q+12*q^2+24*q^3+36*q^4+59*q^5+93*q^6+
137*q^7+202*q^8+287*q^9+415*q^10+568*q^11+793*q^12+1063*q^13+1426*q^14+1886*q^
15+2492*q^16+3229*q^17+4177*q^18+5337*q^19+6772*q^20+8538*q^21+10692*q^22+13272
*q^23+16367*q^24+20103*q^25+24490*q^26+29701*q^27+35761*q^28+42837*q^29+50997*q
^30+60460*q^31+71148*q^32+83374*q^33+97144*q^34+112593*q^35+129848*q^36+148916*
q^37+169851*q^38+192693*q^39+217524*q^40+244125*q^41+272553*q^42+302667*q^43+
334151*q^44+366780*q^45+400264*q^46+433974*q^47+467640*q^48+500483*q^49+532079*
q^50+561333*q^51+587528*q^52+609429*q^53+626105*q^54+636825*q^55+640284*q^56+
635525*q^57+621491*q^58+597920*q^59+564449*q^60+521632*q^61+470158*q^62+411422*
q^63+347946*q^64+282646*q^65+218892*q^66+160093*q^67+109340*q^68+68876*q^69+
39505*q^70+20361*q^71+9297*q^72+3698*q^73+1254*q^74+351*q^75+77*q^76+12*q^77+q^
78, 1+13*q+13*q^2+26*q^3+39*q^4+64*q^5+101*q^6+149*q^7+220*q^8+313*q^9+453*q^10
+621*q^11+868*q^12+1179*q^13+1591*q^14+2102*q^15+2799*q^16+3645*q^17+4753*q^18+
6114*q^19+7819*q^20+9924*q^21+12542*q^22+15711*q^23+19580*q^24+24284*q^25+29897
*q^26+36650*q^27+44681*q^28+54216*q^29+65386*q^30+78620*q^31+93895*q^32+111771*
q^33+132395*q^34+156096*q^35+183196*q^36+214170*q^37+249215*q^38+288768*q^39+
333199*q^40+382811*q^41+437916*q^42+499001*q^43+566052*q^44+639380*q^45+719436*
q^46+805807*q^47+898925*q^48+998436*q^49+1104139*q^50+1215500*q^51+1332404*q^52
+1453709*q^53+1578674*q^54+1706437*q^55+1835240*q^56+1963565*q^57+2089814*q^58+
2211616*q^59+2326546*q^60+2432465*q^61+2526183*q^62+2604582*q^63+2664590*q^64+
2702256*q^65+2714803*q^66+2699573*q^67+2653794*q^68+2575333*q^69+2462821*q^70+
2317012*q^71+2139507*q^72+1934280*q^73+1706584*q^74+1463525*q^75+1214667*q^76+
970392*q^77+741416*q^78+537477*q^79+366267*q^80+232211*q^81+135480*q^82+71922*q
^83+34328*q^84+14534*q^85+5369*q^86+1693*q^87+441*q^88+90*q^89+13*q^90+q^91, 1+
14*q+14*q^2+28*q^3+42*q^4+69*q^5+109*q^6+161*q^7+238*q^8+339*q^9+491*q^10+674*q
^11+943*q^12+1282*q^13+1747*q^14+2319*q^15+3085*q^16+4042*q^17+5290*q^18+6846*q
^19+8801*q^20+11234*q^21+14269*q^22+17995*q^23+22574*q^24+28183*q^25+34990*q^26
+43222*q^27+53098*q^28+64988*q^29+79113*q^30+96001*q^31+115836*q^32+139310*q^33
+166788*q^34+198954*q^35+236322*q^36+279706*q^37+329741*q^38+387330*q^39+453255
*q^40+528546*q^41+614131*q^42+711052*q^43+820386*q^44+943182*q^45+1080754*q^46+
1234021*q^47+1404337*q^48+1592525*q^49+1800111*q^50+2027644*q^51+2276422*q^52+
2546797*q^53+2839861*q^54+3155688*q^55+3494716*q^56+3856756*q^57+4241357*q^58+
4648247*q^59+5075837*q^60+5522876*q^61+5987213*q^62+6466033*q^63+6956391*q^64+
7454209*q^65+7955669*q^66+8454935*q^67+8946392*q^68+9423507*q^69+9877892*q^70+
10302463*q^71+10687436*q^72+11023542*q^73+11300539*q^74+11508790*q^75+11636586*
q^76+11674052*q^77+11611090*q^78+11439442*q^79+11153973*q^80+10750597*q^81+
10228473*q^82+9590812*q^83+8847407*q^84+8012333*q^85+7106241*q^86+6154246*q^87+
5185794*q^88+4234689*q^89+3335140*q^90+2519022*q^91+1812377*q^92+1232420*q^93+
785187*q^94+464346*q^95+252428*q^96+124856*q^97+55564*q^98+21959*q^99+7581*q^
100+2236*q^101+545*q^102+104*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+3*q^4+6*q^3+5*q^2+3*q+1, q^10+4*q^9+4*q^8+8
*q^7+12*q^6+14*q^5+16*q^4+15*q^3+9*q^2+4*q+1, q^15+5*q^14+5*q^13+10*q^12+15*q^
11+24*q^10+31*q^9+37*q^8+47*q^7+50*q^6+51*q^5+43*q^4+29*q^3+14*q^2+5*q+1, q^21+
6*q^20+6*q^19+12*q^18+18*q^17+29*q^16+45*q^15+58*q^14+75*q^13+97*q^12+125*q^11+
143*q^10+159*q^9+174*q^8+174*q^7+163*q^6+133*q^5+90*q^4+49*q^3+20*q^2+6*q+1, 1+
14*q^25+53*q^22+21*q^24+77*q^21+34*q^23+7*q+27*q^2+7*q^26+76*q^3+164*q^4+287*q^
5+420*q^6+529*q^7+599*q^8+633*q^9+626*q^10+603*q^11+559*q^12+487*q^13+422*q^14+
359*q^15+298*q^16+q^28+232*q^17+186*q^18+135*q^19+104*q^20+7*q^27, 1+312*q^25+
670*q^22+413*q^24+820*q^21+529*q^23+8*q+61*q^30+89*q^29+8*q^34+16*q^33+35*q^2+
238*q^26+111*q^3+273*q^4+548*q^5+926*q^6+24*q^32+1348*q^7+1741*q^8+2052*q^9+
2268*q^10+2371*q^11+2401*q^12+2334*q^13+2188*q^14+2027*q^15+1827*q^16+130*q^28+
1621*q^17+1399*q^18+1188*q^19+993*q^20+174*q^27+39*q^31+8*q^35+q^36, 1+3142*q^
25+18*q^42+5038*q^22+3738*q^24+5732*q^21+4373*q^23+9*q+9*q^44+1124*q^30+q^45+
148*q^37+1428*q^29+69*q^39+27*q^41+381*q^34+101*q^38+519*q^33+44*q^2+2627*q^26+
155*q^3+426*q^4+960*q^5+1830*q^6+678*q^32+3015*q^7+4391*q^8+5777*q^9+7025*q^10+
8044*q^11+8773*q^12+9224*q^13+9362*q^14+9225*q^15+8922*q^16+9*q^43+1760*q^28+
8452*q^17+7864*q^18+7178*q^19+6463*q^20+2163*q^27+44*q^40+887*q^31+291*q^35+209
*q^36, 1+21882*q^25+816*q^42+29325*q^22+24435*q^24+31679*q^21+26900*q^23+10*q+
30*q^51+451*q^44+10*q^53+11063*q^30+20*q^52+q^55+10*q^54+339*q^45+2894*q^37+
12897*q^29+235*q^46+1813*q^39+1081*q^41+5421*q^34+49*q^50+2292*q^38+6574*q^33+
54*q^2+19443*q^26+209*q^3+633*q^4+1577*q^5+3335*q^6+7893*q^32+6115*q^7+9910*q^8
+14449*q^9+19305*q^10+24051*q^11+28353*q^12+31934*q^13+34687*q^14+36460*q^15+
37300*q^16+612*q^43+77*q^49+14914*q^28+37394*q^17+36743*q^18+35516*q^19+33754*q
^20+17108*q^27+1401*q^40+9365*q^31+4430*q^35+113*q^48+3607*q^36+166*q^47, 1+
123713*q^25+12956*q^42+145272*q^22+131829*q^24+150002*q^21+139078*q^23+11*q+
1651*q^51+8687*q^44+939*q^53+261*q^57+79087*q^30+1262*q^52+184*q^58+515*q^55+85
*q^60+706*q^54+7040*q^45+31087*q^37+87793*q^29+5658*q^46+11*q^65+22383*q^39+54*
q^61+15674*q^41+48318*q^34+2168*q^50+11*q^64+26451*q^38+55250*q^33+377*q^56+65*
q^2+114964*q^26+274*q^3+905*q^4+2464*q^5+5709*q^6+62697*q^32+11499*q^7+20492*q^
8+32814*q^9+33*q^62+47939*q^10+64861*q^11+82412*q^12+99472*q^13+115044*q^14+
128432*q^15+138996*q^16+q^66+10654*q^43+2778*q^49+96798*q^28+146610*q^17+151408
*q^18+153367*q^19+152823*q^20+105923*q^27+18789*q^40+22*q^63+70666*q^31+42003*q
^35+3561*q^48+125*q^59+36224*q^36+4502*q^47, 1+609429*q^25+129848*q^42+640284*q
^22+626105*q^24+635525*q^21+636825*q^23+12*q+29701*q^51+97144*q^44+20103*q^53+
8538*q^57+467640*q^30+24490*q^52+6772*q^58+13272*q^55+4177*q^60+16367*q^54+
83374*q^45+137*q^71+36*q^74+244125*q^37+568*q^67+500483*q^29+71148*q^46+1063*q^
65+287*q^69+192693*q^39+3229*q^61+148916*q^41+334151*q^34+35761*q^50+1426*q^64+
217524*q^38+366780*q^33+10692*q^56+77*q^2+415*q^68+587528*q^26+351*q^3+1254*q^4
+3698*q^5+9297*q^6+400264*q^32+20361*q^7+39505*q^8+68876*q^9+2492*q^62+109340*q
^10+160093*q^11+218892*q^12+12*q^76+282646*q^13+347946*q^14+411422*q^15+470158*
q^16+793*q^66+202*q^70+12*q^77+q^78+112593*q^43+42837*q^49+532079*q^28+521632*q
^17+564449*q^18+597920*q^19+621491*q^20+561333*q^27+93*q^72+169851*q^40+59*q^73
+1886*q^63+433974*q^31+24*q^75+302667*q^35+50997*q^48+5337*q^59+272553*q^36+
60460*q^47, 1+2714803*q^25+998436*q^42+2575333*q^22+13*q^90+2699573*q^24+
2462821*q^21+2653794*q^23+13*q+333199*q^51+805807*q^44+249215*q^53+132395*q^57+
2432465*q^30+288768*q^52+111771*q^58+183196*q^55+78620*q^60+q^91+214170*q^54+
719436*q^45+7819*q^71+3645*q^74+1578674*q^37+19580*q^67+2526183*q^29+639380*q^
46+29897*q^65+12542*q^69+1332404*q^39+65386*q^61+1104139*q^41+1963565*q^34+868*
q^79+382811*q^50+36650*q^64+1453709*q^38+2089814*q^33+156096*q^56+90*q^2+15711*
q^68+2702256*q^26+441*q^3+1693*q^4+5369*q^5+14534*q^6+2211616*q^32+34328*q^7+
71922*q^8+135480*q^9+54216*q^62+232211*q^10+366267*q^11+537477*q^12+2102*q^76+
741416*q^13+970392*q^14+1214667*q^15+1463525*q^16+24284*q^66+9924*q^70+1591*q^
77+1179*q^78+898925*q^43+437916*q^49+2604582*q^28+1706584*q^17+1934280*q^18+
2139507*q^19+2317012*q^20+2664590*q^27+6114*q^72+1215500*q^40+4753*q^73+44681*q
^63+2326546*q^31+2799*q^75+1835240*q^35+621*q^80+499001*q^48+220*q^83+453*q^81+
101*q^85+64*q^86+39*q^87+26*q^88+13*q^89+149*q^84+313*q^82+93895*q^59+1706437*q
^36+566052*q^47, 1+11153973*q^25+6466033*q^42+9590812*q^22+2319*q^90+10750597*q
^24+8847407*q^21+10228473*q^23+14*q+238*q^97+2839861*q^51+5522876*q^44+2276422*
q^53+1404337*q^57+11508790*q^30+674*q^94+2546797*q^52+1234021*q^58+1800111*q^55
+943182*q^60+1747*q^91+2027644*q^54+5075837*q^45+166788*q^71+96001*q^74+339*q^
96+161*q^98+8946392*q^37+329741*q^67+11636586*q^29+4648247*q^46+943*q^93+109*q^
99+453255*q^65+236322*q^69+7955669*q^39+820386*q^61+6956391*q^41+10302463*q^34+
34990*q^79+3155688*q^50+528546*q^64+69*q^100+8454935*q^38+10687436*q^33+1592525
*q^56+104*q^2+279706*q^68+11439442*q^26+545*q^3+2236*q^4+7581*q^5+21959*q^6+
11023542*q^32+55564*q^7+124856*q^8+252428*q^9+491*q^95+711052*q^62+42*q^101+
464346*q^10+785187*q^11+1232420*q^12+64988*q^76+1812377*q^13+2519022*q^14+
3335140*q^15+4234689*q^16+387330*q^66+198954*q^70+53098*q^77+43222*q^78+5987213
*q^43+28*q^102+3494716*q^49+11674052*q^28+5185794*q^17+6154246*q^18+7106241*q^
19+8012333*q^20+11611090*q^27+139310*q^72+7454209*q^40+115836*q^73+14*q^103+
614131*q^63+11300539*q^31+79113*q^75+9877892*q^35+14*q^104+28183*q^80+3856756*q
^48+q^105+14269*q^83+22574*q^81+8801*q^85+6846*q^86+5290*q^87+4042*q^88+3085*q^
89+11234*q^84+17995*q^82+1080754*q^59+9423507*q^36+1282*q^92+4241357*q^47]


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

[1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496,

    66876096, 317809216]

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

     is given by

[1, 1, 3, 10, 43, 184, 791, 3488, 15597, 70828, 325573, 1511912, 7083255,

    33438052, 158904591]

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

     is given by

[0, 1, 3, 12, 45, 184, 793, 3480, 15595, 70828, 325563, 1511928, 7083241,

    33438044, 158904625]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 43, 184, 793, 3488, 15597, 70828, 325563, 1511912, 7083255,

    33438044, 158904625]

 ODD:, [0, 1, 3, 12, 45, 184, 791, 3480, 15595, 70828, 325573, 1511928, 7083241,

    33438052, 158904591]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.090909091, 5.375000000, 8.407608696,

    12.21906566, 16.82893226, 22.25112208, 28.49614559, 35.57238273,

    43.48684454, 52.24561592, 61.85410844, 72.31720829]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.504813215, 2.138991372, 2.845122970,

    3.610536648, 4.426593639, 5.287781167, 6.190643426, 7.132980338,

    8.113308537, 9.130525168, 10.18371479, 11.27204854]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.26446, 4.57528, 8.09472, 13.0360, 19.5947,

    27.9606, 38.3241, 50.8794, 65.8258, 83.3665, 103.708, 127.059]

Skewness: , [Float(undefined), 0., 0., -0.1547810712, -0.3049206640,

    -0.4093751101, -0.4797862510, -0.5286040239, -0.5634461057, -0.5888322226,

    -0.6076241297, -0.6217462714, -0.6325402889, -0.6409508545, -0.6476354078]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.373405676,

    2.605384445, 2.788087244, 2.930951661, 3.043268909, 3.131792251,

    3.201458906, 3.256177444, 3.299072562, 3.332832937, 3.359621257,

    3.381115930]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, 1+4*q+7*q^2+9*q^3+
10*q^4+13*q^5+15*q^6+13*q^7+9*q^8+4*q^9+q^10, 1+5*q+11*q^2+16*q^3+19*q^4+20*q^5
+27*q^6+33*q^7+36*q^8+41*q^9+43*q^10+38*q^11+26*q^12+14*q^13+5*q^14+q^15, 1+6*q
+16*q^2+27*q^3+35*q^4+39*q^5+41*q^6+53*q^7+67*q^8+77*q^9+83*q^10+91*q^11+112*q^
12+124*q^13+129*q^14+128*q^15+110*q^16+79*q^17+45*q^18+20*q^19+6*q^20+q^21, 1+7
*q+22*q^2+43*q^3+62*q^4+74*q^5+80*q^6+85*q^7+104*q^8+132*q^9+158*q^10+173*q^11+
184*q^12+200*q^13+234*q^14+275*q^15+306*q^16+336*q^17+374*q^18+408*q^19+411*q^
20+385*q^21+327*q^22+238*q^23+145*q^24+71*q^25+27*q^26+7*q^27+q^28, 1+8*q+29*q^
2+65*q^3+105*q^4+136*q^5+154*q^6+165*q^7+177*q^8+208*q^9+259*q^10+316*q^11+358*
q^12+382*q^13+401*q^14+435*q^15+501*q^16+573*q^17+642*q^18+690*q^19+763*q^20+
867*q^21+992*q^22+1109*q^23+1198*q^24+1274*q^25+1318*q^26+1297*q^27+1182*q^28+
982*q^29+725*q^30+459*q^31+244*q^32+105*q^33+35*q^34+8*q^35+q^36, 1+9*q+37*q^2+
94*q^3+170*q^4+241*q^5+290*q^6+319*q^7+342*q^8+370*q^9+424*q^10+513*q^11+625*q^
12+730*q^13+798*q^14+834*q^15+878*q^16+944*q^17+1064*q^18+1213*q^19+1355*q^20+
1460*q^21+1547*q^22+1670*q^23+1870*q^24+2128*q^25+2392*q^26+2661*q^27+2932*q^28
+3267*q^29+3622*q^30+3930*q^31+4157*q^32+4278*q^33+4276*q^34+4081*q^35+3648*q^
36+2996*q^37+2218*q^38+1449*q^39+814*q^40+385*q^41+148*q^42+44*q^43+9*q^44+q^45
, 1+10*q+46*q^2+131*q^3+264*q^4+411*q^5+531*q^6+609*q^7+661*q^8+712*q^9+776*q^
10+876*q^11+1030*q^12+1241*q^13+1468*q^14+1648*q^15+1759*q^16+1835*q^17+1921*q^
18+2054*q^19+2273*q^20+2568*q^21+2877*q^22+3119*q^23+3296*q^24+3445*q^25+3684*q
^26+4052*q^27+4530*q^28+5033*q^29+5507*q^30+5980*q^31+6558*q^32+7338*q^33+8291*
q^34+9270*q^35+10194*q^36+11118*q^37+12056*q^38+12977*q^39+13712*q^40+14157*q^
41+14237*q^42+13840*q^43+12886*q^44+11314*q^45+9212*q^46+6844*q^47+4560*q^48+
2676*q^49+1354*q^50+578*q^51+201*q^52+54*q^53+10*q^54+q^55, 1+11*q+56*q^2+177*q
^3+395*q^4+675*q^5+942*q^6+1140*q^7+1270*q^8+1373*q^9+1488*q^10+1631*q^11+1824*
q^12+2099*q^13+2484*q^14+2941*q^15+3366*q^16+3678*q^17+3878*q^18+4033*q^19+4224
*q^20+4498*q^21+4912*q^22+5462*q^23+6101*q^24+6678*q^25+7103*q^26+7401*q^27+
7709*q^28+8166*q^29+8857*q^30+9769*q^31+10768*q^32+11705*q^33+12488*q^34+13301*
q^35+14347*q^36+15811*q^37+17659*q^38+19738*q^39+21874*q^40+24059*q^41+26426*q^
42+29132*q^43+32187*q^44+35329*q^45+38269*q^46+40933*q^47+43459*q^48+45703*q^49
+47263*q^50+47771*q^51+46994*q^52+44725*q^53+40796*q^54+35270*q^55+28506*q^56+
21231*q^57+14369*q^58+8704*q^59+4644*q^60+2141*q^61+834*q^62+265*q^63+65*q^64+
11*q^65+q^66, 1+12*q+67*q^2+233*q^3+572*q^4+1070*q^5+1617*q^6+2082*q^7+2410*q^8
+2643*q^9+2861*q^10+3119*q^11+3431*q^12+3817*q^13+4326*q^14+5025*q^15+5906*q^16
+6831*q^17+7609*q^18+8156*q^19+8540*q^20+8897*q^21+9334*q^22+9910*q^23+10695*q^
24+11744*q^25+13001*q^26+14268*q^27+15335*q^28+16087*q^29+16669*q^30+17296*q^31
+18202*q^32+19527*q^33+21278*q^34+23284*q^35+25245*q^36+26901*q^37+28286*q^38+
29737*q^39+31685*q^40+34351*q^41+37718*q^42+41510*q^43+45430*q^44+49416*q^45+
53702*q^46+58806*q^47+64972*q^48+72074*q^49+79731*q^50+87629*q^51+95623*q^52+
103958*q^53+112788*q^54+122102*q^55+131448*q^56+140148*q^57+147804*q^58+154211*
q^59+159056*q^60+161335*q^61+160005*q^62+154441*q^63+144270*q^64+129439*q^65+
110458*q^66+88675*q^67+66168*q^68+45328*q^69+28157*q^70+15647*q^71+7664*q^72+
3249*q^73+1165*q^74+341*q^75+77*q^76+12*q^77+q^78, 1+13*q+79*q^2+300*q^3+805*q^
4+1642*q^5+2687*q^6+3699*q^7+4492*q^8+5053*q^9+5504*q^10+5980*q^11+6550*q^12+
7221*q^13+8010*q^14+8985*q^15+10271*q^16+11935*q^17+13829*q^18+15615*q^19+17019
*q^20+18032*q^21+18854*q^22+19695*q^23+20690*q^24+21915*q^25+23474*q^26+25467*q
^27+27919*q^28+30597*q^29+33081*q^30+35007*q^31+36406*q^32+37601*q^33+38987*q^
34+40861*q^35+43421*q^36+46772*q^37+50756*q^38+54892*q^39+58581*q^40+61584*q^41
+64135*q^42+66938*q^43+70646*q^44+75720*q^45+82088*q^46+89274*q^47+96637*q^48+
103685*q^49+110613*q^50+118244*q^51+127688*q^52+139718*q^53+154196*q^54+170328*
q^55+187086*q^56+204149*q^57+222160*q^58+242166*q^59+264929*q^60+289999*q^61+
316281*q^62+342915*q^63+369764*q^64+397111*q^65+425222*q^66+453611*q^67+481100*
q^68+505892*q^69+526211*q^70+540858*q^71+548356*q^72+546544*q^73+532901*q^74+
505836*q^75+465045*q^76+411353*q^77+347245*q^78+277135*q^79+207012*q^80+143208*
q^81+90761*q^82+52112*q^83+26772*q^84+12133*q^85+4765*q^86+1584*q^87+430*q^88+
90*q^89+13*q^90+q^91, 1+14*q+92*q^2+379*q^3+1105*q^4+2447*q^5+4329*q^6+6386*q^7
+8191*q^8+9545*q^9+10557*q^10+11484*q^11+12530*q^12+13771*q^13+15201*q^14+16832
*q^15+18754*q^16+21177*q^17+24301*q^18+28034*q^19+31883*q^20+35249*q^21+37854*q
^22+39880*q^23+41717*q^24+43692*q^25+45967*q^26+48608*q^27+51769*q^28+55673*q^
29+60472*q^30+65949*q^31+71422*q^32+76105*q^33+79686*q^34+82493*q^35+85192*q^36
+88298*q^37+92222*q^38+97308*q^39+103783*q^40+111591*q^41+120129*q^42+128352*q^
43+135322*q^44+140883*q^45+145859*q^46+151501*q^47+158928*q^48+168747*q^49+
181037*q^50+195152*q^51+209827*q^52+223797*q^53+236635*q^54+249150*q^55+263031*
q^56+280203*q^57+301822*q^58+328213*q^59+358452*q^60+390930*q^61+424424*q^62+
458949*q^63+496061*q^64+538193*q^65+587236*q^66+643328*q^67+704356*q^68+767807*
q^69+832366*q^70+899232*q^71+971169*q^72+1049844*q^73+1134479*q^74+1222289*q^75
+1310950*q^76+1399520*q^77+1488284*q^78+1576630*q^79+1662122*q^80+1740349*q^81+
1805932*q^82+1853550*q^83+1878345*q^84+1875752*q^85+1840359*q^86+1766993*q^87+
1652698*q^88+1498414*q^89+1309350*q^90+1095029*q^91+869521*q^92+649920*q^93+
453129*q^94+291946*q^95+172138*q^96+91929*q^97+43947*q^98+18548*q^99+6790*q^100
+2105*q^101+533*q^102+104*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+5*q^3+5*q^2+3*q+1, q^10+4*q^9+7*q^8+9
*q^7+10*q^6+13*q^5+15*q^4+13*q^3+9*q^2+4*q+1, q^15+5*q^14+11*q^13+16*q^12+19*q^
11+20*q^10+27*q^9+33*q^8+36*q^7+41*q^6+43*q^5+38*q^4+26*q^3+14*q^2+5*q+1, q^21+
6*q^20+16*q^19+27*q^18+35*q^17+39*q^16+41*q^15+53*q^14+67*q^13+77*q^12+83*q^11+
91*q^10+112*q^9+124*q^8+129*q^7+128*q^6+110*q^5+79*q^4+45*q^3+20*q^2+6*q+1, 1+
43*q^25+80*q^22+62*q^24+85*q^21+74*q^23+7*q+27*q^2+22*q^26+71*q^3+145*q^4+238*q
^5+327*q^6+385*q^7+411*q^8+408*q^9+374*q^10+336*q^11+306*q^12+275*q^13+234*q^14
+200*q^15+184*q^16+q^28+173*q^17+158*q^18+132*q^19+104*q^20+7*q^27, 1+316*q^25+
401*q^22+358*q^24+435*q^21+382*q^23+8*q+154*q^30+165*q^29+29*q^34+65*q^33+35*q^
2+259*q^26+105*q^3+244*q^4+459*q^5+725*q^6+105*q^32+982*q^7+1182*q^8+1297*q^9+
1318*q^10+1274*q^11+1198*q^12+1109*q^13+992*q^14+867*q^15+763*q^16+177*q^28+690
*q^17+642*q^18+573*q^19+501*q^20+208*q^27+136*q^31+8*q^35+q^36, 1+1355*q^25+94*
q^42+1670*q^22+1460*q^24+1870*q^21+1547*q^23+9*q+9*q^44+834*q^30+q^45+342*q^37+
878*q^29+290*q^39+170*q^41+513*q^34+319*q^38+625*q^33+44*q^2+1213*q^26+148*q^3+
385*q^4+814*q^5+1449*q^6+730*q^32+2218*q^7+2996*q^8+3648*q^9+4081*q^10+4276*q^
11+4278*q^12+4157*q^13+3930*q^14+3622*q^15+3267*q^16+37*q^43+944*q^28+2932*q^17
+2661*q^18+2392*q^19+2128*q^20+1064*q^27+241*q^40+798*q^31+424*q^35+370*q^36, 1
+5507*q^25+1241*q^42+7338*q^22+5980*q^24+8291*q^21+6558*q^23+10*q+264*q^51+876*
q^44+46*q^53+3445*q^30+131*q^52+q^55+10*q^54+776*q^45+1921*q^37+3684*q^29+712*q
^46+1759*q^39+1468*q^41+2568*q^34+411*q^50+1835*q^38+2877*q^33+54*q^2+5033*q^26
+201*q^3+578*q^4+1354*q^5+2676*q^6+3119*q^32+4560*q^7+6844*q^8+9212*q^9+11314*q
^10+12886*q^11+13840*q^12+14237*q^13+14157*q^14+13712*q^15+12977*q^16+1030*q^43
+531*q^49+4052*q^28+12056*q^17+11118*q^18+10194*q^19+9270*q^20+4530*q^27+1648*q
^40+3296*q^31+2273*q^35+609*q^48+2054*q^36+661*q^47, 1+24059*q^25+6101*q^42+
32187*q^22+26426*q^24+35329*q^21+29132*q^23+11*q+2941*q^51+4912*q^44+2099*q^53+
1373*q^57+14347*q^30+2484*q^52+1270*q^58+1631*q^55+942*q^60+1824*q^54+4498*q^45
+8166*q^37+15811*q^29+4224*q^46+11*q^65+7401*q^39+675*q^61+6678*q^41+10768*q^34
+3366*q^50+56*q^64+7709*q^38+11705*q^33+1488*q^56+65*q^2+21874*q^26+265*q^3+834
*q^4+2141*q^5+4644*q^6+12488*q^32+8704*q^7+14369*q^8+21231*q^9+395*q^62+28506*q
^10+35270*q^11+40796*q^12+44725*q^13+46994*q^14+47771*q^15+47263*q^16+q^66+5462
*q^43+3678*q^49+17659*q^28+45703*q^17+43459*q^18+40933*q^19+38269*q^20+19738*q^
27+7103*q^40+177*q^63+13301*q^31+9769*q^35+3878*q^48+1140*q^59+8857*q^36+4033*q
^47, 1+103958*q^25+25245*q^42+131448*q^22+112788*q^24+140148*q^21+122102*q^23+
12*q+14268*q^51+21278*q^44+11744*q^53+8897*q^57+64972*q^30+13001*q^52+8540*q^58
+9910*q^55+7609*q^60+10695*q^54+19527*q^45+2082*q^71+572*q^74+34351*q^37+3119*q
^67+72074*q^29+18202*q^46+3817*q^65+2643*q^69+29737*q^39+6831*q^61+26901*q^41+
45430*q^34+15335*q^50+4326*q^64+31685*q^38+49416*q^33+9334*q^56+77*q^2+2861*q^
68+95623*q^26+341*q^3+1165*q^4+3249*q^5+7664*q^6+53702*q^32+15647*q^7+28157*q^8
+45328*q^9+5906*q^62+66168*q^10+88675*q^11+110458*q^12+67*q^76+129439*q^13+
144270*q^14+154441*q^15+160005*q^16+3431*q^66+2410*q^70+12*q^77+q^78+23284*q^43
+16087*q^49+79731*q^28+161335*q^17+159056*q^18+154211*q^19+147804*q^20+87629*q^
27+1617*q^72+28286*q^40+1070*q^73+5025*q^63+58806*q^31+233*q^75+41510*q^35+
16669*q^48+8156*q^59+37718*q^36+17296*q^47, 1+425222*q^25+103685*q^42+505892*q^
22+13*q^90+453611*q^24+526211*q^21+481100*q^23+13*q+58581*q^51+89274*q^44+50756
*q^53+38987*q^57+289999*q^30+54892*q^52+37601*q^58+43421*q^55+35007*q^60+q^91+
46772*q^54+82088*q^45+17019*q^71+11935*q^74+154196*q^37+20690*q^67+316281*q^29+
75720*q^46+23474*q^65+18854*q^69+127688*q^39+33081*q^61+110613*q^41+204149*q^34
+6550*q^79+61584*q^50+25467*q^64+139718*q^38+222160*q^33+40861*q^56+90*q^2+
19695*q^68+397111*q^26+430*q^3+1584*q^4+4765*q^5+12133*q^6+242166*q^32+26772*q^
7+52112*q^8+90761*q^9+30597*q^62+143208*q^10+207012*q^11+277135*q^12+8985*q^76+
347245*q^13+411353*q^14+465045*q^15+505836*q^16+21915*q^66+18032*q^70+8010*q^77
+7221*q^78+96637*q^43+64135*q^49+342915*q^28+532901*q^17+546544*q^18+548356*q^
19+540858*q^20+369764*q^27+15615*q^72+118244*q^40+13829*q^73+27919*q^63+264929*
q^31+10271*q^75+187086*q^35+5980*q^80+66938*q^48+4492*q^83+5504*q^81+2687*q^85+
1642*q^86+805*q^87+300*q^88+79*q^89+3699*q^84+5053*q^82+36406*q^59+170328*q^36+
70646*q^47, 1+1662122*q^25+458949*q^42+1853550*q^22+16832*q^90+1740349*q^24+
1878345*q^21+1805932*q^23+14*q+8191*q^97+236635*q^51+390930*q^44+209827*q^53+
158928*q^57+1222289*q^30+11484*q^94+223797*q^52+151501*q^58+181037*q^55+140883*
q^60+15201*q^91+195152*q^54+358452*q^45+79686*q^71+65949*q^74+9545*q^96+6386*q^
98+704356*q^37+92222*q^67+1310950*q^29+328213*q^46+12530*q^93+4329*q^99+103783*
q^65+85192*q^69+587236*q^39+135322*q^61+496061*q^41+899232*q^34+45967*q^79+
249150*q^50+111591*q^64+2447*q^100+643328*q^38+971169*q^33+168747*q^56+104*q^2+
88298*q^68+1576630*q^26+533*q^3+2105*q^4+6790*q^5+18548*q^6+1049844*q^32+43947*
q^7+91929*q^8+172138*q^9+10557*q^95+128352*q^62+1105*q^101+291946*q^10+453129*q
^11+649920*q^12+55673*q^76+869521*q^13+1095029*q^14+1309350*q^15+1498414*q^16+
97308*q^66+82493*q^70+51769*q^77+48608*q^78+424424*q^43+379*q^102+263031*q^49+
1399520*q^28+1652698*q^17+1766993*q^18+1840359*q^19+1875752*q^20+1488284*q^27+
76105*q^72+538193*q^40+71422*q^73+92*q^103+120129*q^63+1134479*q^31+60472*q^75+
832366*q^35+14*q^104+43692*q^80+280203*q^48+q^105+37854*q^83+41717*q^81+31883*q
^85+28034*q^86+24301*q^87+21177*q^88+18754*q^89+35249*q^84+39880*q^82+145859*q^
59+767807*q^36+13771*q^92+301822*q^47]


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

[1, 2, 6, 22, 86, 336, 1290, 4870, 18164, 67234, 247786, 911120, 3346618,

    12286942, 45104548]

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

     is given by

[1, 1, 3, 11, 43, 168, 645, 2436, 9085, 33617, 123895, 455559, 1673307, 6143472,

    22552273]

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

     is given by

[0, 1, 3, 11, 43, 168, 645, 2434, 9079, 33617, 123891, 455561, 1673311, 6143470,

    22552275]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 43, 168, 645, 2436, 9085, 33617, 123891, 455559, 1673307,

    6143470, 22552275]

 ODD:, [0, 1, 3, 11, 43, 168, 645, 2434, 9079, 33617, 123895, 455561, 1673311,

    6143472, 22552273]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.045454545, 5.220930233, 8.119047619,

    11.81937984, 16.39096509, 21.89148866, 28.36477080, 35.83991428,

    44.33290785, 53.84987142, 64.39068142, 75.95203714]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.521879004, 2.238153116, 3.122135876,

    4.162113683, 5.330659884, 6.592460328, 7.909412711, 9.245136473,

    10.56866067, 11.85656927, 13.09343128, 14.27092508]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.31612, 5.00933, 9.74773, 17.3232, 28.4159,

    43.4605, 62.5588, 85.4725, 111.697, 140.578, 171.438, 203.659]

Skewness: , [Float(undefined), 0., 0., -0.07672933926, -0.2027177403,

    -0.3452353263, -0.4867453710, -0.6252982108, -0.7624649980, -0.8990703424,

    -1.034645828, -1.167777921, -1.296644515, -1.419242171, -1.533825259]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.266194381,

    2.324967127, 2.405632852, 2.542559618, 2.739034456, 2.996741248,

    3.318348459, 3.705033360, 4.154952445, 4.662715733, 5.219286974,

    5.812934111]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+2*q+4*q^2+6*q^3+5*q^4+3*q^5+q^6, 1+2*q+3*q^2+7*q^3+
12*q^4+16*q^5+18*q^6+15*q^7+9*q^8+4*q^9+q^10, 1+2*q+3*q^2+5*q^3+10*q^4+16*q^5+
26*q^6+39*q^7+52*q^8+60*q^9+58*q^10+46*q^11+29*q^12+14*q^13+5*q^14+q^15, 1+2*q+
3*q^2+5*q^3+8*q^4+12*q^5+21*q^6+31*q^7+49*q^8+74*q^9+106*q^10+143*q^11+182*q^12
+209*q^13+215*q^14+193*q^15+147*q^16+94*q^17+49*q^18+20*q^19+6*q^20+q^21, 1+2*q
+3*q^2+5*q^3+8*q^4+10*q^5+18*q^6+26*q^7+35*q^8+54*q^9+80*q^10+118*q^11+173*q^12
+246*q^13+336*q^14+446*q^15+564*q^16+678*q^17+768*q^18+804*q^19+764*q^20+649*q^
21+483*q^22+310*q^23+169*q^24+76*q^25+27*q^26+7*q^27+q^28, 1+2*q+3*q^2+5*q^3+8*
q^4+10*q^5+16*q^6+23*q^7+33*q^8+45*q^9+61*q^10+85*q^11+120*q^12+172*q^13+245*q^
14+342*q^15+482*q^16+669*q^17+906*q^18+1199*q^19+1539*q^20+1917*q^21+2304*q^22+
2665*q^23+2946*q^24+3082*q^25+3010*q^26+2706*q^27+2207*q^28+1611*q^29+1038*q^30
+582*q^31+279*q^32+111*q^33+35*q^34+8*q^35+q^36, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5
+16*q^6+21*q^7+30*q^8+43*q^9+56*q^10+76*q^11+99*q^12+132*q^13+175*q^14+242*q^15
+327*q^16+443*q^17+610*q^18+835*q^19+1144*q^20+1559*q^21+2091*q^22+2766*q^23+
3602*q^24+4606*q^25+5759*q^26+7028*q^27+8356*q^28+9660*q^29+10828*q^30+11712*q^
31+12151*q^32+11982*q^33+11106*q^34+9567*q^35+7570*q^36+5439*q^37+3510*q^38+
2010*q^39+1007*q^40+433*q^41+155*q^42+44*q^43+9*q^44+q^45, 1+2*q+3*q^2+5*q^3+8*
q^4+10*q^5+16*q^6+21*q^7+28*q^8+40*q^9+54*q^10+71*q^11+94*q^12+122*q^13+152*q^
14+201*q^15+262*q^16+335*q^17+441*q^18+576*q^19+751*q^20+999*q^21+1327*q^22+
1772*q^23+2379*q^24+3176*q^25+4208*q^26+5563*q^27+7272*q^28+9405*q^29+12018*q^
30+15127*q^31+18736*q^32+22835*q^33+27319*q^34+32036*q^35+36762*q^36+41209*q^37
+45031*q^38+47814*q^39+49103*q^40+48470*q^41+45634*q^42+40606*q^43+33826*q^44+
26129*q^45+18538*q^46+11967*q^47+6960*q^48+3605*q^49+1639*q^50+641*q^51+209*q^
52+54*q^53+10*q^54+q^55, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38
*q^9+51*q^10+69*q^11+89*q^12+117*q^13+146*q^14+188*q^15+236*q^16+299*q^17+377*q
^18+477*q^19+605*q^20+763*q^21+966*q^22+1226*q^23+1566*q^24+2015*q^25+2617*q^26
+3403*q^27+4439*q^28+5806*q^29+7586*q^30+9906*q^31+12905*q^32+16722*q^33+21528*
q^34+27483*q^35+34753*q^36+43499*q^37+53860*q^38+65868*q^39+79535*q^40+94711*q^
41+111133*q^42+128384*q^43+145848*q^44+162702*q^45+177962*q^46+190499*q^47+
199075*q^48+202401*q^49+199302*q^50+188991*q^51+171417*q^52+147592*q^53+119677*
q^54+90658*q^55+63647*q^56+41080*q^57+24172*q^58+12845*q^59+6094*q^60+2543*q^61
+914*q^62+274*q^63+65*q^64+11*q^65+q^66, 1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+
21*q^7+28*q^8+38*q^9+49*q^10+66*q^11+87*q^12+112*q^13+141*q^14+182*q^15+227*q^
16+283*q^17+353*q^18+435*q^19+541*q^20+676*q^21+836*q^22+1030*q^23+1275*q^24+
1572*q^25+1943*q^26+2425*q^27+3021*q^28+3792*q^29+4786*q^30+6060*q^31+7705*q^32
+9835*q^33+12588*q^34+16143*q^35+20752*q^36+26661*q^37+34171*q^38+43677*q^39+
55569*q^40+70327*q^41+88484*q^42+110528*q^43+136992*q^44+168323*q^45+204896*q^
46+246994*q^47+294730*q^48+347919*q^49+406085*q^50+468306*q^51+533257*q^52+
599066*q^53+663385*q^54+723302*q^55+775448*q^56+816105*q^57+841399*q^58+847552*
q^59+831288*q^60+790491*q^61+725063*q^62+637721*q^63+534399*q^64+423801*q^65+
315902*q^66+219819*q^67+141808*q^68+84202*q^69+45655*q^70+22396*q^71+9825*q^72+
3796*q^73+1264*q^74+351*q^75+77*q^76+12*q^77+q^78, 1+2*q+3*q^2+5*q^3+8*q^4+10*q
^5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64*q^11+84*q^12+110*q^13+136*q^14+177*q^
15+221*q^16+274*q^17+341*q^18+421*q^19+511*q^20+630*q^21+772*q^22+940*q^23+1156
*q^24+1402*q^25+1698*q^26+2065*q^27+2507*q^28+3034*q^29+3698*q^30+4510*q^31+
5508*q^32+6779*q^33+8346*q^34+10286*q^35+12768*q^36+15891*q^37+19847*q^38+24925
*q^39+31372*q^40+39570*q^41+50040*q^42+63286*q^43+80004*q^44+101047*q^45+127327
*q^46+159964*q^47+200291*q^48+249625*q^49+309447*q^50+381379*q^51+467008*q^52+
567956*q^53+685792*q^54+821724*q^55+976676*q^56+1151141*q^57+1344862*q^58+
1556950*q^59+1785505*q^60+2027343*q^61+2277957*q^62+2531422*q^63+2780202*q^64+
3015367*q^65+3226539*q^66+3402179*q^67+3530023*q^68+3597758*q^69+3593968*q^70+
3509413*q^71+3338707*q^72+3082384*q^73+2748951*q^74+2356083*q^75+1929990*q^76+
1502306*q^77+1104724*q^78+762908*q^79+491822*q^80+294137*q^81+162090*q^82+81673
*q^83+37281*q^84+15236*q^85+5488*q^86+1704*q^87+441*q^88+90*q^89+13*q^90+q^91,
1+2*q+3*q^2+5*q^3+8*q^4+10*q^5+16*q^6+21*q^7+28*q^8+38*q^9+49*q^10+64*q^11+82*q
^12+107*q^13+134*q^14+172*q^15+216*q^16+268*q^17+332*q^18+409*q^19+501*q^20+610
*q^21+738*q^22+894*q^23+1085*q^24+1310*q^25+1580*q^26+1899*q^27+2280*q^28+2734*
q^29+3269*q^30+3910*q^31+4669*q^32+5587*q^33+6695*q^34+8043*q^35+9669*q^36+
11641*q^37+14051*q^38+16980*q^39+20591*q^40+25055*q^41+30584*q^42+37468*q^43+
46096*q^44+56886*q^45+70424*q^46+87423*q^47+108725*q^48+135430*q^49+168927*q^50
+210792*q^51+262932*q^52+327686*q^53+407656*q^54+505897*q^55+625991*q^56+771721
*q^57+947514*q^58+1158077*q^59+1408449*q^60+1703909*q^61+2049739*q^62+2450924*q
^63+2912238*q^64+3437887*q^65+4031058*q^66+4693637*q^67+5425718*q^68+6225046*q^
69+7086860*q^70+8003228*q^71+8962594*q^72+9949404*q^73+10943516*q^74+11920150*q
^75+12850046*q^76+13699726*q^77+14432043*q^78+15007348*q^79+15384928*q^80+
15525499*q^81+15394370*q^82+14965496*q^83+14226261*q^84+13182790*q^85+11865021*
q^86+10329876*q^87+8660282*q^88+6958150*q^89+5331052*q^90+3875055*q^91+2658576*
q^92+1712560*q^93+1030149*q^94+575270*q^95+296293*q^96+139679*q^97+59709*q^98+
22869*q^99+7723*q^100+2248*q^101+545*q^102+104*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+4*q^4+2*q^5+q^6, q^10+2*q^9+3*q^8+7
*q^7+12*q^6+16*q^5+18*q^4+15*q^3+9*q^2+4*q+1, q^15+2*q^14+3*q^13+5*q^12+10*q^11
+16*q^10+26*q^9+39*q^8+52*q^7+60*q^6+58*q^5+46*q^4+29*q^3+14*q^2+5*q+1, q^21+2*
q^20+3*q^19+5*q^18+8*q^17+12*q^16+21*q^15+31*q^14+49*q^13+74*q^12+106*q^11+143*
q^10+182*q^9+209*q^8+215*q^7+193*q^6+147*q^5+94*q^4+49*q^3+20*q^2+6*q+1, 1+5*q^
25+18*q^22+8*q^24+26*q^21+10*q^23+7*q+27*q^2+3*q^26+76*q^3+169*q^4+310*q^5+483*
q^6+649*q^7+764*q^8+804*q^9+768*q^10+678*q^11+564*q^12+446*q^13+336*q^14+246*q^
15+173*q^16+q^28+118*q^17+80*q^18+54*q^19+35*q^20+2*q^27, 1+85*q^25+245*q^22+
120*q^24+342*q^21+172*q^23+8*q+16*q^30+23*q^29+3*q^34+5*q^33+35*q^2+61*q^26+111
*q^3+279*q^4+582*q^5+1038*q^6+8*q^32+1611*q^7+2207*q^8+2706*q^9+3010*q^10+3082*
q^11+2946*q^12+2665*q^13+2304*q^14+1917*q^15+1539*q^16+33*q^28+1199*q^17+906*q^
18+669*q^19+482*q^20+45*q^27+10*q^31+2*q^35+q^36, 1+1144*q^25+5*q^42+2766*q^22+
1559*q^24+3602*q^21+2091*q^23+9*q+2*q^44+242*q^30+q^45+30*q^37+327*q^29+16*q^39
+8*q^41+76*q^34+21*q^38+99*q^33+44*q^2+835*q^26+155*q^3+433*q^4+1007*q^5+2010*q
^6+132*q^32+3510*q^7+5439*q^8+7570*q^9+9567*q^10+11106*q^11+11982*q^12+12151*q^
13+11712*q^14+10828*q^15+9660*q^16+3*q^43+443*q^28+8356*q^17+7028*q^18+5759*q^
19+4606*q^20+610*q^27+10*q^40+175*q^31+56*q^35+43*q^36, 1+12018*q^25+122*q^42+
22835*q^22+15127*q^24+27319*q^21+18736*q^23+10*q+8*q^51+71*q^44+3*q^53+3176*q^
30+5*q^52+q^55+2*q^54+54*q^45+441*q^37+4208*q^29+40*q^46+262*q^39+152*q^41+999*
q^34+10*q^50+335*q^38+1327*q^33+54*q^2+9405*q^26+209*q^3+641*q^4+1639*q^5+3605*
q^6+1772*q^32+6960*q^7+11967*q^8+18538*q^9+26129*q^10+33826*q^11+40606*q^12+
45634*q^13+48470*q^14+49103*q^15+47814*q^16+94*q^43+16*q^49+5563*q^28+45031*q^
17+41209*q^18+36762*q^19+32036*q^20+7272*q^27+201*q^40+2379*q^31+751*q^35+21*q^
48+576*q^36+28*q^47, 1+94711*q^25+1566*q^42+145848*q^22+111133*q^24+162702*q^21
+128384*q^23+11*q+188*q^51+966*q^44+117*q^53+38*q^57+34753*q^30+146*q^52+28*q^
58+69*q^55+16*q^60+89*q^54+763*q^45+5806*q^37+43499*q^29+605*q^46+2*q^65+3403*q
^39+10*q^61+2015*q^41+12905*q^34+236*q^50+3*q^64+4439*q^38+16722*q^33+51*q^56+
65*q^2+79535*q^26+274*q^3+914*q^4+2543*q^5+6094*q^6+21528*q^32+12845*q^7+24172*
q^8+41080*q^9+8*q^62+63647*q^10+90658*q^11+119677*q^12+147592*q^13+171417*q^14+
188991*q^15+199302*q^16+q^66+1226*q^43+299*q^49+53860*q^28+202401*q^17+199075*q
^18+190499*q^19+177962*q^20+65868*q^27+2617*q^40+5*q^63+27483*q^31+9906*q^35+
377*q^48+21*q^59+7586*q^36+477*q^47, 1+599066*q^25+20752*q^42+775448*q^22+
663385*q^24+816105*q^21+723302*q^23+12*q+2425*q^51+12588*q^44+1572*q^53+676*q^
57+294730*q^30+1943*q^52+541*q^58+1030*q^55+353*q^60+1275*q^54+9835*q^45+21*q^
71+8*q^74+70327*q^37+66*q^67+347919*q^29+7705*q^46+112*q^65+38*q^69+43677*q^39+
283*q^61+26661*q^41+136992*q^34+3021*q^50+141*q^64+55569*q^38+168323*q^33+836*q
^56+77*q^2+49*q^68+533257*q^26+351*q^3+1264*q^4+3796*q^5+9825*q^6+204896*q^32+
22396*q^7+45655*q^8+84202*q^9+227*q^62+141808*q^10+219819*q^11+315902*q^12+3*q^
76+423801*q^13+534399*q^14+637721*q^15+725063*q^16+87*q^66+28*q^70+2*q^77+q^78+
16143*q^43+3792*q^49+406085*q^28+790491*q^17+831288*q^18+847552*q^19+841399*q^
20+468306*q^27+16*q^72+34171*q^40+10*q^73+182*q^63+246994*q^31+5*q^75+110528*q^
35+4786*q^48+435*q^59+88484*q^36+6060*q^47, 1+3226539*q^25+249625*q^42+3597758*
q^22+2*q^90+3402179*q^24+3593968*q^21+3530023*q^23+13*q+31372*q^51+159964*q^44+
19847*q^53+8346*q^57+2027343*q^30+24925*q^52+6779*q^58+12768*q^55+4510*q^60+q^
91+15891*q^54+127327*q^45+511*q^71+274*q^74+685792*q^37+1156*q^67+2277957*q^29+
101047*q^46+1698*q^65+772*q^69+467008*q^39+3698*q^61+309447*q^41+1151141*q^34+
84*q^79+39570*q^50+2065*q^64+567956*q^38+1344862*q^33+10286*q^56+90*q^2+940*q^
68+3015367*q^26+441*q^3+1704*q^4+5488*q^5+15236*q^6+1556950*q^32+37281*q^7+
81673*q^8+162090*q^9+3034*q^62+294137*q^10+491822*q^11+762908*q^12+177*q^76+
1104724*q^13+1502306*q^14+1929990*q^15+2356083*q^16+1402*q^66+630*q^70+136*q^77
+110*q^78+200291*q^43+50040*q^49+2531422*q^28+2748951*q^17+3082384*q^18+3338707
*q^19+3509413*q^20+2780202*q^27+421*q^72+381379*q^40+341*q^73+2507*q^63+1785505
*q^31+221*q^75+976676*q^35+64*q^80+63286*q^48+28*q^83+49*q^81+16*q^85+10*q^86+8
*q^87+5*q^88+3*q^89+21*q^84+38*q^82+5508*q^59+821724*q^36+80004*q^47, 1+
15384928*q^25+2450924*q^42+14965496*q^22+172*q^90+15525499*q^24+14226261*q^21+
15394370*q^23+14*q+28*q^97+407656*q^51+1703909*q^44+262932*q^53+108725*q^57+
11920150*q^30+64*q^94+327686*q^52+87423*q^58+168927*q^55+56886*q^60+134*q^91+
210792*q^54+1408449*q^45+6695*q^71+3910*q^74+38*q^96+21*q^98+5425718*q^37+14051
*q^67+12850046*q^29+1158077*q^46+82*q^93+16*q^99+20591*q^65+9669*q^69+4031058*q
^39+46096*q^61+2912238*q^41+8003228*q^34+1580*q^79+505897*q^50+25055*q^64+10*q^
100+4693637*q^38+8962594*q^33+135430*q^56+104*q^2+11641*q^68+15007348*q^26+545*
q^3+2248*q^4+7723*q^5+22869*q^6+9949404*q^32+59709*q^7+139679*q^8+296293*q^9+49
*q^95+37468*q^62+8*q^101+575270*q^10+1030149*q^11+1712560*q^12+2734*q^76+
2658576*q^13+3875055*q^14+5331052*q^15+6958150*q^16+16980*q^66+8043*q^70+2280*q
^77+1899*q^78+2049739*q^43+5*q^102+625991*q^49+13699726*q^28+8660282*q^17+
10329876*q^18+11865021*q^19+13182790*q^20+14432043*q^27+5587*q^72+3437887*q^40+
4669*q^73+3*q^103+30584*q^63+10943516*q^31+3269*q^75+7086860*q^35+2*q^104+1310*
q^80+771721*q^48+q^105+738*q^83+1085*q^81+501*q^85+409*q^86+332*q^87+268*q^88+
216*q^89+610*q^84+894*q^82+70424*q^59+6225046*q^36+107*q^92+947514*q^47]


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

[1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230, 13388094,

    62688448, 295504025]

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

     is given by

[1, 1, 3, 11, 44, 184, 787, 3430, 15234, 68613, 312781, 1440622, 6694058,

    31344218, 147752030]

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

     is given by

[0, 1, 3, 11, 44, 183, 784, 3431, 15234, 68616, 312792, 1440608, 6694036,

    31344230, 147751995]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 44, 183, 784, 3430, 15234, 68616, 312792, 1440622, 6694058,

    31344230, 147751995]

 ODD:, [0, 1, 3, 11, 44, 184, 787, 3431, 15234, 68613, 312781, 1440608, 6694036,

    31344218, 147752030]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.136363636, 5.545454545, 8.798365123,

    12.92679822, 17.94446874, 23.85703033, 30.66703102, 38.37631420,

    46.98690767, 56.50116066, 66.92162042, 78.25088203]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.455255509, 1.982360226, 2.534276748,

    3.100951765, 3.674360413, 4.252142399, 4.836342268, 5.430869228,

    6.039615557, 6.665594089, 7.310785033, 7.976307756]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.11777, 3.92975, 6.42256, 9.61590, 13.5009,

    18.0807, 23.3902, 29.4943, 36.4770, 44.4301, 53.4476, 63.6215]

Skewness: , [Float(undefined), 0., 0., -0.1491938012, -0.3211608239,

    -0.4732883689, -0.5868952386, -0.6612960998, -0.7027148802, -0.7195566069,

    -0.7201667203, -0.7115655213, -0.6988890723, -0.6853668981, -0.6728034189]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.547029639,

    2.939894510, 3.293342576, 3.597621621, 3.832630255, 3.984867933,

    4.055518361, 4.059000184, 4.016356794, 3.948351875, 3.871113260,

    3.795270890]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+9*q^2+13*q^3
+14*q^4+14*q^5+13*q^6+10*q^7+7*q^8+4*q^9+q^10, 1+5*q+14*q^2+26*q^3+36*q^4+42*q^
5+45*q^6+46*q^7+45*q^8+40*q^9+33*q^10+26*q^11+18*q^12+11*q^13+5*q^14+q^15, 1+6*
q+20*q^2+45*q^3+76*q^4+105*q^5+128*q^6+145*q^7+156*q^8+162*q^9+163*q^10+157*q^
11+145*q^12+130*q^13+111*q^14+88*q^15+67*q^16+48*q^17+30*q^18+16*q^19+6*q^20+q^
21, 1+7*q+27*q^2+71*q^3+141*q^4+227*q^5+315*q^6+397*q^7+468*q^8+523*q^9+564*q^
10+594*q^11+609*q^12+611*q^13+601*q^14+574*q^15+536*q^16+489*q^17+431*q^18+367*
q^19+302*q^20+238*q^21+178*q^22+127*q^23+83*q^24+47*q^25+22*q^26+7*q^27+q^28, 1
+8*q+35*q^2+105*q^3+239*q^4+440*q^5+690*q^6+965*q^7+1245*q^8+1511*q^9+1748*q^10
+1948*q^11+2109*q^12+2236*q^13+2330*q^14+2389*q^15+2414*q^16+2403*q^17+2357*q^
18+2277*q^19+2160*q^20+2012*q^21+1841*q^22+1651*q^23+1448*q^24+1237*q^25+1026*q
^26+825*q^27+643*q^28+481*q^29+341*q^30+227*q^31+136*q^32+70*q^33+29*q^34+8*q^
35+q^36, 1+9*q+44*q^2+148*q^3+379*q^4+785*q^5+1377*q^6+2129*q^7+2998*q^8+3935*q
^9+4889*q^10+5814*q^11+6669*q^12+7425*q^13+8077*q^14+8629*q^15+9077*q^16+9422*q
^17+9678*q^18+9847*q^19+9913*q^20+9875*q^21+9736*q^22+9497*q^23+9167*q^24+8743*
q^25+8228*q^26+7642*q^27+7003*q^28+6319*q^29+5609*q^30+4893*q^31+4182*q^32+3495
*q^33+2851*q^34+2265*q^35+1748*q^36+1304*q^37+928*q^38+622*q^39+386*q^40+213*q^
41+100*q^42+37*q^43+9*q^44+q^45, 1+10*q+54*q^2+201*q^3+571*q^4+1313*q^5+2550*q^
6+4334*q^7+6644*q^8+9408*q^9+12519*q^10+15849*q^11+19266*q^12+22641*q^13+25856*
q^14+28831*q^15+31520*q^16+33883*q^17+35918*q^18+37654*q^19+39092*q^20+40237*q^
21+41102*q^22+41678*q^23+41969*q^24+41976*q^25+41669*q^26+41041*q^27+40122*q^28
+38923*q^29+37446*q^30+35714*q^31+33744*q^32+31564*q^33+29222*q^34+26758*q^35+
24218*q^36+21654*q^37+19101*q^38+16594*q^39+14185*q^40+11913*q^41+9811*q^42+
7911*q^43+6233*q^44+4782*q^45+3553*q^46+2536*q^47+1716*q^48+1086*q^49+629*q^50+
321*q^51+138*q^52+46*q^53+10*q^54+q^55, 1+11*q+65*q^2+265*q^3+826*q^4+2086*q^5+
4444*q^6+8250*q^7+13720*q^8+20915*q^9+29755*q^10+40038*q^11+51471*q^12+63707*q^
13+76375*q^14+89115*q^15+101593*q^16+113506*q^17+124628*q^18+134825*q^19+144017
*q^20+152168*q^21+159283*q^22+165401*q^23+170583*q^24+174865*q^25+178247*q^26+
180720*q^27+182269*q^28+182887*q^29+182575*q^30+181320*q^31+179109*q^32+175947*
q^33+171840*q^34+166827*q^35+160975*q^36+154335*q^37+146970*q^38+138964*q^39+
130405*q^40+121391*q^41+112053*q^42+102523*q^43+92934*q^44+83417*q^45+74083*q^
46+65038*q^47+56386*q^48+48221*q^49+40629*q^50+33693*q^51+27464*q^52+21964*q^53
+17194*q^54+13132*q^55+9734*q^56+6955*q^57+4746*q^58+3053*q^59+1824*q^60+988*q^
61+468*q^62+185*q^63+56*q^64+11*q^65+q^66, 1+12*q+77*q^2+341*q^3+1156*q^4+3178*
q^5+7367*q^6+14844*q^7+26669*q^8+43659*q^9+66296*q^10+94693*q^11+128599*q^12+
167445*q^13+210412*q^14+256527*q^15+304757*q^16+354026*q^17+403248*q^18+451454*
q^19+497842*q^20+541730*q^21+582581*q^22+620068*q^23+654056*q^24+684537*q^25+
711593*q^26+735318*q^27+755790*q^28+773139*q^29+787474*q^30+798819*q^31+807181*
q^32+812527*q^33+814770*q^34+813865*q^35+809818*q^36+802636*q^37+792357*q^38+
779023*q^39+762632*q^40+743240*q^41+720991*q^42+696047*q^43+668629*q^44+638999*
q^45+607389*q^46+574063*q^47+539320*q^48+503471*q^49+466885*q^50+429964*q^51+
393083*q^52+356607*q^53+320902*q^54+286288*q^55+253049*q^56+221447*q^57+191713*
q^58+164048*q^59+138631*q^60+115581*q^61+94968*q^62+76810*q^63+61041*q^64+47544
*q^65+36175*q^66+26770*q^67+19151*q^68+13147*q^69+8575*q^70+5245*q^71+2960*q^72
+1503*q^73+663*q^74+242*q^75+67*q^76+12*q^77+q^78, 1+13*q+90*q^2+430*q^3+1574*q
^4+4676*q^5+11713*q^6+25465*q^7+49210*q^8+86265*q^9+139560*q^10+211324*q^11+
302877*q^12+414532*q^13+545606*q^14+694530*q^15+859044*q^16+1036400*q^17+
1223500*q^18+1417033*q^19+1613690*q^20+1810307*q^21+2003862*q^22+2191610*q^23+
2371280*q^24+2541080*q^25+2699743*q^26+2846490*q^27+2980821*q^28+3102605*q^29+
3212139*q^30+3309812*q^31+3395997*q^32+3471118*q^33+3535465*q^34+3589182*q^35+
3632408*q^36+3665188*q^37+3687504*q^38+3699378*q^39+3700688*q^40+3691215*q^41+
3670909*q^42+3639789*q^43+3597907*q^44+3545461*q^45+3482662*q^46+3409705*q^47+
3326851*q^48+3234438*q^49+3132922*q^50+3022963*q^51+2905309*q^52+2780713*q^53+
2650065*q^54+2514315*q^55+2374392*q^56+2231293*q^57+2086079*q^58+1939823*q^59+
1793671*q^60+1648772*q^61+1506236*q^62+1367152*q^63+1232503*q^64+1103123*q^65+
979764*q^66+863099*q^67+753678*q^68+651963*q^69+558318*q^70+472985*q^71+396076*
q^72+327563*q^73+267255*q^74+214830*q^75+169839*q^76+131733*q^77+99929*q^78+
73846*q^79+52898*q^80+36496*q^81+24062*q^82+15003*q^83+8730*q^84+4659*q^85+2223
*q^86+916*q^87+310*q^88+79*q^89+13*q^90+q^91, 1+14*q+104*q^2+533*q^3+2094*q^4+
6681*q^5+17976*q^6+41943*q^7+86806*q^8+162439*q^9+279395*q^10+447866*q^11+
676737*q^12+972812*q^13+1340281*q^14+1780472*q^15+2291890*q^16+2870480*q^17+
3509990*q^18+4202399*q^19+4938450*q^20+5708123*q^21+6500914*q^22+7306142*q^23+
8113343*q^24+8912600*q^25+9694873*q^26+10452219*q^27+11177808*q^28+11866087*q^
29+12513012*q^30+13115999*q^31+13673729*q^32+14185852*q^33+14652711*q^34+
15075203*q^35+15454567*q^36+15792176*q^37+16089412*q^38+16347501*q^39+16567345*
q^40+16749449*q^41+16893940*q^42+17000753*q^43+17069853*q^44+17101174*q^45+
17094610*q^46+17050132*q^47+16967669*q^48+16847090*q^49+16688392*q^50+16491768*
q^51+16257658*q^52+15986795*q^53+15680082*q^54+15338600*q^55+14963693*q^56+
14556828*q^57+14119598*q^58+13653908*q^59+13162005*q^60+12646425*q^61+12110008*
q^62+11555840*q^63+10987113*q^64+10407108*q^65+9819102*q^66+9226385*q^67+
8632391*q^68+8040614*q^69+7454521*q^70+6877569*q^71+6313115*q^72+5764299*q^73+
5234034*q^74+4724936*q^75+4239266*q^76+3778927*q^77+3345462*q^78+2940077*q^79+
2563716*q^80+2217017*q^81+1900273*q^82+1613433*q^83+1356101*q^84+1127499*q^85+
926482*q^86+751597*q^87+601113*q^88+473131*q^89+365656*q^90+276680*q^91+204249*
q^92+146476*q^93+101502*q^94+67512*q^95+42746*q^96+25489*q^97+14120*q^98+7136*q
^99+3207*q^100+1238*q^101+390*q^102+92*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, q^10+4*q^9+9*q^8+
13*q^7+14*q^6+14*q^5+13*q^4+10*q^3+7*q^2+4*q+1, q^15+5*q^14+14*q^13+26*q^12+36*
q^11+42*q^10+45*q^9+46*q^8+45*q^7+40*q^6+33*q^5+26*q^4+18*q^3+11*q^2+5*q+1, q^
21+6*q^20+20*q^19+45*q^18+76*q^17+105*q^16+128*q^15+145*q^14+156*q^13+162*q^12+
163*q^11+157*q^10+145*q^9+130*q^8+111*q^7+88*q^6+67*q^5+48*q^4+30*q^3+16*q^2+6*
q+1, 1+71*q^25+315*q^22+141*q^24+397*q^21+227*q^23+7*q+22*q^2+27*q^26+47*q^3+83
*q^4+127*q^5+178*q^6+238*q^7+302*q^8+367*q^9+431*q^10+489*q^11+536*q^12+574*q^
13+601*q^14+611*q^15+609*q^16+q^28+594*q^17+564*q^18+523*q^19+468*q^20+7*q^27,
1+1948*q^25+2330*q^22+2109*q^24+2389*q^21+2236*q^23+8*q+690*q^30+965*q^29+35*q^
34+105*q^33+29*q^2+1748*q^26+70*q^3+136*q^4+227*q^5+341*q^6+239*q^32+481*q^7+
643*q^8+825*q^9+1026*q^10+1237*q^11+1448*q^12+1651*q^13+1841*q^14+2012*q^15+
2160*q^16+1245*q^28+2277*q^17+2357*q^18+2403*q^19+2414*q^20+1511*q^27+440*q^31+
8*q^35+q^36, 1+9913*q^25+148*q^42+9497*q^22+9875*q^24+9167*q^21+9736*q^23+9*q+9
*q^44+8629*q^30+q^45+2998*q^37+9077*q^29+1377*q^39+379*q^41+5814*q^34+2129*q^38
+6669*q^33+37*q^2+9847*q^26+100*q^3+213*q^4+386*q^5+622*q^6+7425*q^32+928*q^7+
1304*q^8+1748*q^9+2265*q^10+2851*q^11+3495*q^12+4182*q^13+4893*q^14+5609*q^15+
6319*q^16+44*q^43+9422*q^28+7003*q^17+7642*q^18+8228*q^19+8743*q^20+9678*q^27+
785*q^40+8077*q^31+4889*q^35+3935*q^36, 1+37446*q^25+22641*q^42+31564*q^22+
35714*q^24+29222*q^21+33744*q^23+10*q+571*q^51+15849*q^44+54*q^53+41976*q^30+
201*q^52+q^55+10*q^54+12519*q^45+35918*q^37+41669*q^29+9408*q^46+31520*q^39+
25856*q^41+40237*q^34+1313*q^50+33883*q^38+41102*q^33+46*q^2+38923*q^26+138*q^3
+321*q^4+629*q^5+1086*q^6+41678*q^32+1716*q^7+2536*q^8+3553*q^9+4782*q^10+6233*
q^11+7911*q^12+9811*q^13+11913*q^14+14185*q^15+16594*q^16+19266*q^43+2550*q^49+
41041*q^28+19101*q^17+21654*q^18+24218*q^19+26758*q^20+40122*q^27+28831*q^40+
41969*q^31+39092*q^35+4334*q^48+37654*q^36+6644*q^47, 1+121391*q^25+170583*q^42
+92934*q^22+112053*q^24+83417*q^21+102523*q^23+11*q+89115*q^51+159283*q^44+
63707*q^53+20915*q^57+160975*q^30+76375*q^52+13720*q^58+40038*q^55+4444*q^60+
51471*q^54+152168*q^45+182887*q^37+154335*q^29+144017*q^46+11*q^65+180720*q^39+
2086*q^61+174865*q^41+179109*q^34+101593*q^50+65*q^64+182269*q^38+175947*q^33+
29755*q^56+56*q^2+130405*q^26+185*q^3+468*q^4+988*q^5+1824*q^6+171840*q^32+3053
*q^7+4746*q^8+6955*q^9+826*q^62+9734*q^10+13132*q^11+17194*q^12+21964*q^13+
27464*q^14+33693*q^15+40629*q^16+q^66+165401*q^43+113506*q^49+146970*q^28+48221
*q^17+56386*q^18+65038*q^19+74083*q^20+138964*q^27+178247*q^40+265*q^63+166827*
q^31+181320*q^35+124628*q^48+8250*q^59+182575*q^36+134825*q^47, 1+356607*q^25+
809818*q^42+253049*q^22+320902*q^24+221447*q^21+286288*q^23+12*q+735318*q^51+
814770*q^44+684537*q^53+541730*q^57+539320*q^30+711593*q^52+497842*q^58+620068*
q^55+403248*q^60+654056*q^54+812527*q^45+14844*q^71+1156*q^74+743240*q^37+94693
*q^67+503471*q^29+807181*q^46+167445*q^65+43659*q^69+779023*q^39+354026*q^61+
802636*q^41+668629*q^34+755790*q^50+210412*q^64+762632*q^38+638999*q^33+582581*
q^56+67*q^2+66296*q^68+393083*q^26+242*q^3+663*q^4+1503*q^5+2960*q^6+607389*q^
32+5245*q^7+8575*q^8+13147*q^9+304757*q^62+19151*q^10+26770*q^11+36175*q^12+77*
q^76+47544*q^13+61041*q^14+76810*q^15+94968*q^16+128599*q^66+26669*q^70+12*q^77
+q^78+813865*q^43+773139*q^49+466885*q^28+115581*q^17+138631*q^18+164048*q^19+
191713*q^20+429964*q^27+7367*q^72+792357*q^40+3178*q^73+256527*q^63+574063*q^31
+341*q^75+696047*q^35+787474*q^48+451454*q^59+720991*q^36+798819*q^47, 1+979764
*q^25+3234438*q^42+651963*q^22+13*q^90+863099*q^24+558318*q^21+753678*q^23+13*q
+3700688*q^51+3409705*q^44+3687504*q^53+3535465*q^57+1648772*q^30+3699378*q^52+
3471118*q^58+3632408*q^55+3309812*q^60+q^91+3665188*q^54+3482662*q^45+1613690*q
^71+1036400*q^74+2650065*q^37+2371280*q^67+1506236*q^29+3545461*q^46+2699743*q^
65+2003862*q^69+2905309*q^39+3212139*q^61+3132922*q^41+2231293*q^34+302877*q^79
+3691215*q^50+2846490*q^64+2780713*q^38+2086079*q^33+3589182*q^56+79*q^2+
2191610*q^68+1103123*q^26+310*q^3+916*q^4+2223*q^5+4659*q^6+1939823*q^32+8730*q
^7+15003*q^8+24062*q^9+3102605*q^62+36496*q^10+52898*q^11+73846*q^12+694530*q^
76+99929*q^13+131733*q^14+169839*q^15+214830*q^16+2541080*q^66+1810307*q^70+
545606*q^77+414532*q^78+3326851*q^43+3670909*q^49+1367152*q^28+267255*q^17+
327563*q^18+396076*q^19+472985*q^20+1232503*q^27+1417033*q^72+3022963*q^40+
1223500*q^73+2980821*q^63+1793671*q^31+859044*q^75+2374392*q^35+211324*q^80+
3639789*q^48+49210*q^83+139560*q^81+11713*q^85+4676*q^86+1574*q^87+430*q^88+90*
q^89+25465*q^84+86265*q^82+3395997*q^59+2514315*q^36+3597907*q^47, 1+2563716*q^
25+11555840*q^42+1613433*q^22+1780472*q^90+2217017*q^24+1356101*q^21+1900273*q^
23+14*q+86806*q^97+15680082*q^51+12646425*q^44+16257658*q^53+16967669*q^57+
4724936*q^30+447866*q^94+15986795*q^52+17050132*q^58+16688392*q^55+17101174*q^
60+1340281*q^91+16491768*q^54+13162005*q^45+14652711*q^71+13115999*q^74+162439*
q^96+41943*q^98+8632391*q^37+16089412*q^67+4239266*q^29+13653908*q^46+676737*q^
93+17976*q^99+16567345*q^65+15454567*q^69+9819102*q^39+17069853*q^61+10987113*q
^41+6877569*q^34+9694873*q^79+15338600*q^50+16749449*q^64+6681*q^100+9226385*q^
38+6313115*q^33+16847090*q^56+92*q^2+15792176*q^68+2940077*q^26+390*q^3+1238*q^
4+3207*q^5+7136*q^6+5764299*q^32+14120*q^7+25489*q^8+42746*q^9+279395*q^95+
17000753*q^62+2094*q^101+67512*q^10+101502*q^11+146476*q^12+11866087*q^76+
204249*q^13+276680*q^14+365656*q^15+473131*q^16+16347501*q^66+15075203*q^70+
11177808*q^77+10452219*q^78+12110008*q^43+533*q^102+14963693*q^49+3778927*q^28+
601113*q^17+751597*q^18+926482*q^19+1127499*q^20+3345462*q^27+14185852*q^72+
10407108*q^40+13673729*q^73+104*q^103+16893940*q^63+5234034*q^31+12513012*q^75+
7454521*q^35+14*q^104+8912600*q^80+14556828*q^48+q^105+6500914*q^83+8113343*q^
81+4938450*q^85+4202399*q^86+3509990*q^87+2870480*q^88+2291890*q^89+5708123*q^
84+7306142*q^82+17094610*q^59+8040614*q^36+972812*q^92+14119598*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.954545455, 4.855555556, 7.200507614,

    9.985049834, 13.20320168, 16.84809792, 20.91248338, 25.38900260,

    30.27036547, 35.54944065, 41.21930652, 47.27327671]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.521879004, 2.208876593, 3.018201929,

    3.948351237, 4.998266025, 6.167146802, 7.454237419, 8.858721734,

    10.37968924, 12.01613361, 13.76696490, 15.63102600]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.31612, 4.87914, 9.10954, 15.5895, 24.9827,

    38.0337, 55.5657, 78.4770, 107.738, 144.387, 189.529, 244.329]

Skewness: , [Float(undefined), 0., 0., 0.07673785025, 0.1150089428,

    0.1311611266, 0.1383347446, 0.1421539302, 0.1449202130, 0.1475905053,

    0.1505625403, 0.1539783305, 0.1578689505, 0.1622102632, 0.1669547284]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.266213022,

    2.332965779, 2.371429027, 2.397959020, 2.416879804, 2.430500168,

    2.440484294, 2.447991952, 2.453814975, 2.458584480, 2.462654281,

    2.466335628]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+4*q^4+3*q^5+q^6, 1+4*q+9*q^2+13*q^3
+15*q^4+15*q^5+12*q^6+9*q^7+7*q^8+4*q^9+q^10, 1+5*q+14*q^2+26*q^3+38*q^4+47*q^5
+50*q^6+48*q^7+44*q^8+38*q^9+29*q^10+21*q^11+16*q^12+11*q^13+5*q^14+q^15, 1+6*q
+20*q^2+45*q^3+79*q^4+116*q^5+148*q^6+169*q^7+179*q^8+180*q^9+169*q^10+150*q^11
+132*q^12+114*q^13+92*q^14+69*q^15+50*q^16+37*q^17+27*q^18+16*q^19+6*q^20+q^21,
1+7*q+27*q^2+71*q^3+145*q^4+246*q^5+362*q^6+475*q^7+571*q^8+644*q^9+685*q^10+
691*q^11+674*q^12+642*q^13+593*q^14+530*q^15+461*q^16+395*q^17+336*q^18+278*q^
19+218*q^20+163*q^21+119*q^22+87*q^23+64*q^24+43*q^25+22*q^26+7*q^27+q^28, 1+8*
q+35*q^2+105*q^3+244*q^4+469*q^5+779*q^6+1150*q^7+1545*q^8+1929*q^9+2266*q^10+
2524*q^11+2694*q^12+2784*q^13+2796*q^14+2733*q^15+2608*q^16+2439*q^17+2248*q^18
+2048*q^19+1834*q^20+1604*q^21+1375*q^22+1167*q^23+984*q^24+817*q^25+658*q^26+
510*q^27+383*q^28+282*q^29+206*q^30+151*q^31+107*q^32+65*q^33+29*q^34+8*q^35+q^
36, 1+9*q+44*q^2+148*q^3+385*q^4+826*q^5+1526*q^6+2498*q^7+3705*q^8+5074*q^9+
6507*q^10+7891*q^11+9128*q^12+10160*q^13+10950*q^14+11472*q^15+11725*q^16+11726
*q^17+11511*q^18+11136*q^19+10634*q^20+10007*q^21+9277*q^22+8498*q^23+7719*q^24
+6956*q^25+6197*q^26+5439*q^27+4705*q^28+4018*q^29+3393*q^30+2844*q^31+2365*q^
32+1931*q^33+1533*q^34+1184*q^35+895*q^36+665*q^37+488*q^38+357*q^39+258*q^40+
172*q^41+94*q^42+37*q^43+9*q^44+q^45, 1+10*q+54*q^2+201*q^3+578*q^4+1368*q^5+
2780*q^6+4992*q^7+8095*q^8+12066*q^9+16767*q^10+21956*q^11+27328*q^12+32586*q^
13+37471*q^14+41762*q^15+45298*q^16+47981*q^17+49772*q^18+50719*q^19+50914*q^20
+50409*q^21+49245*q^22+47523*q^23+45395*q^24+42996*q^25+40389*q^26+37599*q^27+
34683*q^28+31715*q^29+28755*q^30+25873*q^31+23130*q^32+20532*q^33+18056*q^34+
15701*q^35+13499*q^36+11492*q^37+9701*q^38+8125*q^39+6753*q^40+5553*q^41+4484*q
^42+3538*q^43+2735*q^44+2081*q^45+1560*q^46+1153*q^47+845*q^48+615*q^49+430*q^
50+266*q^51+131*q^52+46*q^53+10*q^54+q^55, 1+11*q+65*q^2+265*q^3+834*q^4+2157*q
^5+4779*q^6+9334*q^7+16419*q^8+26459*q^9+39606*q^10+55679*q^11+74166*q^12+94318
*q^13+115266*q^14+136104*q^15+155973*q^16+174128*q^17+189965*q^18+203090*q^19+
213350*q^20+220704*q^21+225133*q^22+226720*q^23+225716*q^24+222481*q^25+217356*
q^26+210592*q^27+202415*q^28+193090*q^29+182883*q^30+172050*q^31+160864*q^32+
149557*q^33+138255*q^34+127019*q^35+115916*q^36+105055*q^37+94594*q^38+84691*q^
39+75428*q^40+66794*q^41+58737*q^42+51233*q^43+44311*q^44+38010*q^45+32340*q^46
+27299*q^47+22883*q^48+19056*q^49+15738*q^50+12838*q^51+10299*q^52+8114*q^53+
6293*q^54+4818*q^55+3641*q^56+2713*q^57+1998*q^58+1460*q^59+1045*q^60+696*q^61+
397*q^62+177*q^63+56*q^64+11*q^65+q^66, 1+12*q+77*q^2+341*q^3+1165*q^4+3267*q^5
+7834*q^6+16527*q^7+31332*q^8+54274*q^9+87072*q^10+130805*q^11+185661*q^12+
250864*q^13+324777*q^14+405091*q^15+489064*q^16+573784*q^17+656379*q^18+734240*
q^19+805274*q^20+867932*q^21+921034*q^22+963756*q^23+995760*q^24+1017213*q^25+
1028654*q^26+1030766*q^27+1024247*q^28+1009881*q^29+988564*q^30+961230*q^31+
928860*q^32+892469*q^33+852978*q^34+811123*q^35+767445*q^36+722393*q^37+676528*
q^38+630551*q^39+585098*q^40+540597*q^41+497281*q^42+455278*q^43+414734*q^44+
375861*q^45+338892*q^46+304034*q^47+271432*q^48+241164*q^49+213253*q^50+187619*
q^51+164081*q^52+142507*q^53+122900*q^54+105291*q^55+89625*q^56+75778*q^57+
63635*q^58+53102*q^59+44029*q^60+36186*q^61+29370*q^62+23491*q^63+18525*q^64+
14429*q^65+11113*q^66+8459*q^67+6354*q^68+4711*q^69+3458*q^70+2505*q^71+1741*q^
72+1093*q^73+574*q^74+233*q^75+67*q^76+12*q^77+q^78, 1+13*q+90*q^2+430*q^3+1584
*q^4+4785*q^5+12342*q^6+27960*q^7+56815*q^8+105263*q^9+180191*q^10+288111*q^11+
434164*q^12+621274*q^13+849649*q^14+1116663*q^15+1417071*q^16+1743503*q^17+
2087082*q^18+2438113*q^19+2786924*q^20+3124579*q^21+3443118*q^22+3735644*q^23+
3996592*q^24+4221974*q^25+4409434*q^26+4558030*q^27+4667799*q^28+4739522*q^29+
4774720*q^30+4775540*q^31+4744610*q^32+4684969*q^33+4599928*q^34+4492843*q^35+
4366785*q^36+4224327*q^37+4067863*q^38+3900073*q^39+3723835*q^40+3541797*q^41+
3356118*q^42+3168466*q^43+2980207*q^44+2792618*q^45+2606998*q^46+2424678*q^47+
2246878*q^48+2074585*q^49+1908655*q^50+1749811*q^51+1598414*q^52+1454501*q^53+
1318135*q^54+1189565*q^55+1069081*q^56+956842*q^57+852869*q^58+757116*q^59+
669397*q^60+589282*q^61+516242*q^62+449863*q^63+389842*q^64+335901*q^65+287766*
q^66+245133*q^67+207626*q^68+174825*q^69+146329*q^70+121747*q^71+100613*q^72+
82401*q^73+66704*q^74+53327*q^75+42150*q^76+32978*q^77+25544*q^78+19572*q^79+
14813*q^80+11065*q^81+8169*q^82+5963*q^83+4246*q^84+2834*q^85+1667*q^86+807*q^
87+300*q^88+79*q^89+13*q^90+q^91, 1+14*q+104*q^2+533*q^3+2105*q^4+6812*q^5+
18800*q^6+45507*q^7+98648*q^8+194634*q^9+354109*q^10+600491*q^11+957645*q^12+
1447078*q^13+2085214*q^14+2881230*q^15+3835732*q^16+4940407*q^17+6178546*q^18+
7526240*q^19+8954264*q^20+10430412*q^21+11921520*q^22+13394932*q^23+14819855*q^
24+16168668*q^25+17417975*q^26+18549230*q^27+19548619*q^28+20406500*q^29+
21117094*q^30+21678285*q^31+22091204*q^32+22359799*q^33+22490486*q^34+22491746*
q^35+22373369*q^36+22145302*q^37+21817105*q^38+21398620*q^39+20900594*q^40+
20334219*q^41+19710256*q^42+19038493*q^43+18327586*q^44+17585258*q^45+16818724*
q^46+16035013*q^47+15240921*q^48+14442693*q^49+13645956*q^50+12855947*q^51+
12077336*q^52+11313720*q^53+10567746*q^54+9841825*q^55+9138466*q^56+8460046*q^
57+7808628*q^58+7186045*q^59+6593841*q^60+6032860*q^61+5503033*q^62+5003835*q^
63+4534879*q^64+4095980*q^65+3686908*q^66+3307251*q^67+2956364*q^68+2633371*q^
69+2337279*q^70+2067040*q^71+1821396*q^72+1598714*q^73+1397181*q^74+1215279*q^
75+1051952*q^76+906249*q^77+776988*q^78+662837*q^79+562527*q^80+474889*q^81+
398789*q^82+333080*q^83+276568*q^84+228028*q^85+186348*q^86+150719*q^87+120631*
q^88+95635*q^89+75154*q^90+58524*q^91+45116*q^92+34385*q^93+25878*q^94+19234*q^
95+14132*q^96+10209*q^97+7080*q^98+4501*q^99+2474*q^100+1107*q^101+379*q^102+92
*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, q^10+4*q^9+9*q^8+
13*q^7+15*q^6+15*q^5+12*q^4+9*q^3+7*q^2+4*q+1, q^15+5*q^14+14*q^13+26*q^12+38*q
^11+47*q^10+50*q^9+48*q^8+44*q^7+38*q^6+29*q^5+21*q^4+16*q^3+11*q^2+5*q+1, q^21
+6*q^20+20*q^19+45*q^18+79*q^17+116*q^16+148*q^15+169*q^14+179*q^13+180*q^12+
169*q^11+150*q^10+132*q^9+114*q^8+92*q^7+69*q^6+50*q^5+37*q^4+27*q^3+16*q^2+6*q
+1, 1+71*q^25+362*q^22+145*q^24+475*q^21+246*q^23+7*q+22*q^2+27*q^26+43*q^3+64*
q^4+87*q^5+119*q^6+163*q^7+218*q^8+278*q^9+336*q^10+395*q^11+461*q^12+530*q^13+
593*q^14+642*q^15+674*q^16+q^28+691*q^17+685*q^18+644*q^19+571*q^20+7*q^27, 1+
2524*q^25+2796*q^22+2694*q^24+2733*q^21+2784*q^23+8*q+779*q^30+1150*q^29+35*q^
34+105*q^33+29*q^2+2266*q^26+65*q^3+107*q^4+151*q^5+206*q^6+244*q^32+282*q^7+
383*q^8+510*q^9+658*q^10+817*q^11+984*q^12+1167*q^13+1375*q^14+1604*q^15+1834*q
^16+1545*q^28+2048*q^17+2248*q^18+2439*q^19+2608*q^20+1929*q^27+469*q^31+8*q^35
+q^36, 1+10634*q^25+148*q^42+8498*q^22+10007*q^24+7719*q^21+9277*q^23+9*q+9*q^
44+11472*q^30+q^45+3705*q^37+11725*q^29+1526*q^39+385*q^41+7891*q^34+2498*q^38+
9128*q^33+37*q^2+11136*q^26+94*q^3+172*q^4+258*q^5+357*q^6+10160*q^32+488*q^7+
665*q^8+895*q^9+1184*q^10+1533*q^11+1931*q^12+2365*q^13+2844*q^14+3393*q^15+
4018*q^16+44*q^43+11726*q^28+4705*q^17+5439*q^18+6197*q^19+6956*q^20+11511*q^27
+826*q^40+10950*q^31+6507*q^35+5074*q^36, 1+28755*q^25+32586*q^42+20532*q^22+
25873*q^24+18056*q^21+23130*q^23+10*q+578*q^51+21956*q^44+54*q^53+42996*q^30+
201*q^52+q^55+10*q^54+16767*q^45+49772*q^37+40389*q^29+12066*q^46+45298*q^39+
37471*q^41+50409*q^34+1368*q^50+47981*q^38+49245*q^33+46*q^2+31715*q^26+131*q^3
+266*q^4+430*q^5+615*q^6+47523*q^32+845*q^7+1153*q^8+1560*q^9+2081*q^10+2735*q^
11+3538*q^12+4484*q^13+5553*q^14+6753*q^15+8125*q^16+27328*q^43+2780*q^49+37599
*q^28+9701*q^17+11492*q^18+13499*q^19+15701*q^20+34683*q^27+41762*q^40+45395*q^
31+50914*q^35+4992*q^48+50719*q^36+8095*q^47, 1+66794*q^25+225716*q^42+44311*q^
22+58737*q^24+38010*q^21+51233*q^23+11*q+136104*q^51+225133*q^44+94318*q^53+
26459*q^57+115916*q^30+115266*q^52+16419*q^58+55679*q^55+4779*q^60+74166*q^54+
220704*q^45+193090*q^37+105055*q^29+213350*q^46+11*q^65+210592*q^39+2157*q^61+
222481*q^41+160864*q^34+155973*q^50+65*q^64+202415*q^38+149557*q^33+39606*q^56+
56*q^2+75428*q^26+177*q^3+397*q^4+696*q^5+1045*q^6+138255*q^32+1460*q^7+1998*q^
8+2713*q^9+834*q^62+3641*q^10+4818*q^11+6293*q^12+8114*q^13+10299*q^14+12838*q^
15+15738*q^16+q^66+226720*q^43+174128*q^49+94594*q^28+19056*q^17+22883*q^18+
27299*q^19+32340*q^20+84691*q^27+217356*q^40+265*q^63+127019*q^31+172050*q^35+
189965*q^48+9334*q^59+182883*q^36+203090*q^47, 1+142507*q^25+767445*q^42+89625*
q^22+122900*q^24+75778*q^21+105291*q^23+12*q+1030766*q^51+852978*q^44+1017213*q
^53+867932*q^57+271432*q^30+1028654*q^52+805274*q^58+963756*q^55+656379*q^60+
995760*q^54+892469*q^45+16527*q^71+1165*q^74+540597*q^37+130805*q^67+241164*q^
29+928860*q^46+250864*q^65+54274*q^69+630551*q^39+573784*q^61+722393*q^41+
414734*q^34+1024247*q^50+324777*q^64+585098*q^38+375861*q^33+921034*q^56+67*q^2
+87072*q^68+164081*q^26+233*q^3+574*q^4+1093*q^5+1741*q^6+338892*q^32+2505*q^7+
3458*q^8+4711*q^9+489064*q^62+6354*q^10+8459*q^11+11113*q^12+77*q^76+14429*q^13
+18525*q^14+23491*q^15+29370*q^16+185661*q^66+31332*q^70+12*q^77+q^78+811123*q^
43+1009881*q^49+213253*q^28+36186*q^17+44029*q^18+53102*q^19+63635*q^20+187619*
q^27+7834*q^72+676528*q^40+3267*q^73+405091*q^63+304034*q^31+341*q^75+455278*q^
35+988564*q^48+734240*q^59+497281*q^36+961230*q^47, 1+287766*q^25+2074585*q^42+
174825*q^22+13*q^90+245133*q^24+146329*q^21+207626*q^23+13*q+3723835*q^51+
2424678*q^44+4067863*q^53+4599928*q^57+589282*q^30+3900073*q^52+4684969*q^58+
4366785*q^55+4775540*q^60+q^91+4224327*q^54+2606998*q^45+2786924*q^71+1743503*q
^74+1318135*q^37+3996592*q^67+516242*q^29+2792618*q^46+4409434*q^65+3443118*q^
69+1598414*q^39+4774720*q^61+1908655*q^41+956842*q^34+434164*q^79+3541797*q^50+
4558030*q^64+1454501*q^38+852869*q^33+4492843*q^56+79*q^2+3735644*q^68+335901*q
^26+300*q^3+807*q^4+1667*q^5+2834*q^6+757116*q^32+4246*q^7+5963*q^8+8169*q^9+
4739522*q^62+11065*q^10+14813*q^11+19572*q^12+1116663*q^76+25544*q^13+32978*q^
14+42150*q^15+53327*q^16+4221974*q^66+3124579*q^70+849649*q^77+621274*q^78+
2246878*q^43+3356118*q^49+449863*q^28+66704*q^17+82401*q^18+100613*q^19+121747*
q^20+389842*q^27+2438113*q^72+1749811*q^40+2087082*q^73+4667799*q^63+669397*q^
31+1417071*q^75+1069081*q^35+288111*q^80+3168466*q^48+56815*q^83+180191*q^81+
12342*q^85+4785*q^86+1584*q^87+430*q^88+90*q^89+27960*q^84+105263*q^82+4744610*
q^59+1189565*q^36+2980207*q^47, 1+562527*q^25+5003835*q^42+333080*q^22+2881230*
q^90+474889*q^24+276568*q^21+398789*q^23+14*q+98648*q^97+10567746*q^51+6032860*
q^44+12077336*q^53+15240921*q^57+1215279*q^30+600491*q^94+11313720*q^52+
16035013*q^58+13645956*q^55+17585258*q^60+2085214*q^91+12855947*q^54+6593841*q^
45+22490486*q^71+21678285*q^74+194634*q^96+45507*q^98+2956364*q^37+21817105*q^
67+1051952*q^29+7186045*q^46+957645*q^93+18800*q^99+20900594*q^65+22373369*q^69
+3686908*q^39+18327586*q^61+4534879*q^41+2067040*q^34+17417975*q^79+9841825*q^
50+20334219*q^64+6812*q^100+3307251*q^38+1821396*q^33+14442693*q^56+92*q^2+
22145302*q^68+662837*q^26+379*q^3+1107*q^4+2474*q^5+4501*q^6+1598714*q^32+7080*
q^7+10209*q^8+14132*q^9+354109*q^95+19038493*q^62+2105*q^101+19234*q^10+25878*q
^11+34385*q^12+20406500*q^76+45116*q^13+58524*q^14+75154*q^15+95635*q^16+
21398620*q^66+22491746*q^70+19548619*q^77+18549230*q^78+5503033*q^43+533*q^102+
9138466*q^49+906249*q^28+120631*q^17+150719*q^18+186348*q^19+228028*q^20+776988
*q^27+22359799*q^72+4095980*q^40+22091204*q^73+104*q^103+19710256*q^63+1397181*
q^31+21117094*q^75+2337279*q^35+14*q^104+16168668*q^80+8460046*q^48+q^105+
11921520*q^83+14819855*q^81+8954264*q^85+7526240*q^86+6178546*q^87+4940407*q^88
+3835732*q^89+10430412*q^84+13394932*q^82+16818724*q^59+2633371*q^36+1447078*q^
92+7808628*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.954545455, 4.811111111, 7.027918782,

    9.573089701, 12.42171068, 15.55372000, 18.95254199, 22.60419208,

    26.49666947, 30.61953049, 34.96357965, 39.52063983]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.521879004, 2.195421723, 2.968828279,

    3.832732787, 4.779369739, 5.802465181, 6.896899268, 8.058421475,

    9.283441858, 10.56888152, 11.91206448, 13.31063821]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.31612, 4.81988, 8.81394, 14.6898, 22.8424,

    33.6686, 47.5672, 64.9382, 86.1823, 111.701, 141.897, 177.173]

Skewness: , [Float(undefined), 0., 0., 0.07673785025, 0.1652952223,

    0.2423045238, 0.3045291450, 0.3541162700, 0.3938654598, 0.4261214179,

    0.4526585115, 0.4747894024, 0.4934745519, 0.5094307953, 0.5231937372]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.266213022,

    2.390926714, 2.502808170, 2.603584118, 2.691829543, 2.768084190,

    2.833696127, 2.890298134, 2.939364088, 2.982164128, 3.019723314,

    3.052882123]

                                end of this data



For the equivalence class of patterns,

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

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

                                  here it is:

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 3*q+4*q^2+6*q^3+5*q^4+3*q^5+q^6, 7*q^3+14*q^4+18*q^5+
18*q^6+15*q^7+9*q^8+4*q^9+q^10, 17*q^6+40*q^7+62*q^8+67*q^9+61*q^10+46*q^11+29*
q^12+14*q^13+5*q^14+q^15, 41*q^10+114*q^11+194*q^12+244*q^13+245*q^14+207*q^15+
151*q^16+94*q^17+49*q^18+20*q^19+6*q^20+q^21, 99*q^15+316*q^16+598*q^17+829*q^
18+931*q^19+876*q^20+712*q^21+506*q^22+315*q^23+169*q^24+76*q^25+27*q^26+7*q^27
+q^28, 239*q^21+862*q^22+1794*q^23+2734*q^24+3351*q^25+3473*q^26+3116*q^27+2460
*q^28+1723*q^29+1072*q^30+588*q^31+279*q^32+111*q^33+35*q^34+8*q^35+q^36, 577*q
^28+2320*q^29+5278*q^30+8747*q^31+11645*q^32+13090*q^33+12797*q^34+11062*q^35+
8541*q^36+5922*q^37+3690*q^38+2057*q^39+1014*q^40+433*q^41+155*q^42+44*q^43+9*q
^44+q^45, 1393*q^36+6178*q^37+15266*q^38+27336*q^39+39213*q^40+47495*q^41+50051
*q^42+46794*q^43+39265*q^44+29784*q^45+20494*q^46+12798*q^47+7230*q^48+3667*q^
49+1647*q^50+641*q^51+209*q^52+54*q^53+10*q^54+q^55, 3363*q^45+16308*q^46+43526
*q^47+83733*q^48+128707*q^49+166840*q^50+188268*q^51+188706*q^52+170259*q^53+
139467*q^54+104260*q^55+71311*q^56+44632*q^57+25502*q^58+13230*q^59+6173*q^60+
2552*q^61+914*q^62+274*q^63+65*q^64+11*q^65+q^66, 8119*q^55+42734*q^56+122578*q
^57+252138*q^58+413279*q^59+570489*q^60+685324*q^61+731832*q^62+704571*q^63+
617508*q^64+495836*q^65+366205*q^66+249240*q^67+156329*q^68+90194*q^69+47672*q^
70+22924*q^71+9923*q^72+3806*q^73+1264*q^74+351*q^75+77*q^76+12*q^77+q^78, 
19601*q^66+111288*q^67+341534*q^68+748083*q^69+1302205*q^70+1906038*q^71+
2426469*q^72+2746302*q^73+2805045*q^74+2612516*q^75+2234890*q^76+1764669*q^77+
1290039*q^78+874395*q^79+549505*q^80+319701*q^81+171650*q^82+84606*q^83+37983*q
^84+15355*q^85+5499*q^86+1704*q^87+441*q^88+90*q^89+13*q^90+q^91, 47321*q^78+
288274*q^79+942754*q^80+2190972*q^81+4036005*q^82+6241439*q^83+8387895*q^84+
10020374*q^85+10807505*q^86+10640364*q^87+9638150*q^88+8077372*q^89+6287163*q^
90+4556089*q^91+3077369*q^92+1937323*q^93+1135342*q^94+617807*q^95+310889*q^96+
143802*q^97+60619*q^98+23011*q^99+7735*q^100+2248*q^101+545*q^102+104*q^103+14*
q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 3*q^5+4*q^4+6*q^3+5*q^2+3*q+1, 7*q^7+14*q^6+18*q^5+18
*q^4+15*q^3+9*q^2+4*q+1, 17*q^9+40*q^8+62*q^7+67*q^6+61*q^5+46*q^4+29*q^3+14*q^
2+5*q+1, 41*q^11+114*q^10+194*q^9+244*q^8+245*q^7+207*q^6+151*q^5+94*q^4+49*q^3
+20*q^2+6*q+1, 99*q^13+316*q^12+598*q^11+829*q^10+931*q^9+876*q^8+712*q^7+506*q
^6+315*q^5+169*q^4+76*q^3+27*q^2+7*q+1, 239*q^15+862*q^14+1794*q^13+2734*q^12+
3351*q^11+3473*q^10+3116*q^9+2460*q^8+1723*q^7+1072*q^6+588*q^5+279*q^4+111*q^3
+35*q^2+8*q+1, 577*q^17+2320*q^16+5278*q^15+8747*q^14+11645*q^13+13090*q^12+
12797*q^11+11062*q^10+8541*q^9+5922*q^8+3690*q^7+2057*q^6+1014*q^5+433*q^4+155*
q^3+44*q^2+9*q+1, 1393*q^19+6178*q^18+15266*q^17+27336*q^16+39213*q^15+47495*q^
14+50051*q^13+46794*q^12+39265*q^11+29784*q^10+20494*q^9+12798*q^8+7230*q^7+
3667*q^6+1647*q^5+641*q^4+209*q^3+54*q^2+10*q+1, 3363*q^21+16308*q^20+43526*q^
19+83733*q^18+128707*q^17+166840*q^16+188268*q^15+188706*q^14+170259*q^13+
139467*q^12+104260*q^11+71311*q^10+44632*q^9+25502*q^8+13230*q^7+6173*q^6+2552*
q^5+914*q^4+274*q^3+65*q^2+11*q+1, 8119*q^23+42734*q^22+122578*q^21+252138*q^20
+413279*q^19+570489*q^18+685324*q^17+731832*q^16+704571*q^15+617508*q^14+495836
*q^13+366205*q^12+249240*q^11+156329*q^10+90194*q^9+47672*q^8+22924*q^7+9923*q^
6+3806*q^5+1264*q^4+351*q^3+77*q^2+12*q+1, 1+19601*q^25+748083*q^22+111288*q^24
+1302205*q^21+341534*q^23+13*q+90*q^2+441*q^3+1704*q^4+5499*q^5+15355*q^6+37983
*q^7+84606*q^8+171650*q^9+319701*q^10+549505*q^11+874395*q^12+1290039*q^13+
1764669*q^14+2234890*q^15+2612516*q^16+2805045*q^17+2746302*q^18+2426469*q^19+
1906038*q^20, 1+942754*q^25+6241439*q^22+2190972*q^24+8387895*q^21+4036005*q^23
+14*q+104*q^2+288274*q^26+545*q^3+2248*q^4+7735*q^5+23011*q^6+60619*q^7+143802*
q^8+310889*q^9+617807*q^10+1135342*q^11+1937323*q^12+3077369*q^13+4556089*q^14+
6287163*q^15+8077372*q^16+9638150*q^17+10640364*q^18+10807505*q^19+10020374*q^
20+47321*q^27]


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

[1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406,

    22369622, 89478486]

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

     is given by

[1, 1, 3, 10, 42, 174, 686, 2724, 10916, 43706, 174778, 699020, 2796172,

    11184874, 44739306]

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

     is given by

[0, 1, 3, 12, 44, 168, 680, 2738, 10930, 43676, 174748, 699082, 2796234,

    11184748, 44739180]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 42, 168, 680, 2724, 10916, 43676, 174748, 699020, 2796172,

    11184748, 44739180]

 ODD:, [0, 1, 3, 12, 44, 174, 686, 2738, 10930, 43706, 174778, 699082, 2796234,

    11184874, 44739306]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.181818182, 5.790697674, 9.391812865,

    13.99414348, 19.59758330, 26.20150142, 33.80558925, 42.40973204,

    52.01389169, 62.61805634, 74.22222244, 86.82638895]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.369683561, 1.657374493, 1.883676475,

    2.079663989, 2.257234924, 2.421513471, 2.575284588, 2.720374587,

    2.858110837, 2.989509254, 3.115371136, 3.236342139]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.87603, 2.74689, 3.54824, 4.32500, 5.09511,

    5.86373, 6.63209, 7.40044, 8.16880, 8.93717, 9.70554, 10.4739]

Skewness: , [Float(undefined), 0., 0., 0.09473197454, 0.2276247095,

    0.2891869363, 0.3120335941, 0.3179153340, 0.3165622232, 0.3120228524,

    0.3060902365, 0.2995973080, 0.2929624448, 0.2864001830, 0.2800268468]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.245866086,

    2.404589388, 2.570538662, 2.685700157, 2.759953671, 2.809044423,

    2.843039370, 2.867664814, 2.886228672, 2.900629473, 2.912062215,

    2.921325975]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, 1+4*q+7*q^2+11*q^3
+13*q^4+15*q^5+14*q^6+11*q^7+7*q^8+4*q^9+q^10, 1+5*q+11*q^2+19*q^3+26*q^4+33*q^
5+39*q^6+44*q^7+46*q^8+42*q^9+36*q^10+28*q^11+19*q^12+11*q^13+5*q^14+q^15, 1+6*
q+16*q^2+31*q^3+48*q^4+64*q^5+83*q^6+100*q^7+116*q^8+133*q^9+141*q^10+146*q^11+
146*q^12+133*q^13+116*q^14+95*q^15+72*q^16+51*q^17+31*q^18+16*q^19+6*q^20+q^21,
1+7*q+22*q^2+48*q^3+83*q^4+120*q^5+162*q^6+206*q^7+253*q^8+302*q^9+348*q^10+392
*q^11+435*q^12+465*q^13+484*q^14+493*q^15+493*q^16+472*q^17+430*q^18+377*q^19+
315*q^20+253*q^21+191*q^22+135*q^23+87*q^24+48*q^25+22*q^26+7*q^27+q^28, 1+8*q+
29*q^2+71*q^3+136*q^4+215*q^5+306*q^6+406*q^7+516*q^8+637*q^9+768*q^10+904*q^11
+1038*q^12+1172*q^13+1293*q^14+1412*q^15+1528*q^16+1623*q^17+1683*q^18+1716*q^
19+1730*q^20+1710*q^21+1651*q^22+1551*q^23+1408*q^24+1235*q^25+1048*q^26+859*q^
27+678*q^28+512*q^29+363*q^30+239*q^31+141*q^32+71*q^33+29*q^34+8*q^35+q^36, 1+
9*q+37*q^2+101*q^3+213*q^4+368*q^5+558*q^6+777*q^7+1021*q^8+1296*q^9+1602*q^10+
1940*q^11+2312*q^12+2697*q^13+3078*q^14+3463*q^15+3850*q^16+4244*q^17+4629*q^18
+4985*q^19+5314*q^20+5621*q^21+5869*q^22+6050*q^23+6172*q^24+6211*q^25+6184*q^
26+6074*q^27+5864*q^28+5557*q^29+5140*q^30+4634*q^31+4076*q^32+3489*q^33+2902*q
^34+2346*q^35+1835*q^36+1379*q^37+985*q^38+657*q^39+403*q^40+219*q^41+101*q^42+
37*q^43+9*q^44+q^45, 1+10*q+46*q^2+139*q^3+321*q^4+604*q^5+981*q^6+1441*q^7+
1971*q^8+2576*q^9+3259*q^10+4028*q^11+4898*q^12+5860*q^13+6893*q^14+7979*q^15+
9090*q^16+10243*q^17+11413*q^18+12593*q^19+13784*q^20+15008*q^21+16221*q^22+
17377*q^23+18453*q^24+19441*q^25+20343*q^26+21164*q^27+21854*q^28+22376*q^29+
22693*q^30+22806*q^31+22703*q^32+22394*q^33+21873*q^34+21089*q^35+20030*q^36+
18692*q^37+17115*q^38+15363*q^39+13507*q^40+11622*q^41+9780*q^42+8033*q^43+6427
*q^44+4995*q^45+3742*q^46+2681*q^47+1813*q^48+1139*q^49+652*q^50+328*q^51+139*q
^52+46*q^53+10*q^54+q^55, 1+11*q+56*q^2+186*q^3+468*q^4+955*q^5+1664*q^6+2588*q
^7+3704*q^8+5008*q^9+6504*q^10+8206*q^11+10145*q^12+12344*q^13+14801*q^14+17500
*q^15+20420*q^16+23520*q^17+26772*q^18+30133*q^19+33575*q^20+37146*q^21+40868*q
^22+44686*q^23+48523*q^24+52359*q^25+56140*q^26+59875*q^27+63552*q^28+67087*q^
29+70416*q^30+73496*q^31+76330*q^32+78853*q^33+81053*q^34+82849*q^35+84146*q^36
+84894*q^37+85064*q^38+84615*q^39+83560*q^40+81876*q^41+79530*q^42+76462*q^43+
72654*q^44+68107*q^45+62892*q^46+57171*q^47+51115*q^48+44932*q^49+38805*q^50+
32892*q^51+27338*q^52+22244*q^53+17669*q^54+13655*q^55+10211*q^56+7330*q^57+
5006*q^58+3209*q^59+1901*q^60+1018*q^61+476*q^62+186*q^63+56*q^64+11*q^65+q^66,
1+12*q+67*q^2+243*q^3+663*q^4+1461*q^5+2729*q^6+4503*q^7+6768*q^8+9508*q^9+
12725*q^10+16436*q^11+20695*q^12+25566*q^13+31096*q^14+37328*q^15+44277*q^16+
51916*q^17+60217*q^18+69075*q^19+78377*q^20+88151*q^21+98400*q^22+109129*q^23+
120288*q^24+131824*q^25+143664*q^26+155798*q^27+168136*q^28+180601*q^29+193054*
q^30+205378*q^31+217552*q^32+229636*q^33+241538*q^34+253126*q^35+264173*q^36+
274516*q^37+284035*q^38+292747*q^39+300538*q^40+307350*q^41+313077*q^42+317505*
q^43+320489*q^44+321818*q^45+321304*q^46+318949*q^47+314761*q^48+308744*q^49+
300869*q^50+291002*q^51+279048*q^52+264980*q^53+248857*q^54+230914*q^55+211570*
q^56+191271*q^57+170535*q^58+149883*q^59+129768*q^60+110611*q^61+92732*q^62+
76368*q^63+61669*q^64+48704*q^65+37479*q^66+27974*q^67+20124*q^68+13842*q^69+
9015*q^70+5485*q^71+3068*q^72+1541*q^73+672*q^74+243*q^75+67*q^76+12*q^77+q^78,
1+13*q+79*q^2+311*q^3+916*q^4+2171*q^5+4339*q^6+7600*q^7+12022*q^8+17610*q^9+
24372*q^10+32333*q^11+41579*q^12+52240*q^13+64449*q^14+78355*q^15+94100*q^16+
111796*q^17+131524*q^18+153198*q^19+176670*q^20+201805*q^21+228591*q^22+257037*
q^23+287078*q^24+318632*q^25+351684*q^26+386245*q^27+422249*q^28+459616*q^29+
498088*q^30+537324*q^31+577092*q^32+617359*q^33+658192*q^34+699538*q^35+741181*
q^36+782725*q^37+823845*q^38+864359*q^39+904139*q^40+942978*q^41+980670*q^42+
1016884*q^43+1051273*q^44+1083548*q^45+1113353*q^46+1140389*q^47+1164450*q^48+
1185298*q^49+1202731*q^50+1216387*q^51+1225891*q^52+1230657*q^53+1230260*q^54+
1224348*q^55+1212796*q^56+1195561*q^57+1172642*q^58+1143977*q^59+1109505*q^60+
1069089*q^61+1022596*q^62+970065*q^63+911809*q^64+848530*q^65+781328*q^66+
711536*q^67+640609*q^68+569973*q^69+500901*q^70+434582*q^71+372012*q^72+313956*
q^73+260967*q^74+213354*q^75+171229*q^76+134560*q^77+103197*q^78+76910*q^79+
55419*q^80+38351*q^81+25282*q^82+15714*q^83+9086*q^84+4806*q^85+2270*q^86+926*q
^87+311*q^88+79*q^89+13*q^90+q^91, 1+14*q+92*q^2+391*q^3+1238*q^4+3144*q^5+6707
*q^6+12464*q^7+20772*q^8+31804*q^9+45652*q^10+62393*q^11+82179*q^12+105259*q^13
+131935*q^14+162554*q^15+197503*q^16+237196*q^17+282068*q^18+332416*q^19+388257
*q^20+449463*q^21+515945*q^22+587654*q^23+664538*q^24+746513*q^25+833471*q^26+
925510*q^27+1022734*q^28+1125212*q^29+1232859*q^30+1345315*q^31+1461945*q^32+
1582332*q^33+1706247*q^34+1833539*q^35+1963981*q^36+2097228*q^37+2232628*q^38+
2369829*q^39+2508643*q^40+2648867*q^41+2790064*q^42+2931757*q^43+3072855*q^44+
3212520*q^45+3350003*q^46+3484945*q^47+3616991*q^48+3745847*q^49+3870877*q^50+
3991234*q^51+4106004*q^52+4214235*q^53+4314821*q^54+4406855*q^55+4489547*q^56+
4562274*q^57+4624541*q^58+4675952*q^59+4715631*q^60+4742567*q^61+4755345*q^62+
4752562*q^63+4733093*q^64+4696163*q^65+4641386*q^66+4568739*q^67+4478361*q^68+
4370276*q^69+4244417*q^70+4100554*q^71+3938496*q^72+3758386*q^73+3560948*q^74+
3347605*q^75+3120642*q^76+2883209*q^77+2639085*q^78+2392383*q^79+2147230*q^80+
1907402*q^81+1676325*q^82+1456898*q^83+1251443*q^84+1061780*q^85+889064*q^86+
733887*q^87+596344*q^88+476108*q^89+372561*q^90+284887*q^91+212086*q^92+153028*
q^93+106434*q^94+70866*q^95+44794*q^96+26595*q^97+14632*q^98+7331*q^99+3264*q^
100+1249*q^101+391*q^102+92*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+4*q^4+3*q^5+q^6, q^10+4*q^9+7*q^8+
11*q^7+13*q^6+15*q^5+14*q^4+11*q^3+7*q^2+4*q+1, q^15+5*q^14+11*q^13+19*q^12+26*
q^11+33*q^10+39*q^9+44*q^8+46*q^7+42*q^6+36*q^5+28*q^4+19*q^3+11*q^2+5*q+1, q^
21+6*q^20+16*q^19+31*q^18+48*q^17+64*q^16+83*q^15+100*q^14+116*q^13+133*q^12+
141*q^11+146*q^10+146*q^9+133*q^8+116*q^7+95*q^6+72*q^5+51*q^4+31*q^3+16*q^2+6*
q+1, 1+48*q^25+162*q^22+83*q^24+206*q^21+120*q^23+7*q+22*q^2+22*q^26+48*q^3+87*
q^4+135*q^5+191*q^6+253*q^7+315*q^8+377*q^9+430*q^10+472*q^11+493*q^12+493*q^13
+484*q^14+465*q^15+435*q^16+q^28+392*q^17+348*q^18+302*q^19+253*q^20+7*q^27, 1+
904*q^25+1293*q^22+1038*q^24+1412*q^21+1172*q^23+8*q+306*q^30+406*q^29+29*q^34+
71*q^33+29*q^2+768*q^26+71*q^3+141*q^4+239*q^5+363*q^6+136*q^32+512*q^7+678*q^8
+859*q^9+1048*q^10+1235*q^11+1408*q^12+1551*q^13+1651*q^14+1710*q^15+1730*q^16+
516*q^28+1716*q^17+1683*q^18+1623*q^19+1528*q^20+637*q^27+215*q^31+8*q^35+q^36,
1+5314*q^25+101*q^42+6050*q^22+5621*q^24+6172*q^21+5869*q^23+9*q+9*q^44+3463*q^
30+q^45+1021*q^37+3850*q^29+558*q^39+213*q^41+1940*q^34+777*q^38+2312*q^33+37*q
^2+4985*q^26+101*q^3+219*q^4+403*q^5+657*q^6+2697*q^32+985*q^7+1379*q^8+1835*q^
9+2346*q^10+2902*q^11+3489*q^12+4076*q^13+4634*q^14+5140*q^15+5557*q^16+37*q^43
+4244*q^28+5864*q^17+6074*q^18+6184*q^19+6211*q^20+4629*q^27+368*q^40+3078*q^31
+1602*q^35+1296*q^36, 1+22693*q^25+5860*q^42+22394*q^22+22806*q^24+21873*q^21+
22703*q^23+10*q+321*q^51+4028*q^44+46*q^53+19441*q^30+139*q^52+q^55+10*q^54+
3259*q^45+11413*q^37+20343*q^29+2576*q^46+9090*q^39+6893*q^41+15008*q^34+604*q^
50+10243*q^38+16221*q^33+46*q^2+22376*q^26+139*q^3+328*q^4+652*q^5+1139*q^6+
17377*q^32+1813*q^7+2681*q^8+3742*q^9+4995*q^10+6427*q^11+8033*q^12+9780*q^13+
11622*q^14+13507*q^15+15363*q^16+4898*q^43+981*q^49+21164*q^28+17115*q^17+18692
*q^18+20030*q^19+21089*q^20+21854*q^27+7979*q^40+18453*q^31+13784*q^35+1441*q^
48+12593*q^36+1971*q^47, 1+81876*q^25+48523*q^42+72654*q^22+79530*q^24+68107*q^
21+76462*q^23+11*q+17500*q^51+40868*q^44+12344*q^53+5008*q^57+84146*q^30+14801*
q^52+3704*q^58+8206*q^55+1664*q^60+10145*q^54+37146*q^45+67087*q^37+84894*q^29+
33575*q^46+11*q^65+59875*q^39+955*q^61+52359*q^41+76330*q^34+20420*q^50+56*q^64
+63552*q^38+78853*q^33+6504*q^56+56*q^2+83560*q^26+186*q^3+476*q^4+1018*q^5+
1901*q^6+81053*q^32+3209*q^7+5006*q^8+7330*q^9+468*q^62+10211*q^10+13655*q^11+
17669*q^12+22244*q^13+27338*q^14+32892*q^15+38805*q^16+q^66+44686*q^43+23520*q^
49+85064*q^28+44932*q^17+51115*q^18+57171*q^19+62892*q^20+84615*q^27+56140*q^40
+186*q^63+82849*q^31+73496*q^35+26772*q^48+2588*q^59+70416*q^36+30133*q^47, 1+
264980*q^25+264173*q^42+211570*q^22+248857*q^24+191271*q^21+230914*q^23+12*q+
155798*q^51+241538*q^44+131824*q^53+88151*q^57+314761*q^30+143664*q^52+78377*q^
58+109129*q^55+60217*q^60+120288*q^54+229636*q^45+4503*q^71+663*q^74+307350*q^
37+16436*q^67+308744*q^29+217552*q^46+25566*q^65+9508*q^69+292747*q^39+51916*q^
61+274516*q^41+320489*q^34+168136*q^50+31096*q^64+300538*q^38+321818*q^33+98400
*q^56+67*q^2+12725*q^68+279048*q^26+243*q^3+672*q^4+1541*q^5+3068*q^6+321304*q^
32+5485*q^7+9015*q^8+13842*q^9+44277*q^62+20124*q^10+27974*q^11+37479*q^12+67*q
^76+48704*q^13+61669*q^14+76368*q^15+92732*q^16+20695*q^66+6768*q^70+12*q^77+q^
78+253126*q^43+180601*q^49+300869*q^28+110611*q^17+129768*q^18+149883*q^19+
170535*q^20+291002*q^27+2729*q^72+284035*q^40+1461*q^73+37328*q^63+318949*q^31+
243*q^75+317505*q^35+193054*q^48+69075*q^59+313077*q^36+205378*q^47, 1+781328*q
^25+1185298*q^42+569973*q^22+13*q^90+711536*q^24+500901*q^21+640609*q^23+13*q+
904139*q^51+1140389*q^44+823845*q^53+658192*q^57+1069089*q^30+864359*q^52+
617359*q^58+741181*q^55+537324*q^60+q^91+782725*q^54+1113353*q^45+176670*q^71+
111796*q^74+1230260*q^37+287078*q^67+1022596*q^29+1083548*q^46+351684*q^65+
228591*q^69+1225891*q^39+498088*q^61+1202731*q^41+1195561*q^34+41579*q^79+
942978*q^50+386245*q^64+1230657*q^38+1172642*q^33+699538*q^56+79*q^2+257037*q^
68+848530*q^26+311*q^3+926*q^4+2270*q^5+4806*q^6+1143977*q^32+9086*q^7+15714*q^
8+25282*q^9+459616*q^62+38351*q^10+55419*q^11+76910*q^12+78355*q^76+103197*q^13
+134560*q^14+171229*q^15+213354*q^16+318632*q^66+201805*q^70+64449*q^77+52240*q
^78+1164450*q^43+980670*q^49+970065*q^28+260967*q^17+313956*q^18+372012*q^19+
434582*q^20+911809*q^27+153198*q^72+1216387*q^40+131524*q^73+422249*q^63+
1109505*q^31+94100*q^75+1212796*q^35+32333*q^80+1016884*q^48+12022*q^83+24372*q
^81+4339*q^85+2171*q^86+916*q^87+311*q^88+79*q^89+7600*q^84+17610*q^82+577092*q
^59+1224348*q^36+1051273*q^47, 1+2147230*q^25+4752562*q^42+1456898*q^22+162554*
q^90+1907402*q^24+1251443*q^21+1676325*q^23+14*q+20772*q^97+4314821*q^51+
4742567*q^44+4106004*q^53+3616991*q^57+3347605*q^30+62393*q^94+4214235*q^52+
3484945*q^58+3870877*q^55+3212520*q^60+131935*q^91+3991234*q^54+4715631*q^45+
1706247*q^71+1345315*q^74+31804*q^96+12464*q^98+4478361*q^37+2232628*q^67+
3120642*q^29+4675952*q^46+82179*q^93+6707*q^99+2508643*q^65+1963981*q^69+
4641386*q^39+3072855*q^61+4733093*q^41+4100554*q^34+833471*q^79+4406855*q^50+
2648867*q^64+3144*q^100+4568739*q^38+3938496*q^33+3745847*q^56+92*q^2+2097228*q
^68+2392383*q^26+391*q^3+1249*q^4+3264*q^5+7331*q^6+3758386*q^32+14632*q^7+
26595*q^8+44794*q^9+45652*q^95+2931757*q^62+1238*q^101+70866*q^10+106434*q^11+
153028*q^12+1125212*q^76+212086*q^13+284887*q^14+372561*q^15+476108*q^16+
2369829*q^66+1833539*q^70+1022734*q^77+925510*q^78+4755345*q^43+391*q^102+
4489547*q^49+2883209*q^28+596344*q^17+733887*q^18+889064*q^19+1061780*q^20+
2639085*q^27+1582332*q^72+4696163*q^40+1461945*q^73+92*q^103+2790064*q^63+
3560948*q^31+1232859*q^75+4244417*q^35+14*q^104+746513*q^80+4562274*q^48+q^105+
515945*q^83+664538*q^81+388257*q^85+332416*q^86+282068*q^87+237196*q^88+197503*
q^89+449463*q^84+587654*q^82+3350003*q^59+4370276*q^36+105259*q^92+4624541*q^47
]


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

[1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248,

    44725516, 195354368]

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

     is given by

[1, 1, 3, 10, 43, 183, 776, 3327, 14347, 62155, 270012, 1175416, 5124098,

    22362790, 97677084]

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

     is given by

[0, 1, 3, 12, 45, 183, 776, 3325, 14349, 62155, 270028, 1175404, 5124150,

    22362726, 97677284]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 10, 43, 183, 776, 3327, 14347, 62155, 270028, 1175416, 5124098,

    22362726, 97677284]

 ODD:, [0, 1, 3, 12, 45, 183, 776, 3325, 14349, 62155, 270012, 1175404, 5124150,

    22362790, 97677084]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.011363636, 7.554644809,

    10.65335052, 14.32997595, 18.60443964, 23.49378972, 29.01242501,

    35.17246748, 41.98413836, 49.45608842, 57.59567270]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.507556723, 2.182025321, 2.987408505,

    3.921713164, 4.981032918, 6.161261577, 7.458489261, 8.869086364,

    10.38970872, 12.01728025, 13.74896900, 15.58216189]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.27273, 4.76123, 8.92461, 15.3798, 24.8107,

    37.9611, 55.6291, 78.6607, 107.946, 144.415, 189.034, 242.804]

                                                      -5
Skewness: , [Float(undefined), 0., 0., 0.2918624564 10  , -0.01453443670,

    -0.04022660960, -0.07202840090, -0.1058859734, -0.1393016393, -0.1709096889,

    -0.2000921459, -0.2266222242, -0.2505013927, -0.2718476845, -0.2908273534]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.340792622,

    2.411780390, 2.440153935, 2.461847101, 2.483174826, 2.505029140,

    2.527195169, 2.549330975, 2.571054636, 2.592102886, 2.612256775,

    2.631332438]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+5*q^3+5*q^4+3*q^5+q^6, 1+4*q+7*q^2+10*q^3
+13*q^4+14*q^5+14*q^6+13*q^7+9*q^8+4*q^9+q^10, 1+5*q+11*q^2+18*q^3+26*q^4+33*q^
5+40*q^6+45*q^7+47*q^8+45*q^9+41*q^10+36*q^11+26*q^12+14*q^13+5*q^14+q^15, 1+6*
q+16*q^2+30*q^3+48*q^4+67*q^5+88*q^6+111*q^7+131*q^8+146*q^9+158*q^10+166*q^11+
166*q^12+157*q^13+141*q^14+123*q^15+103*q^16+76*q^17+45*q^18+20*q^19+6*q^20+q^
21, 1+7*q+22*q^2+47*q^3+83*q^4+127*q^5+178*q^6+238*q^7+303*q^8+369*q^9+433*q^10
+494*q^11+546*q^12+587*q^13+613*q^14+625*q^15+624*q^16+605*q^17+564*q^18+506*q^
19+440*q^20+372*q^21+303*q^22+224*q^23+141*q^24+71*q^25+27*q^26+7*q^27+q^28, 1+
8*q+29*q^2+70*q^3+136*q^4+227*q^5+341*q^6+481*q^7+644*q^8+828*q^9+1030*q^10+
1245*q^11+1464*q^12+1676*q^13+1874*q^14+2057*q^15+2222*q^16+2356*q^17+2445*q^18
+2487*q^19+2487*q^20+2448*q^21+2368*q^22+2241*q^23+2064*q^24+1846*q^25+1609*q^
26+1369*q^27+1135*q^28+905*q^29+669*q^30+436*q^31+239*q^32+105*q^33+35*q^34+8*q
^35+q^36, 1+9*q+37*q^2+100*q^3+213*q^4+386*q^5+622*q^6+928*q^7+1305*q^8+1752*q^
9+2272*q^10+2864*q^11+3521*q^12+4226*q^13+4958*q^14+5700*q^15+6450*q^16+7189*q^
17+7890*q^18+8529*q^19+9094*q^20+9573*q^21+9955*q^22+10220*q^23+10352*q^24+
10339*q^25+10181*q^26+9894*q^27+9498*q^28+8995*q^29+8374*q^30+7636*q^31+6808*q^
32+5942*q^33+5084*q^34+4262*q^35+3480*q^36+2736*q^37+2017*q^38+1345*q^39+780*q^
40+379*q^41+148*q^42+44*q^43+9*q^44+q^45, 1+10*q+46*q^2+138*q^3+321*q^4+629*q^5
+1086*q^6+1716*q^7+2537*q^8+3558*q^9+4793*q^10+6254*q^11+7953*q^12+9886*q^13+
12033*q^14+14362*q^15+16851*q^16+19471*q^17+22173*q^18+24908*q^19+27638*q^20+
30329*q^21+32937*q^22+35395*q^23+37635*q^24+39605*q^25+41270*q^26+42604*q^27+
43589*q^28+44199*q^29+44390*q^30+44123*q^31+43392*q^32+42230*q^33+40683*q^34+
38792*q^35+36597*q^36+34132*q^37+31403*q^38+28426*q^39+25276*q^40+22082*q^41+
18971*q^42+16028*q^43+13278*q^44+10718*q^45+8338*q^46+6133*q^47+4150*q^48+2505*
q^49+1307*q^50+571*q^51+201*q^52+54*q^53+10*q^54+q^55, 1+11*q+56*q^2+185*q^3+
468*q^4+988*q^5+1824*q^6+3053*q^7+4747*q^8+6961*q^9+9750*q^10+13165*q^11+17261*
q^12+22089*q^13+27676*q^14+34024*q^15+41123*q^16+48945*q^17+57423*q^18+66469*q^
19+75985*q^20+85899*q^21+96134*q^22+106583*q^23+117090*q^24+127487*q^25+137606*
q^26+147318*q^27+156504*q^28+165041*q^29+172776*q^30+179552*q^31+185217*q^32+
189659*q^33+192805*q^34+194625*q^35+195101*q^36+194214*q^37+191916*q^38+188156*
q^39+182942*q^40+176376*q^41+168636*q^42+159920*q^43+150390*q^44+140154*q^45+
129306*q^46+117938*q^47+106170*q^48+94208*q^49+82377*q^50+71006*q^51+60341*q^52
+50479*q^53+41421*q^54+33123*q^55+25569*q^56+18769*q^57+12832*q^58+7971*q^59+
4384*q^60+2079*q^61+826*q^62+265*q^63+65*q^64+11*q^65+q^66, 1+12*q+67*q^2+242*q
^3+663*q^4+1503*q^5+2960*q^6+5245*q^7+8576*q^8+13154*q^9+19173*q^10+26820*q^11+
36280*q^12+47748*q^13+61404*q^14+77407*q^15+95893*q^16+116970*q^17+140664*q^18+
166933*q^19+195667*q^20+226736*q^21+260009*q^22+295322*q^23+332452*q^24+371120*
q^25+410985*q^26+451683*q^27+492855*q^28+534125*q^29+575048*q^30+615128*q^31+
653861*q^32+690802*q^33+725553*q^34+757722*q^35+786910*q^36+812722*q^37+834758*
q^38+852622*q^39+865968*q^40+874565*q^41+878329*q^42+877267*q^43+871389*q^44+
860709*q^45+845273*q^46+825148*q^47+800441*q^48+771367*q^49+738322*q^50+701866*
q^51+662641*q^52+621254*q^53+578232*q^54+533981*q^55+488824*q^56+443094*q^57+
397249*q^58+351936*q^59+307981*q^60+266226*q^61+227313*q^62+191548*q^63+158950*
q^64+129374*q^65+102671*q^66+78778*q^67+57742*q^68+39773*q^69+25241*q^70+14444*
q^71+7290*q^72+3170*q^73+1156*q^74+341*q^75+77*q^76+12*q^77+q^78, 1+13*q+79*q^2
+310*q^3+916*q^4+2223*q^5+4659*q^6+8730*q^7+15004*q^8+24070*q^9+36525*q^10+
52971*q^11+74007*q^12+100255*q^13+132339*q^14+170883*q^15+216515*q^16+269861*q^
17+331471*q^18+401763*q^19+480999*q^20+569305*q^21+666722*q^22+773180*q^23+
888487*q^24+1012320*q^25+1144247*q^26+1283715*q^27+1430113*q^28+1582731*q^29+
1740714*q^30+1902966*q^31+2068211*q^32+2235082*q^33+2402282*q^34+2568550*q^35+
2732631*q^36+2893176*q^37+3048730*q^38+3197695*q^39+3338468*q^40+3469530*q^41+
3589558*q^42+3697373*q^43+3791913*q^44+3872196*q^45+3937338*q^46+3986450*q^47+
4018649*q^48+4033181*q^49+4029650*q^50+4008061*q^51+3968743*q^52+3912209*q^53+
3839091*q^54+3750027*q^55+3645584*q^56+3526362*q^57+3393180*q^58+3247255*q^59+
3090236*q^60+2924126*q^61+2751047*q^62+2573039*q^63+2391897*q^64+2209160*q^65+
2026146*q^66+1844058*q^67+1664065*q^68+1487588*q^69+1316471*q^70+1152918*q^71+
999040*q^72+856449*q^73+725911*q^74+607446*q^75+500555*q^76+404602*q^77+319048*
q^78+243639*q^79+178383*q^80+123583*q^81+79723*q^82+47040*q^83+24915*q^84+11617
*q^85+4667*q^86+1574*q^87+430*q^88+90*q^89+13*q^90+q^91, 1+14*q+92*q^2+390*q^3+
1238*q^4+3207*q^5+7136*q^6+14120*q^7+25490*q^8+42755*q^9+67549*q^10+101605*q^11
+146717*q^12+204759*q^13+277669*q^14+367434*q^15+476120*q^16+605890*q^17+758956
*q^18+937462*q^19+1143379*q^20+1378416*q^21+1644064*q^22+1941519*q^23+2271655*q
^24+2634985*q^25+3031679*q^26+3461584*q^27+3924311*q^28+4419244*q^29+4945503*q^
30+5501767*q^31+6086125*q^32+6696094*q^33+7328819*q^34+7981203*q^35+8649989*q^
36+9331683*q^37+10022475*q^38+10718154*q^39+11414167*q^40+12105760*q^41+
12788213*q^42+13456864*q^43+14106981*q^44+14733682*q^45+15332060*q^46+15897259*
q^47+16424499*q^48+16909118*q^49+17346751*q^50+17733501*q^51+18066000*q^52+
18341353*q^53+18557113*q^54+18711170*q^55+18801602*q^56+18826699*q^57+18785214*
q^58+18676657*q^59+18501429*q^60+18260826*q^61+17956921*q^62+17592430*q^63+
17170526*q^64+16694625*q^65+16168247*q^66+15595012*q^67+14978651*q^68+14323223*
q^69+13633581*q^70+12915732*q^71+12176609*q^72+11423508*q^73+10663364*q^74+
9902322*q^75+9145592*q^76+8397627*q^77+7662374*q^78+6943636*q^79+6245311*q^80+
5571672*q^81+4927681*q^82+4318881*q^83+3750487*q^84+3226272*q^85+2747865*q^86+
2314832*q^87+1925297*q^88+1576780*q^89+1266855*q^90+993612*q^91+755843*q^92+
552971*q^93+384871*q^94+251503*q^95+152019*q^96+83649*q^97+41211*q^98+17859*q^
99+6671*q^100+2094*q^101+533*q^102+104*q^103+14*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+5*q^3+5*q^2+3*q+1, q^10+4*q^9+7*q^8+
10*q^7+13*q^6+14*q^5+14*q^4+13*q^3+9*q^2+4*q+1, q^15+5*q^14+11*q^13+18*q^12+26*
q^11+33*q^10+40*q^9+45*q^8+47*q^7+45*q^6+41*q^5+36*q^4+26*q^3+14*q^2+5*q+1, q^
21+6*q^20+16*q^19+30*q^18+48*q^17+67*q^16+88*q^15+111*q^14+131*q^13+146*q^12+
158*q^11+166*q^10+166*q^9+157*q^8+141*q^7+123*q^6+103*q^5+76*q^4+45*q^3+20*q^2+
6*q+1, 1+47*q^25+178*q^22+83*q^24+238*q^21+127*q^23+7*q+27*q^2+22*q^26+71*q^3+
141*q^4+224*q^5+303*q^6+372*q^7+440*q^8+506*q^9+564*q^10+605*q^11+624*q^12+625*
q^13+613*q^14+587*q^15+546*q^16+q^28+494*q^17+433*q^18+369*q^19+303*q^20+7*q^27
, 1+1245*q^25+1874*q^22+1464*q^24+2057*q^21+1676*q^23+8*q+341*q^30+481*q^29+29*
q^34+70*q^33+35*q^2+1030*q^26+105*q^3+239*q^4+436*q^5+669*q^6+136*q^32+905*q^7+
1135*q^8+1369*q^9+1609*q^10+1846*q^11+2064*q^12+2241*q^13+2368*q^14+2448*q^15+
2487*q^16+644*q^28+2487*q^17+2445*q^18+2356*q^19+2222*q^20+828*q^27+227*q^31+8*
q^35+q^36, 1+9094*q^25+100*q^42+10220*q^22+9573*q^24+10352*q^21+9955*q^23+9*q+9
*q^44+5700*q^30+q^45+1305*q^37+6450*q^29+622*q^39+213*q^41+2864*q^34+928*q^38+
3521*q^33+44*q^2+8529*q^26+148*q^3+379*q^4+780*q^5+1345*q^6+4226*q^32+2017*q^7+
2736*q^8+3480*q^9+4262*q^10+5084*q^11+5942*q^12+6808*q^13+7636*q^14+8374*q^15+
8995*q^16+37*q^43+7189*q^28+9498*q^17+9894*q^18+10181*q^19+10339*q^20+7890*q^27
+386*q^40+4958*q^31+2272*q^35+1752*q^36, 1+44390*q^25+9886*q^42+42230*q^22+
44123*q^24+40683*q^21+43392*q^23+10*q+321*q^51+6254*q^44+46*q^53+39605*q^30+138
*q^52+q^55+10*q^54+4793*q^45+22173*q^37+41270*q^29+3558*q^46+16851*q^39+12033*q
^41+30329*q^34+629*q^50+19471*q^38+32937*q^33+54*q^2+44199*q^26+201*q^3+571*q^4
+1307*q^5+2505*q^6+35395*q^32+4150*q^7+6133*q^8+8338*q^9+10718*q^10+13278*q^11+
16028*q^12+18971*q^13+22082*q^14+25276*q^15+28426*q^16+7953*q^43+1086*q^49+
42604*q^28+31403*q^17+34132*q^18+36597*q^19+38792*q^20+43589*q^27+14362*q^40+
37635*q^31+27638*q^35+1716*q^48+24908*q^36+2537*q^47, 1+176376*q^25+117090*q^42
+150390*q^22+168636*q^24+140154*q^21+159920*q^23+11*q+34024*q^51+96134*q^44+
22089*q^53+6961*q^57+195101*q^30+27676*q^52+4747*q^58+13165*q^55+1824*q^60+
17261*q^54+85899*q^45+165041*q^37+194214*q^29+75985*q^46+11*q^65+147318*q^39+
988*q^61+127487*q^41+185217*q^34+41123*q^50+56*q^64+156504*q^38+189659*q^33+
9750*q^56+65*q^2+182942*q^26+265*q^3+826*q^4+2079*q^5+4384*q^6+192805*q^32+7971
*q^7+12832*q^8+18769*q^9+468*q^62+25569*q^10+33123*q^11+41421*q^12+50479*q^13+
60341*q^14+71006*q^15+82377*q^16+q^66+106583*q^43+48945*q^49+191916*q^28+94208*
q^17+106170*q^18+117938*q^19+129306*q^20+188156*q^27+137606*q^40+185*q^63+
194625*q^31+179552*q^35+57423*q^48+3053*q^59+172776*q^36+66469*q^47, 1+621254*q
^25+786910*q^42+488824*q^22+578232*q^24+443094*q^21+533981*q^23+12*q+451683*q^
51+725553*q^44+371120*q^53+226736*q^57+800441*q^30+410985*q^52+195667*q^58+
295322*q^55+140664*q^60+332452*q^54+690802*q^45+5245*q^71+663*q^74+874565*q^37+
26820*q^67+771367*q^29+653861*q^46+47748*q^65+13154*q^69+852622*q^39+116970*q^
61+812722*q^41+871389*q^34+492855*q^50+61404*q^64+865968*q^38+860709*q^33+
260009*q^56+77*q^2+19173*q^68+662641*q^26+341*q^3+1156*q^4+3170*q^5+7290*q^6+
845273*q^32+14444*q^7+25241*q^8+39773*q^9+95893*q^62+57742*q^10+78778*q^11+
102671*q^12+67*q^76+129374*q^13+158950*q^14+191548*q^15+227313*q^16+36280*q^66+
8576*q^70+12*q^77+q^78+757722*q^43+534125*q^49+738322*q^28+266226*q^17+307981*q
^18+351936*q^19+397249*q^20+701866*q^27+2960*q^72+834758*q^40+1503*q^73+77407*q
^63+825148*q^31+242*q^75+877267*q^35+575048*q^48+166933*q^59+878329*q^36+615128
*q^47, 1+2026146*q^25+4033181*q^42+1487588*q^22+13*q^90+1844058*q^24+1316471*q^
21+1664065*q^23+13*q+3338468*q^51+3986450*q^44+3048730*q^53+2402282*q^57+
2924126*q^30+3197695*q^52+2235082*q^58+2732631*q^55+1902966*q^60+q^91+2893176*q
^54+3937338*q^45+480999*q^71+269861*q^74+3839091*q^37+888487*q^67+2751047*q^29+
3872196*q^46+1144247*q^65+666722*q^69+3968743*q^39+1740714*q^61+4029650*q^41+
3526362*q^34+74007*q^79+3469530*q^50+1283715*q^64+3912209*q^38+3393180*q^33+
2568550*q^56+90*q^2+773180*q^68+2209160*q^26+430*q^3+1574*q^4+4667*q^5+11617*q^
6+3247255*q^32+24915*q^7+47040*q^8+79723*q^9+1582731*q^62+123583*q^10+178383*q^
11+243639*q^12+170883*q^76+319048*q^13+404602*q^14+500555*q^15+607446*q^16+
1012320*q^66+569305*q^70+132339*q^77+100255*q^78+4018649*q^43+3589558*q^49+
2573039*q^28+725911*q^17+856449*q^18+999040*q^19+1152918*q^20+2391897*q^27+
401763*q^72+4008061*q^40+331471*q^73+1430113*q^63+3090236*q^31+216515*q^75+
3645584*q^35+52971*q^80+3697373*q^48+15004*q^83+36525*q^81+4659*q^85+2223*q^86+
916*q^87+310*q^88+79*q^89+8730*q^84+24070*q^82+2068211*q^59+3750027*q^36+
3791913*q^47, 1+6245311*q^25+17592430*q^42+4318881*q^22+367434*q^90+5571672*q^
24+3750487*q^21+4927681*q^23+14*q+25490*q^97+18557113*q^51+18260826*q^44+
18066000*q^53+16424499*q^57+9902322*q^30+101605*q^94+18341353*q^52+15897259*q^
58+17346751*q^55+14733682*q^60+277669*q^91+17733501*q^54+18501429*q^45+7328819*
q^71+5501767*q^74+42755*q^96+14120*q^98+14978651*q^37+10022475*q^67+9145592*q^
29+18676657*q^46+146717*q^93+7136*q^99+11414167*q^65+8649989*q^69+16168247*q^39
+14106981*q^61+17170526*q^41+12915732*q^34+3031679*q^79+18711170*q^50+12105760*
q^64+3207*q^100+15595012*q^38+12176609*q^33+16909118*q^56+104*q^2+9331683*q^68+
6943636*q^26+533*q^3+2094*q^4+6671*q^5+17859*q^6+11423508*q^32+41211*q^7+83649*
q^8+152019*q^9+67549*q^95+13456864*q^62+1238*q^101+251503*q^10+384871*q^11+
552971*q^12+4419244*q^76+755843*q^13+993612*q^14+1266855*q^15+1576780*q^16+
10718154*q^66+7981203*q^70+3924311*q^77+3461584*q^78+17956921*q^43+390*q^102+
18801602*q^49+8397627*q^28+1925297*q^17+2314832*q^18+2747865*q^19+3226272*q^20+
7662374*q^27+6696094*q^72+16694625*q^40+6086125*q^73+92*q^103+12788213*q^63+
10663364*q^31+4945503*q^75+13633581*q^35+14*q^104+2634985*q^80+18826699*q^48+q^
105+1644064*q^83+2271655*q^81+1143379*q^85+937462*q^86+758956*q^87+605890*q^88+
476120*q^89+1378416*q^84+1941519*q^82+15332060*q^59+14323223*q^36+204759*q^92+
18785214*q^47]


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

[1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738,

    142078746, 745387038]

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

     is given by

[1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

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

     is given by

[0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869,

    71039373, 372693519]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]

 ODD:, [0, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723,

    13648869, 71039373, 372693519]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.045454545, 5.144444444, 7.794416244,

    10.99169435, 14.73299836, 19.01555812, 23.83703384, 29.19542785,

    35.08901366, 41.51628194, 48.47589966, 55.96667887]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.521879004, 2.208876593, 3.016177917,

    3.938260757, 4.969607339, 6.105240438, 7.340738364, 8.672164331,

    10.09599248, 11.60904481, 13.20843956, 14.89154919]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.31612, 4.87914, 9.09733, 15.5099, 24.6970,

    37.2740, 53.8864, 75.2064, 101.929, 134.770, 174.463, 221.758]

Skewness: , [Float(undefined), 0., 0., -0.07672933926, -0.1150089428,

    -0.1273617380, -0.1279901695, -0.1238853985, -0.1181235307, -0.1120218128,

    -0.1061296779, -0.1006574752, -0.09566320250, -0.09114015974,

    -0.08705399224]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.266194381,

    2.332965779, 2.376203989, 2.413199387, 2.446364519, 2.476206381,

    2.503078693, 2.527314962, 2.549251298, 2.569169246, 2.587351727,

    2.604022492]

                                end of this data



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

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

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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%1 := {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+4*q^2+6*q^3+5*q^4+2*q^5+q^6, 1+4*q+7*q^2+11*q^3
+15*q^4+16*q^5+14*q^6+10*q^7+5*q^8+2*q^9+q^10, 1+5*q+11*q^2+19*q^3+29*q^4+37*q^
5+45*q^6+47*q^7+44*q^8+36*q^9+27*q^10+18*q^11+10*q^12+5*q^13+2*q^14+q^15, 1+6*q
+16*q^2+31*q^3+52*q^4+73*q^5+96*q^6+120*q^7+131*q^8+137*q^9+138*q^10+124*q^11+
110*q^12+88*q^13+63*q^14+46*q^15+31*q^16+18*q^17+10*q^18+5*q^19+2*q^20+q^21, 1+
7*q+22*q^2+48*q^3+88*q^4+136*q^5+189*q^6+251*q^7+307*q^8+353*q^9+390*q^10+412*q
^11+415*q^12+409*q^13+380*q^14+336*q^15+295*q^16+243*q^17+189*q^18+145*q^19+104
*q^20+73*q^21+50*q^22+31*q^23+18*q^24+10*q^25+5*q^26+2*q^27+q^28, 1+8*q+29*q^2+
71*q^3+142*q^4+240*q^5+357*q^6+497*q^7+644*q^8+789*q^9+933*q^10+1053*q^11+1153*
q^12+1224*q^13+1262*q^14+1270*q^15+1237*q^16+1173*q^17+1084*q^18+980*q^19+859*q
^20+736*q^21+612*q^22+495*q^23+392*q^24+298*q^25+220*q^26+161*q^27+114*q^28+77*
q^29+50*q^30+31*q^31+18*q^32+10*q^33+5*q^34+2*q^35+q^36, 1+9*q+37*q^2+101*q^3+
220*q^4+404*q^5+646*q^6+948*q^7+1293*q^8+1655*q^9+2042*q^10+2424*q^11+2782*q^12
+3123*q^13+3402*q^14+3621*q^15+3800*q^16+3880*q^17+3878*q^18+3818*q^19+3666*q^
20+3468*q^21+3248*q^22+2961*q^23+2660*q^24+2362*q^25+2034*q^26+1732*q^27+1451*q
^28+1184*q^29+959*q^30+759*q^31+581*q^32+443*q^33+329*q^34+236*q^35+171*q^36+
118*q^37+77*q^38+50*q^39+31*q^40+18*q^41+10*q^42+5*q^43+2*q^44+q^45, 1+10*q+46*
q^2+139*q^3+329*q^4+653*q^5+1122*q^6+1743*q^7+2503*q^8+3353*q^9+4285*q^10+5270*
q^11+6268*q^12+7285*q^13+8269*q^14+9173*q^15+10006*q^16+10737*q^17+11284*q^18+
11692*q^19+11919*q^20+11934*q^21+11821*q^22+11537*q^23+11107*q^24+10602*q^25+
9986*q^26+9246*q^27+8486*q^28+7681*q^29+6852*q^30+6080*q^31+5305*q^32+4549*q^33
+3875*q^34+3249*q^35+2677*q^36+2193*q^37+1766*q^38+1396*q^39+1096*q^40+841*q^41
+632*q^42+474*q^43+345*q^44+246*q^45+175*q^46+118*q^47+77*q^48+50*q^49+31*q^50+
18*q^51+10*q^52+5*q^53+2*q^54+q^55, 1+11*q+56*q^2+186*q^3+477*q^4+1019*q^5+1877
*q^6+3093*q^7+4679*q^8+6573*q^9+8729*q^10+11097*q^11+13592*q^12+16225*q^13+
18938*q^14+21643*q^15+24342*q^16+26947*q^17+29369*q^18+31544*q^19+33381*q^20+
34851*q^21+35959*q^22+36620*q^23+36864*q^24+36739*q^25+36255*q^26+35461*q^27+
34335*q^28+32910*q^29+31266*q^30+29424*q^31+27416*q^32+25325*q^33+23200*q^34+
21045*q^35+18955*q^36+16902*q^37+14912*q^38+13048*q^39+11302*q^40+9690*q^41+
8251*q^42+6935*q^43+5762*q^44+4758*q^45+3887*q^46+3132*q^47+2500*q^48+1968*q^49
+1529*q^50+1178*q^51+892*q^52+663*q^53+490*q^54+355*q^55+250*q^56+175*q^57+118*
q^58+77*q^59+50*q^60+31*q^61+18*q^62+10*q^63+5*q^64+2*q^65+q^66, 1+12*q+67*q^2+
243*q^3+673*q^4+1542*q^5+3036*q^6+5308*q^7+8462*q^8+12475*q^9+17266*q^10+22741*
q^11+28719*q^12+35143*q^13+41972*q^14+49052*q^15+56389*q^16+63915*q^17+71412*q^
18+78811*q^19+85866*q^20+92287*q^21+98123*q^22+103151*q^23+107221*q^24+110437*q
^25+112669*q^26+113889*q^27+114311*q^28+113803*q^29+112350*q^30+110206*q^31+
107238*q^32+103506*q^33+99275*q^34+94420*q^35+89160*q^36+83821*q^37+78168*q^38+
72393*q^39+66682*q^40+60901*q^41+55284*q^42+49898*q^43+44611*q^44+39638*q^45+
35045*q^46+30684*q^47+26719*q^48+23110*q^49+19785*q^50+16835*q^51+14222*q^52+
11890*q^53+9891*q^54+8155*q^55+6648*q^56+5394*q^57+4334*q^58+3429*q^59+2698*q^
60+2101*q^61+1611*q^62+1229*q^63+923*q^64+679*q^65+500*q^66+359*q^67+250*q^68+
175*q^69+118*q^70+77*q^71+50*q^72+31*q^73+18*q^74+10*q^75+5*q^76+2*q^77+q^78, 1
+13*q+79*q^2+311*q^3+927*q^4+2271*q^5+4765*q^6+8831*q^7+14835*q^8+22953*q^9+
33164*q^10+45348*q^11+59192*q^12+74437*q^13+90959*q^14+108499*q^15+127009*q^16+
146496*q^17+166734*q^18+187579*q^19+208762*q^20+229676*q^21+249952*q^22+269266*
q^23+286974*q^24+303009*q^25+317185*q^26+329131*q^27+338937*q^28+346592*q^29+
351856*q^30+354996*q^31+355944*q^32+354443*q^33+350891*q^34+345293*q^35+337485*
q^36+328038*q^37+317102*q^38+304672*q^39+291324*q^40+277117*q^41+262094*q^42+
246771*q^43+231145*q^44+215174*q^45+199321*q^46+183763*q^47+168340*q^48+153582*
q^49+139408*q^50+125759*q^51+112942*q^52+100857*q^53+89483*q^54+79079*q^55+
69448*q^56+60559*q^57+52594*q^58+45416*q^59+38911*q^60+33210*q^61+28141*q^62+
23657*q^63+19804*q^64+16448*q^65+13546*q^66+11118*q^67+9039*q^68+7276*q^69+5831
*q^70+4627*q^71+3627*q^72+2831*q^73+2183*q^74+1662*q^75+1260*q^76+939*q^77+689*
q^78+504*q^79+359*q^80+250*q^81+175*q^82+118*q^83+77*q^84+50*q^85+31*q^86+18*q^
87+10*q^88+5*q^89+2*q^90+q^91, 1+14*q+92*q^2+391*q^3+1250*q^4+3265*q^5+7280*q^6
+14280*q^7+25264*q^8+41010*q^9+61903*q^10+88003*q^11+118938*q^12+154127*q^13+
193147*q^14+235430*q^15+280659*q^16+328875*q^17+379870*q^18+433636*q^19+490029*
q^20+548265*q^21+607666*q^22+667300*q^23+725806*q^24+782273*q^25+835847*q^26+
885605*q^27+931266*q^28+972315*q^29+1008336*q^30+1039431*q^31+1065211*q^32+
1085513*q^33+1100257*q^34+1109137*q^35+1112120*q^36+1109394*q^37+1100907*q^38+
1087006*q^39+1068117*q^40+1044532*q^41+1016819*q^42+985618*q^43+951220*q^44+
914145*q^45+874977*q^46+833808*q^47+791334*q^48+747809*q^49+703533*q^50+659351*
q^51+615419*q^52+571948*q^53+529476*q^54+488215*q^55+448284*q^56+410077*q^57+
373491*q^58+338622*q^59+305803*q^60+275007*q^61+246123*q^62+219421*q^63+194762*
q^64+172050*q^65+151345*q^66+132525*q^67+115439*q^68+100168*q^69+86466*q^70+
74210*q^71+63433*q^72+53937*q^73+45569*q^74+38350*q^75+32077*q^76+26660*q^77+
22050*q^78+18111*q^79+14771*q^80+11994*q^81+9657*q^82+7709*q^83+6124*q^84+4825*
q^85+3760*q^86+2913*q^87+2234*q^88+1693*q^89+1276*q^90+949*q^91+693*q^92+504*q^
93+359*q^94+250*q^95+175*q^96+118*q^97+77*q^98+50*q^99+31*q^100+18*q^101+10*q^
102+5*q^103+2*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+4*q^4+6*q^3+5*q^2+2*q+1, q^10+4*q^9+7*q^8+
11*q^7+15*q^6+16*q^5+14*q^4+10*q^3+5*q^2+2*q+1, q^15+5*q^14+11*q^13+19*q^12+29*
q^11+37*q^10+45*q^9+47*q^8+44*q^7+36*q^6+27*q^5+18*q^4+10*q^3+5*q^2+2*q+1, q^21
+6*q^20+16*q^19+31*q^18+52*q^17+73*q^16+96*q^15+120*q^14+131*q^13+137*q^12+138*
q^11+124*q^10+110*q^9+88*q^8+63*q^7+46*q^6+31*q^5+18*q^4+10*q^3+5*q^2+2*q+1, 1+
48*q^25+189*q^22+88*q^24+251*q^21+136*q^23+2*q+5*q^2+22*q^26+10*q^3+18*q^4+31*q
^5+50*q^6+73*q^7+104*q^8+145*q^9+189*q^10+243*q^11+295*q^12+336*q^13+380*q^14+
409*q^15+415*q^16+q^28+412*q^17+390*q^18+353*q^19+307*q^20+7*q^27, 1+1053*q^25+
1262*q^22+1153*q^24+1270*q^21+1224*q^23+2*q+357*q^30+497*q^29+29*q^34+71*q^33+5
*q^2+933*q^26+10*q^3+18*q^4+31*q^5+50*q^6+142*q^32+77*q^7+114*q^8+161*q^9+220*q
^10+298*q^11+392*q^12+495*q^13+612*q^14+736*q^15+859*q^16+644*q^28+980*q^17+
1084*q^18+1173*q^19+1237*q^20+789*q^27+240*q^31+8*q^35+q^36, 1+3666*q^25+101*q^
42+2961*q^22+3468*q^24+2660*q^21+3248*q^23+2*q+9*q^44+3621*q^30+q^45+1293*q^37+
3800*q^29+646*q^39+220*q^41+2424*q^34+948*q^38+2782*q^33+5*q^2+3818*q^26+10*q^3
+18*q^4+31*q^5+50*q^6+3123*q^32+77*q^7+118*q^8+171*q^9+236*q^10+329*q^11+443*q^
12+581*q^13+759*q^14+959*q^15+1184*q^16+37*q^43+3880*q^28+1451*q^17+1732*q^18+
2034*q^19+2362*q^20+3878*q^27+404*q^40+3402*q^31+2042*q^35+1655*q^36, 1+6852*q^
25+7285*q^42+4549*q^22+6080*q^24+3875*q^21+5305*q^23+2*q+329*q^51+5270*q^44+46*
q^53+10602*q^30+139*q^52+q^55+10*q^54+4285*q^45+11284*q^37+9986*q^29+3353*q^46+
10006*q^39+8269*q^41+11934*q^34+653*q^50+10737*q^38+11821*q^33+5*q^2+7681*q^26+
10*q^3+18*q^4+31*q^5+50*q^6+11537*q^32+77*q^7+118*q^8+175*q^9+246*q^10+345*q^11
+474*q^12+632*q^13+841*q^14+1096*q^15+1396*q^16+6268*q^43+1122*q^49+9246*q^28+
1766*q^17+2193*q^18+2677*q^19+3249*q^20+8486*q^27+9173*q^40+11107*q^31+11919*q^
35+1743*q^48+11692*q^36+2503*q^47, 1+9690*q^25+36864*q^42+5762*q^22+8251*q^24+
4758*q^21+6935*q^23+2*q+21643*q^51+35959*q^44+16225*q^53+6573*q^57+18955*q^30+
18938*q^52+4679*q^58+11097*q^55+1877*q^60+13592*q^54+34851*q^45+32910*q^37+
16902*q^29+33381*q^46+11*q^65+35461*q^39+1019*q^61+36739*q^41+27416*q^34+24342*
q^50+56*q^64+34335*q^38+25325*q^33+8729*q^56+5*q^2+11302*q^26+10*q^3+18*q^4+31*
q^5+50*q^6+23200*q^32+77*q^7+118*q^8+175*q^9+477*q^62+250*q^10+355*q^11+490*q^
12+663*q^13+892*q^14+1178*q^15+1529*q^16+q^66+36620*q^43+26947*q^49+14912*q^28+
1968*q^17+2500*q^18+3132*q^19+3887*q^20+13048*q^27+36255*q^40+186*q^63+21045*q^
31+29424*q^35+29369*q^48+3093*q^59+31266*q^36+31544*q^47, 1+11890*q^25+89160*q^
42+6648*q^22+9891*q^24+5394*q^21+8155*q^23+2*q+113889*q^51+99275*q^44+110437*q^
53+92287*q^57+26719*q^30+112669*q^52+85866*q^58+103151*q^55+71412*q^60+107221*q
^54+103506*q^45+5308*q^71+673*q^74+60901*q^37+22741*q^67+23110*q^29+107238*q^46
+35143*q^65+12475*q^69+72393*q^39+63915*q^61+83821*q^41+44611*q^34+114311*q^50+
41972*q^64+66682*q^38+39638*q^33+98123*q^56+5*q^2+17266*q^68+14222*q^26+10*q^3+
18*q^4+31*q^5+50*q^6+35045*q^32+77*q^7+118*q^8+175*q^9+56389*q^62+250*q^10+359*
q^11+500*q^12+67*q^76+679*q^13+923*q^14+1229*q^15+1611*q^16+28719*q^66+8462*q^
70+12*q^77+q^78+94420*q^43+113803*q^49+19785*q^28+2101*q^17+2698*q^18+3429*q^19
+4334*q^20+16835*q^27+3036*q^72+78168*q^40+1542*q^73+49052*q^63+30684*q^31+243*
q^75+49898*q^35+112350*q^48+78811*q^59+55284*q^36+110206*q^47, 1+13546*q^25+
153582*q^42+7276*q^22+13*q^90+11118*q^24+5831*q^21+9039*q^23+2*q+291324*q^51+
183763*q^44+317102*q^53+350891*q^57+33210*q^30+304672*q^52+354443*q^58+337485*q
^55+354996*q^60+q^91+328038*q^54+199321*q^45+208762*q^71+146496*q^74+89483*q^37
+286974*q^67+28141*q^29+215174*q^46+317185*q^65+249952*q^69+112942*q^39+351856*
q^61+139408*q^41+60559*q^34+59192*q^79+277117*q^50+329131*q^64+100857*q^38+
52594*q^33+345293*q^56+5*q^2+269266*q^68+16448*q^26+10*q^3+18*q^4+31*q^5+50*q^6
+45416*q^32+77*q^7+118*q^8+175*q^9+346592*q^62+250*q^10+359*q^11+504*q^12+
108499*q^76+689*q^13+939*q^14+1260*q^15+1662*q^16+303009*q^66+229676*q^70+90959
*q^77+74437*q^78+168340*q^43+262094*q^49+23657*q^28+2183*q^17+2831*q^18+3627*q^
19+4627*q^20+19804*q^27+187579*q^72+125759*q^40+166734*q^73+338937*q^63+38911*q
^31+127009*q^75+69448*q^35+45348*q^80+246771*q^48+14835*q^83+33164*q^81+4765*q^
85+2271*q^86+927*q^87+311*q^88+79*q^89+8831*q^84+22953*q^82+355944*q^59+79079*q
^36+231145*q^47, 1+14771*q^25+219421*q^42+7709*q^22+235430*q^90+11994*q^24+6124
*q^21+9657*q^23+2*q+25264*q^97+529476*q^51+275007*q^44+615419*q^53+791334*q^57+
38350*q^30+88003*q^94+571948*q^52+833808*q^58+703533*q^55+914145*q^60+193147*q^
91+659351*q^54+305803*q^45+1100257*q^71+1039431*q^74+41010*q^96+14280*q^98+
115439*q^37+1100907*q^67+32077*q^29+338622*q^46+118938*q^93+7280*q^99+1068117*q
^65+1112120*q^69+151345*q^39+951220*q^61+194762*q^41+74210*q^34+835847*q^79+
488215*q^50+1044532*q^64+3265*q^100+132525*q^38+63433*q^33+747809*q^56+5*q^2+
1109394*q^68+18111*q^26+10*q^3+18*q^4+31*q^5+50*q^6+53937*q^32+77*q^7+118*q^8+
175*q^9+61903*q^95+985618*q^62+1250*q^101+250*q^10+359*q^11+504*q^12+972315*q^
76+693*q^13+949*q^14+1276*q^15+1693*q^16+1087006*q^66+1109137*q^70+931266*q^77+
885605*q^78+246123*q^43+391*q^102+448284*q^49+26660*q^28+2234*q^17+2913*q^18+
3760*q^19+4825*q^20+22050*q^27+1085513*q^72+172050*q^40+1065211*q^73+92*q^103+
1016819*q^63+45569*q^31+1008336*q^75+86466*q^35+14*q^104+782273*q^80+410077*q^
48+q^105+607666*q^83+725806*q^81+490029*q^85+433636*q^86+379870*q^87+328875*q^
88+280659*q^89+548265*q^84+667300*q^82+874977*q^59+100168*q^36+154127*q^92+
373491*q^47]


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

[1, 2, 6, 22, 86, 337, 1299, 4910, 18228, 66640, 240550, 859295, 3043525,

    10705182, 37441618]

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

     is given by

[1, 1, 3, 11, 43, 169, 650, 2454, 9113, 33320, 120275, 429658, 1521773, 5352537,

    18720755]

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

     is given by

[0, 1, 3, 11, 43, 168, 649, 2456, 9115, 33320, 120275, 429637, 1521752, 5352645,

    18720863]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 11, 43, 168, 649, 2454, 9113, 33320, 120275, 429658, 1521773,

    5352645, 18720863]

 ODD:, [0, 1, 3, 11, 43, 169, 650, 2456, 9115, 33320, 120275, 429637, 1521752,

    5352537, 18720755]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.954545455, 4.802325581, 6.991097923,

    9.483448807, 12.25417515, 15.28653720, 18.56935774, 22.09512783,

    25.85883428, 29.85722739, 34.08834348, 38.55117567]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.460923516, 2.056299270, 2.763138721,

    3.578482670, 4.493552507, 5.499957188, 6.590515564, 7.758973104,

    8.999765060, 10.30792277, 11.67904361, 13.10926978]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.13430, 4.22837, 7.63494, 12.8055, 20.1920,

    30.2495, 43.4349, 60.2017, 80.9958, 106.253, 136.400, 171.853]

Skewness: , [Float(undefined), 0., 0., -0.008672084295, 0.03384782665,

    0.09448055499, 0.1505143851, 0.1961165942, 0.2319847149, 0.2601100967,

    0.2822286635, 0.2996158957, 0.3131834320, 0.3236039794, 0.3314065090]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.497740792,

    2.638928473, 2.681245660, 2.705767494, 2.729516727, 2.753553221,

    2.777041385, 2.799645412, 2.821127792, 2.841306675, 2.859968525,

    2.877011985]

                                end of this data



For the equivalence class of patterns,

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

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

                                  here it is:

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+6*q^3+5*q^4+q^5+q^6, 1+4*q+9*q^2+15*q^3+
20*q^4+18*q^5+12*q^6+5*q^7+2*q^8+q^10, 1+5*q+14*q^2+29*q^3+49*q^4+65*q^5+70*q^6
+57*q^7+35*q^8+16*q^9+5*q^10+5*q^11+2*q^12+q^15, 1+6*q+20*q^2+49*q^3+98*q^4+161
*q^5+223*q^6+257*q^7+242*q^8+181*q^9+109*q^10+53*q^11+28*q^12+17*q^13+3*q^14+4*
q^15+4*q^16+2*q^17+q^21, 1+7*q+27*q^2+76*q^3+174*q^4+333*q^5+546*q^6+771*q^7+
935*q^8+960*q^9+827*q^10+593*q^11+363*q^12+200*q^13+113*q^14+51*q^15+28*q^16+22
*q^17+14*q^18+4*q^19+2*q^20+2*q^21+4*q^22+2*q^23+q^28, 1+8*q+35*q^2+111*q^3+285
*q^4+616*q^5+1150*q^6+1877*q^7+2691*q^8+3379*q^9+3698*q^10+3495*q^11+2850*q^12+
2015*q^13+1273*q^14+755*q^15+423*q^16+228*q^17+145*q^18+84*q^19+49*q^20+22*q^21
+16*q^22+16*q^23+16*q^24+3*q^25+2*q^26+2*q^28+4*q^29+2*q^30+q^36, 1+9*q+44*q^2+
155*q^3+440*q^4+1054*q^5+2190*q^6+4009*q^7+6521*q^8+9450*q^9+12200*q^10+13977*q
^11+14154*q^12+12631*q^13+9972*q^14+7055*q^15+4605*q^16+2846*q^17+1690*q^18+
1021*q^19+612*q^20+368*q^21+213*q^22+118*q^23+89*q^24+82*q^25+44*q^26+16*q^27+9
*q^28+11*q^29+18*q^30+14*q^31+3*q^32+2*q^33+2*q^36+4*q^37+2*q^38+q^45, 1+10*q+
54*q^2+209*q^3+649*q^4+1701*q^5+3875*q^6+7810*q^7+14078*q^8+22825*q^9+33379*q^
10+44023*q^11+52278*q^12+55742*q^13+53280*q^14+45714*q^15+35485*q^16+25325*q^17
+16997*q^18+10926*q^19+6890*q^20+4315*q^21+2681*q^22+1668*q^23+982*q^24+627*q^
25+472*q^26+325*q^27+161*q^28+91*q^29+66*q^30+84*q^31+72*q^32+38*q^33+16*q^34+8
*q^35+3*q^36+12*q^37+16*q^38+14*q^39+3*q^40+2*q^41+2*q^45+4*q^46+2*q^47+q^55, 1
+11*q+65*q^2+274*q^3+923*q^4+2622*q^5+6479*q^6+14197*q^7+27930*q^8+49707*q^9+
80396*q^10+118425*q^11+158942*q^12+194185*q^13+215639*q^14+217399*q^15+199151*q
^16+166516*q^17+128391*q^18+92713*q^19+63810*q^20+42578*q^21+27977*q^22+18187*q
^23+11804*q^24+7493*q^25+4773*q^26+3250*q^27+2252*q^28+1392*q^29+819*q^30+526*q
^31+444*q^32+389*q^33+257*q^34+146*q^35+78*q^36+40*q^37+57*q^38+72*q^39+66*q^40
+39*q^41+18*q^42+6*q^43+5*q^45+10*q^46+16*q^47+14*q^48+3*q^49+2*q^50+2*q^55+4*q
^56+2*q^57+q^66, 1+12*q+77*q^2+351*q^3+1274*q^4+3894*q^5+10353*q^6+24438*q^7+
51911*q^8+100113*q^9+176318*q^10+284590*q^11+421775*q^12+574281*q^13+718127*q^
14+824016*q^15+867066*q^16+837081*q^17+743651*q^18+611933*q^19+471497*q^20+
344873*q^21+243051*q^22+167342*q^23+113671*q^24+76659*q^25+51312*q^26+34060*q^
27+23068*q^28+15884*q^29+10704*q^30+6888*q^31+4476*q^32+3131*q^33+2458*q^34+
1752*q^35+1143*q^36+686*q^37+404*q^38+319*q^39+347*q^40+312*q^41+251*q^42+147*q
^43+67*q^44+20*q^45+33*q^46+48*q^47+68*q^48+66*q^49+40*q^50+16*q^51+6*q^52+2*q^
54+3*q^55+10*q^56+16*q^57+14*q^58+3*q^59+2*q^60+2*q^66+4*q^67+2*q^68+q^78, 1+13
*q+90*q^2+441*q^3+1715*q^4+5607*q^5+15938*q^6+40242*q^7+91562*q^8+189579*q^9+
359614*q^10+627804*q^11+1011614*q^12+1507012*q^13+2076921*q^14+2647921*q^15+
3121740*q^16+3402149*q^17+3428973*q^18+3202555*q^19+2784185*q^20+2270094*q^21+
1754172*q^22+1300369*q^23+936355*q^24+661908*q^25+462964*q^26+321783*q^27+
222474*q^28+154335*q^29+108076*q^30+75592*q^31+52053*q^32+35392*q^33+24469*q^34
+17793*q^35+12980*q^36+8984*q^37+5972*q^38+3813*q^39+2612*q^40+2114*q^41+1783*q
^42+1456*q^43+1035*q^44+581*q^45+278*q^46+206*q^47+246*q^48+288*q^49+310*q^50+
245*q^51+138*q^52+61*q^53+16*q^54+20*q^55+26*q^56+42*q^57+66*q^58+68*q^59+38*q^
60+16*q^61+6*q^62+2*q^64+3*q^66+10*q^67+16*q^68+14*q^69+3*q^70+2*q^71+2*q^78+4*
q^79+2*q^80+q^91, 1+14*q+104*q^2+545*q^3+2260*q^4+7865*q^5+23779*q^6+63863*q^7+
154676*q^8+341410*q^9+691900*q^10+1294220*q^11+2242617*q^12+3608521*q^13+
5399557*q^14+7518700*q^15+9744361*q^16+11752559*q^17+13189913*q^18+13780169*q^
19+13421710*q^20+12225771*q^21+10471545*q^22+8499539*q^23+6601673*q^24+4959169*
q^25+3640014*q^26+2632946*q^27+1888719*q^28+1348447*q^29+962174*q^30+689459*q^
31+495560*q^32+355286*q^33+252835*q^34+179995*q^35+130394*q^36+95860*q^37+69355
*q^38+49054*q^39+33833*q^40+23447*q^41+17094*q^42+13089*q^43+10163*q^44+7610*q^
45+5009*q^46+2996*q^47+1891*q^48+1565*q^49+1497*q^50+1510*q^51+1312*q^52+918*q^
53+501*q^54+229*q^55+148*q^56+164*q^57+190*q^58+273*q^59+306*q^60+239*q^61+132*
q^62+63*q^63+16*q^64+17*q^65+10*q^66+18*q^67+42*q^68+68*q^69+66*q^70+38*q^71+16
*q^72+6*q^73+2*q^75+3*q^78+10*q^79+16*q^80+14*q^81+3*q^82+2*q^83+2*q^91+4*q^92+
2*q^93+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+6*q^3+5*q^2+q+1, q^10+4*q^9+9*q^8+15*
q^7+20*q^6+18*q^5+12*q^4+5*q^3+2*q^2+1, q^15+5*q^14+14*q^13+29*q^12+49*q^11+65*
q^10+70*q^9+57*q^8+35*q^7+16*q^6+5*q^5+5*q^4+2*q^3+1, q^21+6*q^20+20*q^19+49*q^
18+98*q^17+161*q^16+223*q^15+257*q^14+242*q^13+181*q^12+109*q^11+53*q^10+28*q^9
+17*q^8+3*q^7+4*q^6+4*q^5+2*q^4+1, 1+76*q^25+546*q^22+174*q^24+771*q^21+333*q^
23+27*q^26+2*q^5+4*q^6+2*q^7+2*q^8+4*q^9+14*q^10+22*q^11+28*q^12+51*q^13+113*q^
14+200*q^15+363*q^16+q^28+593*q^17+827*q^18+960*q^19+935*q^20+7*q^27, 1+3495*q^
25+1273*q^22+2850*q^24+755*q^21+2015*q^23+1150*q^30+1877*q^29+35*q^34+111*q^33+
3698*q^26+2*q^6+285*q^32+4*q^7+2*q^8+2*q^10+3*q^11+16*q^12+16*q^13+16*q^14+22*q
^15+49*q^16+2691*q^28+84*q^17+145*q^18+228*q^19+423*q^20+3379*q^27+616*q^31+8*q
^35+q^36, 1+612*q^25+155*q^42+118*q^22+368*q^24+89*q^21+213*q^23+9*q^44+7055*q^
30+q^45+6521*q^37+4605*q^29+2190*q^39+440*q^41+13977*q^34+4009*q^38+14154*q^33+
1021*q^26+12631*q^32+2*q^7+4*q^8+2*q^9+2*q^12+3*q^13+14*q^14+18*q^15+11*q^16+44
*q^43+2846*q^28+9*q^17+16*q^18+44*q^19+82*q^20+1690*q^27+1054*q^40+9972*q^31+
12200*q^35+9450*q^36, 1+66*q^25+55742*q^42+38*q^22+84*q^24+16*q^21+72*q^23+649*
q^51+44023*q^44+54*q^53+627*q^30+209*q^52+q^55+10*q^54+33379*q^45+16997*q^37+
472*q^29+22825*q^46+35485*q^39+53280*q^41+4315*q^34+1701*q^50+25325*q^38+2681*q
^33+91*q^26+1668*q^32+2*q^8+4*q^9+2*q^10+2*q^14+3*q^15+14*q^16+52278*q^43+3875*
q^49+325*q^28+16*q^17+12*q^18+3*q^19+8*q^20+161*q^27+45714*q^40+982*q^31+6890*q
^35+7810*q^48+10926*q^36+14078*q^47, 1+39*q^25+11804*q^42+18*q^24+5*q^21+6*q^23
+217399*q^51+27977*q^44+194185*q^53+49707*q^57+78*q^30+215639*q^52+27930*q^58+
118425*q^55+6479*q^60+158942*q^54+42578*q^45+1392*q^37+40*q^29+63810*q^46+11*q^
65+3250*q^39+2622*q^61+7493*q^41+444*q^34+199151*q^50+65*q^64+2252*q^38+389*q^
33+80396*q^56+66*q^26+257*q^32+2*q^9+923*q^62+4*q^10+2*q^11+2*q^16+q^66+18187*q
^43+166516*q^49+57*q^28+3*q^17+14*q^18+16*q^19+10*q^20+72*q^27+4773*q^40+274*q^
63+146*q^31+526*q^35+128391*q^48+14197*q^59+819*q^36+92713*q^47, 1+1143*q^42+10
*q^22+2*q^24+16*q^21+3*q^23+34060*q^51+2458*q^44+76659*q^53+344873*q^57+68*q^30
+51312*q^52+471497*q^58+167342*q^55+743651*q^60+113671*q^54+3131*q^45+24438*q^
71+1274*q^74+312*q^37+284590*q^67+66*q^29+4476*q^46+574281*q^65+100113*q^69+319
*q^39+837081*q^61+686*q^41+67*q^34+23068*q^50+718127*q^64+347*q^38+20*q^33+
243051*q^56+176318*q^68+6*q^26+33*q^32+867066*q^62+2*q^10+4*q^11+2*q^12+77*q^76
+421775*q^66+51911*q^70+12*q^77+q^78+1752*q^43+15884*q^49+40*q^28+2*q^18+3*q^19
+14*q^20+16*q^27+10353*q^72+404*q^40+3894*q^73+824016*q^63+48*q^31+351*q^75+147
*q^35+10704*q^48+611933*q^59+251*q^36+6888*q^47, 1+3*q^25+288*q^42+14*q^22+13*q
^90+10*q^24+3*q^21+16*q^23+2612*q^51+206*q^44+5972*q^53+24469*q^57+16*q^30+3813
*q^52+35392*q^58+12980*q^55+75592*q^60+q^91+8984*q^54+278*q^45+2784185*q^71+
3402149*q^74+16*q^37+936355*q^67+6*q^29+581*q^46+462964*q^65+1754172*q^69+138*q
^39+108076*q^61+310*q^41+42*q^34+1011614*q^79+2114*q^50+321783*q^64+61*q^38+66*
q^33+17793*q^56+1300369*q^68+68*q^32+154335*q^62+2*q^11+4*q^12+2647921*q^76+2*q
^13+661908*q^66+2270094*q^70+2076921*q^77+1507012*q^78+246*q^43+1783*q^49+2*q^
20+2*q^27+3202555*q^72+245*q^40+3428973*q^73+222474*q^63+38*q^31+3121740*q^75+
26*q^35+627804*q^80+1456*q^48+91562*q^83+359614*q^81+15938*q^85+5607*q^86+1715*
q^87+441*q^88+90*q^89+40242*q^84+189579*q^82+52053*q^59+20*q^36+1035*q^47, 1+16
*q^25+63*q^42+2*q^22+7518700*q^90+14*q^24+3*q^23+154676*q^97+501*q^51+239*q^44+
1312*q^53+1891*q^57+2*q^30+1294220*q^94+918*q^52+2996*q^58+1497*q^55+7610*q^60+
5399557*q^91+1510*q^54+306*q^45+252835*q^71+689459*q^74+341410*q^96+63863*q^98+
42*q^37+69355*q^67+273*q^46+2242617*q^93+23779*q^99+33833*q^65+130394*q^69+10*q
^39+10163*q^61+16*q^41+38*q^34+3640014*q^79+229*q^50+23447*q^64+7865*q^100+18*q
^38+16*q^33+1565*q^56+95860*q^68+10*q^26+6*q^32+691900*q^95+13089*q^62+2260*q^
101+2*q^12+1348447*q^76+4*q^13+2*q^14+49054*q^66+179995*q^70+1888719*q^77+
2632946*q^78+132*q^43+545*q^102+148*q^49+3*q^27+355286*q^72+17*q^40+495560*q^73
+104*q^103+17094*q^63+962174*q^75+66*q^35+14*q^104+4959169*q^80+164*q^48+q^105+
10471545*q^83+6601673*q^81+13421710*q^85+13780169*q^86+13189913*q^87+11752559*q
^88+9744361*q^89+12225771*q^84+8499539*q^82+5009*q^59+68*q^36+3608521*q^92+190*
q^47]


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

[1, 2, 6, 22, 87, 354, 1459, 6056, 25252, 105632, 442916, 1860498, 7826120,

    32956964, 138911074]

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

     is given by

[1, 1, 3, 12, 45, 176, 728, 3035, 12639, 52809, 221442, 930303, 3913182,

    16478450, 69455399]

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

     is given by

[0, 1, 3, 10, 42, 178, 731, 3021, 12613, 52823, 221474, 930195, 3912938,

    16478514, 69455675]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 45, 178, 731, 3035, 12639, 52823, 221474, 930303, 3913182,

    16478514, 69455675]

 ODD:, [0, 1, 3, 10, 42, 176, 728, 3021, 12613, 52809, 221442, 930195, 3912938,

    16478450, 69455399]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.818181818, 4.252873563, 5.728813559,

    7.226182317, 8.740587847, 10.27114684, 11.81747008, 13.37903801,

    14.95516201, 16.54505349, 18.14788753, 19.76284403]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.402477147, 1.782535910, 2.112624216,

    2.407060644, 2.676758656, 2.930665994, 3.175857207, 3.417584804,

    3.659550269, 3.904255270, 4.153320511, 4.407735535]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 1.96694, 3.17743, 4.46318, 5.79394, 7.16504,

    8.58880, 10.0861, 11.6799, 13.3923, 15.2432, 17.2501, 19.4281]

Skewness: , [Float(undefined), 0., 0., 0.1274601866, 0.2493750189, 0.3566436002,

    0.4299758762, 0.4702361485, 0.4878371563, 0.4944264812, 0.4993427267,

    0.5087351939, 0.5259376704, 0.5521901020, 0.5873669939]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.737684812,

    3.337232081, 3.866626201, 4.262231311, 4.497115344, 4.587183608,

    4.575304087, 4.511308542, 4.437884488, 4.385226739, 4.371064633,

    4.403431687]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

    }

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+5*q^3+5*q^4+2*q^5+q^6, 1+4*q+9*q^2+13*q^3
+17*q^4+15*q^5+13*q^6+9*q^7+5*q^8+2*q^9+q^10, 1+5*q+14*q^2+26*q^3+41*q^4+50*q^5
+55*q^6+52*q^7+45*q^8+35*q^9+24*q^10+17*q^11+9*q^12+5*q^13+2*q^14+q^15, 1+6*q+
20*q^2+45*q^3+83*q^4+124*q^5+163*q^6+187*q^7+199*q^8+191*q^9+171*q^10+147*q^11+
117*q^12+90*q^13+62*q^14+43*q^15+28*q^16+17*q^17+9*q^18+5*q^19+2*q^20+q^21, 1+7
*q+27*q^2+71*q^3+150*q^4+261*q^5+396*q^6+528*q^7+647*q^8+721*q^9+755*q^10+746*q
^11+701*q^12+632*q^13+541*q^14+452*q^15+362*q^16+281*q^17+209*q^18+151*q^19+104
*q^20+70*q^21+47*q^22+28*q^23+17*q^24+9*q^25+5*q^26+2*q^27+q^28, 1+8*q+35*q^2+
105*q^3+250*q^4+493*q^5+844*q^6+1274*q^7+1753*q^8+2204*q^9+2592*q^10+2863*q^11+
3012*q^12+3037*q^13+2938*q^14+2759*q^15+2502*q^16+2213*q^17+1901*q^18+1597*q^19
+1304*q^20+1043*q^21+814*q^22+622*q^23+465*q^24+335*q^25+239*q^26+165*q^27+112*
q^28+74*q^29+47*q^30+28*q^31+17*q^32+9*q^33+5*q^34+2*q^35+q^36, 1+9*q+44*q^2+
148*q^3+392*q^4+861*q^5+1637*q^6+2748*q^7+4183*q^8+5826*q^9+7557*q^10+9188*q^11
+10606*q^12+11690*q^13+12377*q^14+12680*q^15+12583*q^16+12175*q^17+11474*q^18+
10594*q^19+9558*q^20+8466*q^21+7352*q^22+6285*q^23+5276*q^24+4352*q^25+3532*q^
26+2818*q^27+2216*q^28+1711*q^29+1304*q^30+972*q^31+716*q^32+515*q^33+365*q^34+
253*q^35+173*q^36+116*q^37+74*q^38+47*q^39+28*q^40+17*q^41+9*q^42+5*q^43+2*q^44
+q^45, 1+10*q+54*q^2+201*q^3+586*q^4+1416*q^5+2955*q^6+5446*q^7+9071*q^8+13821*
q^9+19550*q^10+25879*q^11+32414*q^12+38649*q^13+44188*q^14+48716*q^15+51989*q^
16+53976*q^17+54612*q^18+54089*q^19+52451*q^20+49940*q^21+46721*q^22+43038*q^23
+39040*q^24+34915*q^25+30791*q^26+26804*q^27+23038*q^28+19556*q^29+16405*q^30+
13589*q^31+11132*q^32+9001*q^33+7201*q^34+5683*q^35+4439*q^36+3423*q^37+2608*q^
38+1961*q^39+1454*q^40+1062*q^41+766*q^42+545*q^43+379*q^44+261*q^45+177*q^46+
116*q^47+74*q^48+47*q^49+28*q^50+17*q^51+9*q^52+5*q^53+2*q^54+q^55, 1+11*q+65*q
^2+265*q^3+843*q^4+2220*q^5+5039*q^6+10097*q^7+18244*q^8+30129*q^9+46086*q^10+
65871*q^11+88838*q^12+113788*q^13+139402*q^14+164209*q^15+186882*q^16+206391*q^
17+221849*q^18+232968*q^19+239424*q^20+241446*q^21+239251*q^22+233396*q^23+
224341*q^24+212758*q^25+199193*q^26+184287*q^27+168567*q^28+152508*q^29+136564*
q^30+121049*q^31+106272*q^32+92397*q^33+79618*q^34+67942*q^35+57492*q^36+48192*
q^37+40064*q^38+32990*q^39+26939*q^40+21789*q^41+17475*q^42+13881*q^43+10923*q^
44+8522*q^45+6580*q^46+5033*q^47+3807*q^48+2850*q^49+2107*q^50+1544*q^51+1112*q
^52+796*q^53+559*q^54+387*q^55+265*q^56+177*q^57+116*q^58+74*q^59+47*q^60+28*q^
61+17*q^62+9*q^63+5*q^64+2*q^65+q^66, 1+12*q+77*q^2+341*q^3+1175*q^4+3347*q^5+
8203*q^6+17735*q^7+34518*q^8+61341*q^9+100787*q^10+154524*q^11+223061*q^12+
305210*q^13+398508*q^14+499124*q^15+602665*q^16+704473*q^17+800017*q^18+885700*
q^19+958276*q^20+1015911*q^21+1057194*q^22+1082074*q^23+1090763*q^24+1084426*q^
25+1064382*q^26+1032433*q^27+990418*q^28+940282*q^29+884009*q^30+823415*q^31+
760213*q^32+695919*q^33+631906*q^34+569204*q^35+508848*q^36+451459*q^37+397687*
q^38+347762*q^39+302031*q^40+260428*q^41+223056*q^42+189712*q^43+160267*q^44+
134469*q^45+112065*q^46+92747*q^47+76240*q^48+62230*q^49+50437*q^50+40590*q^51+
32422*q^52+25718*q^53+20236*q^54+15804*q^55+12242*q^56+9412*q^57+7166*q^58+5409
*q^59+4045*q^60+2996*q^61+2197*q^62+1594*q^63+1142*q^64+810*q^65+567*q^66+391*q
^67+265*q^68+177*q^69+116*q^70+74*q^71+47*q^72+28*q^73+17*q^74+9*q^75+5*q^76+2*
q^77+q^78, 1+13*q+90*q^2+430*q^3+1595*q^4+4884*q^5+12847*q^6+29784*q^7+62078*q^
8+118014*q^9+207136*q^10+338820*q^11+520921*q^12+757895*q^13+1050157*q^14+
1393084*q^15+1777908*q^16+2192054*q^17+2620667*q^18+3048227*q^19+3459081*q^20+
3839731*q^21+4177989*q^22+4465369*q^23+4695224*q^24+4864486*q^25+4972070*q^26+
5019497*q^27+5009714*q^28+4947322*q^29+4837799*q^30+4687308*q^31+4502331*q^32+
4289356*q^33+4054931*q^34+3804948*q^35+3545204*q^36+3280666*q^37+3016107*q^38+
2755187*q^39+2501487*q^40+2257358*q^41+2025241*q^42+1806440*q^43+1602266*q^44+
1413197*q^45+1239670*q^46+1081489*q^47+938453*q^48+809951*q^49+695327*q^50+
593737*q^51+504274*q^52+426038*q^53+357965*q^54+299192*q^55+248666*q^56+205584*
q^57+168984*q^58+138147*q^59+112261*q^60+90721*q^61+72854*q^62+58169*q^63+46141
*q^64+36379*q^65+28489*q^66+22161*q^67+17116*q^68+13124*q^69+9990*q^70+7538*q^
71+5647*q^72+4191*q^73+3086*q^74+2247*q^75+1624*q^76+1156*q^77+818*q^78+571*q^
79+391*q^80+265*q^81+177*q^82+116*q^83+74*q^84+47*q^85+28*q^86+17*q^87+9*q^88+5
*q^89+2*q^90+q^91, 1+14*q+104*q^2+533*q^3+2117*q^4+6932*q^5+19471*q^6+48157*q^7
+106956*q^8+216453*q^9+403949*q^10+701764*q^11+1144293*q^12+1763236*q^13+
2583345*q^14+3617513*q^15+4864858*q^16+6308848*q^17+7919020*q^18+9653209*q^19+
11460761*q^20+13287672*q^21+15078360*q^22+16782060*q^23+18352027*q^24+19750814*
q^25+20948434*q^26+21924926*q^27+22668192*q^28+23174825*q^29+23448187*q^30+
23497673*q^31+23337557*q^32+22985400*q^33+22461956*q^34+21788832*q^35+20988827*
q^36+20084433*q^37+19098100*q^38+18050639*q^39+16962214*q^40+15850461*q^41+
14732094*q^42+13621053*q^43+12530265*q^44+11469707*q^45+10448567*q^46+9473227*q
^47+8549367*q^48+7680377*q^49+6868895*q^50+6115876*q^51+5421622*q^52+4785339*q^
53+4205513*q^54+3680177*q^55+3206671*q^56+2782279*q^57+2403771*q^58+2067966*q^
59+1771430*q^60+1510997*q^61+1283253*q^62+1085181*q^63+913626*q^64+765866*q^65+
639105*q^66+530974*q^67+439081*q^68+361451*q^69+296124*q^70+241446*q^71+195900*
q^72+158160*q^73+127033*q^74+101505*q^75+80669*q^76+63756*q^77+50112*q^78+39152
*q^79+30407*q^80+23465*q^81+17990*q^82+13698*q^83+10362*q^84+7776*q^85+5793*q^
86+4281*q^87+3136*q^88+2277*q^89+1638*q^90+1164*q^91+822*q^92+571*q^93+391*q^94
+265*q^95+177*q^96+116*q^97+74*q^98+47*q^99+28*q^100+17*q^101+9*q^102+5*q^103+2
*q^104+q^105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, q^6+3*q^5+5*q^4+5*q^3+5*q^2+2*q+1, q^10+4*q^9+9*q^8+
13*q^7+17*q^6+15*q^5+13*q^4+9*q^3+5*q^2+2*q+1, q^15+5*q^14+14*q^13+26*q^12+41*q
^11+50*q^10+55*q^9+52*q^8+45*q^7+35*q^6+24*q^5+17*q^4+9*q^3+5*q^2+2*q+1, q^21+6
*q^20+20*q^19+45*q^18+83*q^17+124*q^16+163*q^15+187*q^14+199*q^13+191*q^12+171*
q^11+147*q^10+117*q^9+90*q^8+62*q^7+43*q^6+28*q^5+17*q^4+9*q^3+5*q^2+2*q+1, 1+
71*q^25+396*q^22+150*q^24+528*q^21+261*q^23+2*q+5*q^2+27*q^26+9*q^3+17*q^4+28*q
^5+47*q^6+70*q^7+104*q^8+151*q^9+209*q^10+281*q^11+362*q^12+452*q^13+541*q^14+
632*q^15+701*q^16+q^28+746*q^17+755*q^18+721*q^19+647*q^20+7*q^27, 1+2863*q^25+
2938*q^22+3012*q^24+2759*q^21+3037*q^23+2*q+844*q^30+1274*q^29+35*q^34+105*q^33
+5*q^2+2592*q^26+9*q^3+17*q^4+28*q^5+47*q^6+250*q^32+74*q^7+112*q^8+165*q^9+239
*q^10+335*q^11+465*q^12+622*q^13+814*q^14+1043*q^15+1304*q^16+1753*q^28+1597*q^
17+1901*q^18+2213*q^19+2502*q^20+2204*q^27+493*q^31+8*q^35+q^36, 1+9558*q^25+
148*q^42+6285*q^22+8466*q^24+5276*q^21+7352*q^23+2*q+9*q^44+12680*q^30+q^45+
4183*q^37+12583*q^29+1637*q^39+392*q^41+9188*q^34+2748*q^38+10606*q^33+5*q^2+
10594*q^26+9*q^3+17*q^4+28*q^5+47*q^6+11690*q^32+74*q^7+116*q^8+173*q^9+253*q^
10+365*q^11+515*q^12+716*q^13+972*q^14+1304*q^15+1711*q^16+44*q^43+12175*q^28+
2216*q^17+2818*q^18+3532*q^19+4352*q^20+11474*q^27+861*q^40+12377*q^31+7557*q^
35+5826*q^36, 1+16405*q^25+38649*q^42+9001*q^22+13589*q^24+7201*q^21+11132*q^23
+2*q+586*q^51+25879*q^44+54*q^53+34915*q^30+201*q^52+q^55+10*q^54+19550*q^45+
54612*q^37+30791*q^29+13821*q^46+51989*q^39+44188*q^41+49940*q^34+1416*q^50+
53976*q^38+46721*q^33+5*q^2+19556*q^26+9*q^3+17*q^4+28*q^5+47*q^6+43038*q^32+74
*q^7+116*q^8+177*q^9+261*q^10+379*q^11+545*q^12+766*q^13+1062*q^14+1454*q^15+
1961*q^16+32414*q^43+2955*q^49+26804*q^28+2608*q^17+3423*q^18+4439*q^19+5683*q^
20+23038*q^27+48716*q^40+39040*q^31+52451*q^35+5446*q^48+54089*q^36+9071*q^47,
1+21789*q^25+224341*q^42+10923*q^22+17475*q^24+8522*q^21+13881*q^23+2*q+164209*
q^51+239251*q^44+113788*q^53+30129*q^57+57492*q^30+139402*q^52+18244*q^58+65871
*q^55+5039*q^60+88838*q^54+241446*q^45+152508*q^37+48192*q^29+239424*q^46+11*q^
65+184287*q^39+2220*q^61+212758*q^41+106272*q^34+186882*q^50+65*q^64+168567*q^
38+92397*q^33+46086*q^56+5*q^2+26939*q^26+9*q^3+17*q^4+28*q^5+47*q^6+79618*q^32
+74*q^7+116*q^8+177*q^9+843*q^62+265*q^10+387*q^11+559*q^12+796*q^13+1112*q^14+
1544*q^15+2107*q^16+q^66+233396*q^43+206391*q^49+40064*q^28+2850*q^17+3807*q^18
+5033*q^19+6580*q^20+32990*q^27+199193*q^40+265*q^63+67942*q^31+121049*q^35+
221849*q^48+10097*q^59+136564*q^36+232968*q^47, 1+25718*q^25+508848*q^42+12242*
q^22+20236*q^24+9412*q^21+15804*q^23+2*q+1032433*q^51+631906*q^44+1084426*q^53+
1015911*q^57+76240*q^30+1064382*q^52+958276*q^58+1082074*q^55+800017*q^60+
1090763*q^54+695919*q^45+17735*q^71+1175*q^74+260428*q^37+154524*q^67+62230*q^
29+760213*q^46+305210*q^65+61341*q^69+347762*q^39+704473*q^61+451459*q^41+
160267*q^34+990418*q^50+398508*q^64+302031*q^38+134469*q^33+1057194*q^56+5*q^2+
100787*q^68+32422*q^26+9*q^3+17*q^4+28*q^5+47*q^6+112065*q^32+74*q^7+116*q^8+
177*q^9+602665*q^62+265*q^10+391*q^11+567*q^12+77*q^76+810*q^13+1142*q^14+1594*
q^15+2197*q^16+223061*q^66+34518*q^70+12*q^77+q^78+569204*q^43+940282*q^49+
50437*q^28+2996*q^17+4045*q^18+5409*q^19+7166*q^20+40590*q^27+8203*q^72+397687*
q^40+3347*q^73+499124*q^63+92747*q^31+341*q^75+189712*q^35+884009*q^48+885700*q
^59+223056*q^36+823415*q^47, 1+28489*q^25+809951*q^42+13124*q^22+13*q^90+22161*
q^24+9990*q^21+17116*q^23+2*q+2501487*q^51+1081489*q^44+3016107*q^53+4054931*q^
57+90721*q^30+2755187*q^52+4289356*q^58+3545204*q^55+4687308*q^60+q^91+3280666*
q^54+1239670*q^45+3459081*q^71+2192054*q^74+357965*q^37+4695224*q^67+72854*q^29
+1413197*q^46+4972070*q^65+4177989*q^69+504274*q^39+4837799*q^61+695327*q^41+
205584*q^34+520921*q^79+2257358*q^50+5019497*q^64+426038*q^38+168984*q^33+
3804948*q^56+5*q^2+4465369*q^68+36379*q^26+9*q^3+17*q^4+28*q^5+47*q^6+138147*q^
32+74*q^7+116*q^8+177*q^9+4947322*q^62+265*q^10+391*q^11+571*q^12+1393084*q^76+
818*q^13+1156*q^14+1624*q^15+2247*q^16+4864486*q^66+3839731*q^70+1050157*q^77+
757895*q^78+938453*q^43+2025241*q^49+58169*q^28+3086*q^17+4191*q^18+5647*q^19+
7538*q^20+46141*q^27+3048227*q^72+593737*q^40+2620667*q^73+5009714*q^63+112261*
q^31+1777908*q^75+248666*q^35+338820*q^80+1806440*q^48+62078*q^83+207136*q^81+
12847*q^85+4884*q^86+1595*q^87+430*q^88+90*q^89+29784*q^84+118014*q^82+4502331*
q^59+299192*q^36+1602266*q^47, 1+30407*q^25+1085181*q^42+13698*q^22+3617513*q^
90+23465*q^24+10362*q^21+17990*q^23+2*q+106956*q^97+4205513*q^51+1510997*q^44+
5421622*q^53+8549367*q^57+101505*q^30+701764*q^94+4785339*q^52+9473227*q^58+
6868895*q^55+11469707*q^60+2583345*q^91+6115876*q^54+1771430*q^45+22461956*q^71
+23497673*q^74+216453*q^96+48157*q^98+439081*q^37+19098100*q^67+80669*q^29+
2067966*q^46+1144293*q^93+19471*q^99+16962214*q^65+20988827*q^69+639105*q^39+
12530265*q^61+913626*q^41+241446*q^34+20948434*q^79+3680177*q^50+15850461*q^64+
6932*q^100+530974*q^38+195900*q^33+7680377*q^56+5*q^2+20084433*q^68+39152*q^26+
9*q^3+17*q^4+28*q^5+47*q^6+158160*q^32+74*q^7+116*q^8+177*q^9+403949*q^95+
13621053*q^62+2117*q^101+265*q^10+391*q^11+571*q^12+23174825*q^76+822*q^13+1164
*q^14+1638*q^15+2277*q^16+18050639*q^66+21788832*q^70+22668192*q^77+21924926*q^
78+1283253*q^43+533*q^102+3206671*q^49+63756*q^28+3136*q^17+4281*q^18+5793*q^19
+7776*q^20+50112*q^27+22985400*q^72+765866*q^40+23337557*q^73+104*q^103+
14732094*q^63+127033*q^31+23448187*q^75+296124*q^35+14*q^104+19750814*q^80+
2782279*q^48+q^105+15078360*q^83+18352027*q^81+11460761*q^85+9653209*q^86+
7919020*q^87+6308848*q^88+4864858*q^89+13287672*q^84+16782060*q^82+10448567*q^
59+361451*q^36+1763236*q^92+2403771*q^47]


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

[1, 2, 6, 22, 89, 382, 1711, 7922, 37663, 182936, 904302, 4535994, 23034564,

    118209806, 612165222]

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

     is given by

[1, 1, 3, 12, 46, 191, 855, 3963, 18832, 91459, 452135, 2267988, 11517272,

    59104890, 306082656]

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

     is given by

[0, 1, 3, 10, 43, 191, 856, 3959, 18831, 91477, 452167, 2268006, 11517292,

    59104916, 306082566]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 46, 191, 856, 3963, 18832, 91477, 452167, 2267988,

    11517272, 59104916, 306082566]

 ODD:, [0, 1, 3, 10, 43, 191, 855, 3959, 18831, 91459, 452135, 2268006,

    11517292, 59104890, 306082656]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 2.909090909, 4.640449438, 6.641361257,

    8.883693746, 11.35155264, 14.03358734, 16.91976976, 20.00062037,

    23.26726314, 26.71157240, 30.32621938, 34.10462854]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.474297704, 2.062218676, 2.712134111,

    3.413783648, 4.162138131, 4.955518660, 5.793079199, 6.673707166,

    7.595909189, 8.558014931, 9.558371031, 10.59544686]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.17355, 4.25275, 7.35567, 11.6539, 17.3234,

    24.5572, 33.5598, 44.5384, 57.6978, 73.2396, 91.3625, 112.263]

Skewness: , [Float(undefined), 0., 0., 0.07174070299, 0.1798724243,

    0.2763268748, 0.3486841043, 0.3999714487, 0.4364250457, 0.4632721245,

    0.4839717648, 0.5005942600, 0.5143466091, 0.5259694953, 0.5359436040]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.419438345,

    2.614774250, 2.768127915, 2.893396696, 2.989145082, 3.059450168,

    3.111412509, 3.151241343, 3.183203667, 3.209838701, 3.232605251,

    3.252420951]

                                end of this data



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

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

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

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

                                  here it is:

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

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

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

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

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

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

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

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

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

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+4*q^3+5*q^4+3*q^5+q^6, 1+4*q+9*q^2+11*q^3
+12*q^4+12*q^5+12*q^6+11*q^7+9*q^8+4*q^9+q^10, 1+5*q+14*q^2+23*q^3+29*q^4+30*q^
5+34*q^6+35*q^7+35*q^8+34*q^9+30*q^10+29*q^11+23*q^12+14*q^13+5*q^14+q^15, 1+6*
q+20*q^2+41*q^3+62*q^4+73*q^5+82*q^6+89*q^7+99*q^8+106*q^9+104*q^10+104*q^11+
106*q^12+99*q^13+89*q^14+82*q^15+73*q^16+62*q^17+41*q^18+20*q^19+6*q^20+q^21, 1
+7*q+27*q^2+66*q^3+118*q^4+162*q^5+194*q^6+215*q^7+242*q^8+269*q^9+297*q^10+314
*q^11+324*q^12+330*q^13+330*q^14+330*q^15+324*q^16+314*q^17+297*q^18+269*q^19+
242*q^20+215*q^21+194*q^22+162*q^23+118*q^24+66*q^25+27*q^26+7*q^27+q^28, 1+8*q
+35*q^2+99*q^3+205*q^4+326*q^5+433*q^6+509*q^7+578*q^8+646*q^9+730*q^10+815*q^
11+894*q^12+959*q^13+993*q^14+1017*q^15+1044*q^16+1080*q^17+1102*q^18+1080*q^19
+1044*q^20+1017*q^21+993*q^22+959*q^23+894*q^24+815*q^25+730*q^26+646*q^27+578*
q^28+509*q^29+433*q^30+326*q^31+205*q^32+99*q^33+35*q^34+8*q^35+q^36, 1+9*q+44*
q^2+141*q^3+332*q^4+603*q^5+898*q^6+1150*q^7+1358*q^8+1539*q^9+1741*q^10+1962*q
^11+2217*q^12+2479*q^13+2717*q^14+2895*q^15+3027*q^16+3157*q^17+3301*q^18+3435*
q^19+3521*q^20+3569*q^21+3595*q^22+3595*q^23+3569*q^24+3521*q^25+3435*q^26+3301
*q^27+3157*q^28+3027*q^29+2895*q^30+2717*q^31+2479*q^32+2217*q^33+1962*q^34+
1741*q^35+1539*q^36+1358*q^37+1150*q^38+898*q^39+603*q^40+332*q^41+141*q^42+44*
q^43+9*q^44+q^45, 1+10*q+54*q^2+193*q^3+509*q^4+1041*q^5+1734*q^6+2446*q^7+3080
*q^8+3614*q^9+4129*q^10+4666*q^11+5289*q^12+5990*q^13+6764*q^14+7526*q^15+8189*
q^16+8759*q^17+9269*q^18+9772*q^19+10236*q^20+10676*q^21+11128*q^22+11556*q^23+
11885*q^24+12055*q^25+12101*q^26+12091*q^27+12091*q^28+12101*q^29+12055*q^30+
11885*q^31+11556*q^32+11128*q^33+10676*q^34+10236*q^35+9772*q^36+9269*q^37+8759
*q^38+8189*q^39+7526*q^40+6764*q^41+5990*q^42+5289*q^43+4666*q^44+4129*q^45+
3614*q^46+3080*q^47+2446*q^48+1734*q^49+1041*q^50+509*q^51+193*q^52+54*q^53+10*
q^54+q^55, 1+11*q+65*q^2+256*q^3+747*q^4+1699*q^5+3143*q^6+4889*q^7+6657*q^8+
8242*q^9+9673*q^10+11051*q^11+12547*q^12+14221*q^13+16160*q^14+18304*q^15+20544
*q^16+22714*q^17+24725*q^18+26571*q^19+28264*q^20+29842*q^21+31417*q^22+33098*q
^23+34869*q^24+36568*q^25+37966*q^26+38975*q^27+39691*q^28+40330*q^29+41008*q^
30+41634*q^31+42061*q^32+42216*q^33+42061*q^34+41634*q^35+41008*q^36+40330*q^37
+39691*q^38+38975*q^39+37966*q^40+36568*q^41+34869*q^42+33098*q^43+31417*q^44+
29842*q^45+28264*q^46+26571*q^47+24725*q^48+22714*q^49+20544*q^50+18304*q^51+
16160*q^52+14221*q^53+12547*q^54+11051*q^55+9673*q^56+8242*q^57+6657*q^58+4889*
q^59+3143*q^60+1699*q^61+747*q^62+256*q^63+65*q^64+11*q^65+q^66, 1+12*q+77*q^2+
331*q^3+1058*q^4+2648*q^5+5396*q^6+9222*q^7+13648*q^8+18054*q^9+22126*q^10+
25886*q^11+29657*q^12+33692*q^13+38281*q^14+43494*q^15+49353*q^16+55647*q^17+
62137*q^18+68559*q^19+74630*q^20+80244*q^21+85476*q^22+90711*q^23+96214*q^24+
102022*q^25+107907*q^26+113555*q^27+118722*q^28+123326*q^29+127497*q^30+131321*
q^31+134735*q^32+137632*q^33+140078*q^34+142285*q^35+144277*q^36+145861*q^37+
146844*q^38+147174*q^39+146844*q^40+145861*q^41+144277*q^42+142285*q^43+140078*
q^44+137632*q^45+134735*q^46+131321*q^47+127497*q^48+123326*q^49+118722*q^50+
113555*q^51+107907*q^52+102022*q^53+96214*q^54+90711*q^55+85476*q^56+80244*q^57
+74630*q^58+68559*q^59+62137*q^60+55647*q^61+49353*q^62+43494*q^63+38281*q^64+
33692*q^65+29657*q^66+25886*q^67+22126*q^68+18054*q^69+13648*q^70+9222*q^71+
5396*q^72+2648*q^73+1058*q^74+331*q^75+77*q^76+12*q^77+q^78, 1+13*q+90*q^2+419*
q^3+1455*q^4+3972*q^5+8846*q^6+16519*q^7+26567*q^8+37781*q^9+48939*q^10+59409*q
^11+69427*q^12+79562*q^13+90581*q^14+102941*q^15+117026*q^16+132784*q^17+150098
*q^18+168584*q^19+187638*q^20+206411*q^21+224225*q^22+241154*q^23+257909*q^24+
275180*q^25+293169*q^26+311596*q^27+330135*q^28+348630*q^29+366993*q^30+385028*
q^31+402102*q^32+417533*q^33+431051*q^34+443134*q^35+454615*q^36+466072*q^37+
477533*q^38+488621*q^39+498477*q^40+506160*q^41+511187*q^42+513968*q^43+515345*
q^44+515931*q^45+515931*q^46+515345*q^47+513968*q^48+511187*q^49+506160*q^50+
498477*q^51+488621*q^52+477533*q^53+466072*q^54+454615*q^55+443134*q^56+431051*
q^57+417533*q^58+402102*q^59+385028*q^60+366993*q^61+348630*q^62+330135*q^63+
311596*q^64+293169*q^65+275180*q^66+257909*q^67+241154*q^68+224225*q^69+206411*
q^70+187638*q^71+168584*q^72+150098*q^73+132784*q^74+117026*q^75+102941*q^76+
90581*q^77+79562*q^78+69427*q^79+59409*q^80+48939*q^81+37781*q^82+26567*q^83+
16519*q^84+8846*q^85+3972*q^86+1455*q^87+419*q^88+90*q^89+13*q^90+q^91, 1+14*q+
104*q^2+521*q^3+1952*q^4+5769*q^5+13942*q^6+28279*q^7+49284*q^8+75471*q^9+
104065*q^10+132481*q^11+159721*q^12+186272*q^13+213684*q^14+243429*q^15+276879*
q^16+314612*q^17+357002*q^18+403928*q^19+455075*q^20+509237*q^21+564380*q^22+
618655*q^23+671442*q^24+723721*q^25+776695*q^26+831100*q^27+886885*q^28+944032*
q^29+1002761*q^30+1063415*q^31+1125603*q^32+1187642*q^33+1247106*q^34+1302197*q
^35+1352923*q^36+1400922*q^37+1448087*q^38+1495527*q^39+1542946*q^40+1588939*q^
41+1631764*q^42+1670371*q^43+1704769*q^44+1735368*q^45+1762200*q^46+1784989*q^
47+1803907*q^48+1819573*q^49+1832227*q^50+1841321*q^51+1846054*q^52+1846054*q^
53+1841321*q^54+1832227*q^55+1819573*q^56+1803907*q^57+1784989*q^58+1762200*q^
59+1735368*q^60+1704769*q^61+1670371*q^62+1631764*q^63+1588939*q^64+1542946*q^
65+1495527*q^66+1448087*q^67+1400922*q^68+1352923*q^69+1302197*q^70+1247106*q^
71+1187642*q^72+1125603*q^73+1063415*q^74+1002761*q^75+944032*q^76+886885*q^77+
831100*q^78+776695*q^79+723721*q^80+671442*q^81+618655*q^82+564380*q^83+509237*
q^84+455075*q^85+403928*q^86+357002*q^87+314612*q^88+276879*q^89+243429*q^90+
213684*q^91+186272*q^92+159721*q^93+132481*q^94+104065*q^95+75471*q^96+49284*q^
97+28279*q^98+13942*q^99+5769*q^100+1952*q^101+521*q^102+104*q^103+14*q^104+q^
105]
     with the reverse patterns and complement patterns having distributions

[1, 1+q, 1+2*q+2*q^2+q^3, 1+3*q+5*q^2+4*q^3+5*q^4+3*q^5+q^6, 1+4*q+9*q^2+11*q^3
+12*q^4+12*q^5+12*q^6+11*q^7+9*q^8+4*q^9+q^10, 1+5*q+14*q^2+23*q^3+29*q^4+30*q^
5+34*q^6+35*q^7+35*q^8+34*q^9+30*q^10+29*q^11+23*q^12+14*q^13+5*q^14+q^15, 1+6*
q+20*q^2+41*q^3+62*q^4+73*q^5+82*q^6+89*q^7+99*q^8+106*q^9+104*q^10+104*q^11+
106*q^12+99*q^13+89*q^14+82*q^15+73*q^16+62*q^17+41*q^18+20*q^19+6*q^20+q^21, 1
+7*q+27*q^2+66*q^3+118*q^4+162*q^5+194*q^6+215*q^7+242*q^8+269*q^9+297*q^10+314
*q^11+324*q^12+330*q^13+330*q^14+330*q^15+324*q^16+314*q^17+297*q^18+269*q^19+
242*q^20+215*q^21+194*q^22+162*q^23+118*q^24+66*q^25+27*q^26+7*q^27+q^28, 1+8*q
+35*q^2+99*q^3+205*q^4+326*q^5+433*q^6+509*q^7+578*q^8+646*q^9+730*q^10+815*q^
11+894*q^12+959*q^13+993*q^14+1017*q^15+1044*q^16+1080*q^17+1102*q^18+1080*q^19
+1044*q^20+1017*q^21+993*q^22+959*q^23+894*q^24+815*q^25+730*q^26+646*q^27+578*
q^28+509*q^29+433*q^30+326*q^31+205*q^32+99*q^33+35*q^34+8*q^35+q^36, 1+9*q+44*
q^2+141*q^3+332*q^4+603*q^5+898*q^6+1150*q^7+1358*q^8+1539*q^9+1741*q^10+1962*q
^11+2217*q^12+2479*q^13+2717*q^14+2895*q^15+3027*q^16+3157*q^17+3301*q^18+3435*
q^19+3521*q^20+3569*q^21+3595*q^22+3595*q^23+3569*q^24+3521*q^25+3435*q^26+3301
*q^27+3157*q^28+3027*q^29+2895*q^30+2717*q^31+2479*q^32+2217*q^33+1962*q^34+
1741*q^35+1539*q^36+1358*q^37+1150*q^38+898*q^39+603*q^40+332*q^41+141*q^42+44*
q^43+9*q^44+q^45, 1+10*q+54*q^2+193*q^3+509*q^4+1041*q^5+1734*q^6+2446*q^7+3080
*q^8+3614*q^9+4129*q^10+4666*q^11+5289*q^12+5990*q^13+6764*q^14+7526*q^15+8189*
q^16+8759*q^17+9269*q^18+9772*q^19+10236*q^20+10676*q^21+11128*q^22+11556*q^23+
11885*q^24+12055*q^25+12101*q^26+12091*q^27+12091*q^28+12101*q^29+12055*q^30+
11885*q^31+11556*q^32+11128*q^33+10676*q^34+10236*q^35+9772*q^36+9269*q^37+8759
*q^38+8189*q^39+7526*q^40+6764*q^41+5990*q^42+5289*q^43+4666*q^44+4129*q^45+
3614*q^46+3080*q^47+2446*q^48+1734*q^49+1041*q^50+509*q^51+193*q^52+54*q^53+10*
q^54+q^55, 1+11*q+65*q^2+256*q^3+747*q^4+1699*q^5+3143*q^6+4889*q^7+6657*q^8+
8242*q^9+9673*q^10+11051*q^11+12547*q^12+14221*q^13+16160*q^14+18304*q^15+20544
*q^16+22714*q^17+24725*q^18+26571*q^19+28264*q^20+29842*q^21+31417*q^22+33098*q
^23+34869*q^24+36568*q^25+37966*q^26+38975*q^27+39691*q^28+40330*q^29+41008*q^
30+41634*q^31+42061*q^32+42216*q^33+42061*q^34+41634*q^35+41008*q^36+40330*q^37
+39691*q^38+38975*q^39+37966*q^40+36568*q^41+34869*q^42+33098*q^43+31417*q^44+
29842*q^45+28264*q^46+26571*q^47+24725*q^48+22714*q^49+20544*q^50+18304*q^51+
16160*q^52+14221*q^53+12547*q^54+11051*q^55+9673*q^56+8242*q^57+6657*q^58+4889*
q^59+3143*q^60+1699*q^61+747*q^62+256*q^63+65*q^64+11*q^65+q^66, 1+12*q+77*q^2+
331*q^3+1058*q^4+2648*q^5+5396*q^6+9222*q^7+13648*q^8+18054*q^9+22126*q^10+
25886*q^11+29657*q^12+33692*q^13+38281*q^14+43494*q^15+49353*q^16+55647*q^17+
62137*q^18+68559*q^19+74630*q^20+80244*q^21+85476*q^22+90711*q^23+96214*q^24+
102022*q^25+107907*q^26+113555*q^27+118722*q^28+123326*q^29+127497*q^30+131321*
q^31+134735*q^32+137632*q^33+140078*q^34+142285*q^35+144277*q^36+145861*q^37+
146844*q^38+147174*q^39+146844*q^40+145861*q^41+144277*q^42+142285*q^43+140078*
q^44+137632*q^45+134735*q^46+131321*q^47+127497*q^48+123326*q^49+118722*q^50+
113555*q^51+107907*q^52+102022*q^53+96214*q^54+90711*q^55+85476*q^56+80244*q^57
+74630*q^58+68559*q^59+62137*q^60+55647*q^61+49353*q^62+43494*q^63+38281*q^64+
33692*q^65+29657*q^66+25886*q^67+22126*q^68+18054*q^69+13648*q^70+9222*q^71+
5396*q^72+2648*q^73+1058*q^74+331*q^75+77*q^76+12*q^77+q^78, 1+13*q+90*q^2+419*
q^3+1455*q^4+3972*q^5+8846*q^6+16519*q^7+26567*q^8+37781*q^9+48939*q^10+59409*q
^11+69427*q^12+79562*q^13+90581*q^14+102941*q^15+117026*q^16+132784*q^17+150098
*q^18+168584*q^19+187638*q^20+206411*q^21+224225*q^22+241154*q^23+257909*q^24+
275180*q^25+293169*q^26+311596*q^27+330135*q^28+348630*q^29+366993*q^30+385028*
q^31+402102*q^32+417533*q^33+431051*q^34+443134*q^35+454615*q^36+466072*q^37+
477533*q^38+488621*q^39+498477*q^40+506160*q^41+511187*q^42+513968*q^43+515345*
q^44+515931*q^45+515931*q^46+515345*q^47+513968*q^48+511187*q^49+506160*q^50+
498477*q^51+488621*q^52+477533*q^53+466072*q^54+454615*q^55+443134*q^56+431051*
q^57+417533*q^58+402102*q^59+385028*q^60+366993*q^61+348630*q^62+330135*q^63+
311596*q^64+293169*q^65+275180*q^66+257909*q^67+241154*q^68+224225*q^69+206411*
q^70+187638*q^71+168584*q^72+150098*q^73+132784*q^74+117026*q^75+102941*q^76+
90581*q^77+79562*q^78+69427*q^79+59409*q^80+48939*q^81+37781*q^82+26567*q^83+
16519*q^84+8846*q^85+3972*q^86+1455*q^87+419*q^88+90*q^89+13*q^90+q^91, 1+14*q+
104*q^2+521*q^3+1952*q^4+5769*q^5+13942*q^6+28279*q^7+49284*q^8+75471*q^9+
104065*q^10+132481*q^11+159721*q^12+186272*q^13+213684*q^14+243429*q^15+276879*
q^16+314612*q^17+357002*q^18+403928*q^19+455075*q^20+509237*q^21+564380*q^22+
618655*q^23+671442*q^24+723721*q^25+776695*q^26+831100*q^27+886885*q^28+944032*
q^29+1002761*q^30+1063415*q^31+1125603*q^32+1187642*q^33+1247106*q^34+1302197*q
^35+1352923*q^36+1400922*q^37+1448087*q^38+1495527*q^39+1542946*q^40+1588939*q^
41+1631764*q^42+1670371*q^43+1704769*q^44+1735368*q^45+1762200*q^46+1784989*q^
47+1803907*q^48+1819573*q^49+1832227*q^50+1841321*q^51+1846054*q^52+1846054*q^
53+1841321*q^54+1832227*q^55+1819573*q^56+1803907*q^57+1784989*q^58+1762200*q^
59+1735368*q^60+1704769*q^61+1670371*q^62+1631764*q^63+1588939*q^64+1542946*q^
65+1495527*q^66+1448087*q^67+1400922*q^68+1352923*q^69+1302197*q^70+1247106*q^
71+1187642*q^72+1125603*q^73+1063415*q^74+1002761*q^75+944032*q^76+886885*q^77+
831100*q^78+776695*q^79+723721*q^80+671442*q^81+618655*q^82+564380*q^83+509237*
q^84+455075*q^85+403928*q^86+357002*q^87+314612*q^88+276879*q^89+243429*q^90+
213684*q^91+186272*q^92+159721*q^93+132481*q^94+104065*q^95+75471*q^96+49284*q^
97+28279*q^98+13942*q^99+5769*q^100+1952*q^101+521*q^102+104*q^103+14*q^104+q^
105]


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

[1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406,

    22369622, 89478486]

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

     is given by

[1, 1, 3, 12, 44, 171, 683, 2736, 10928, 43691, 174763, 699076, 2796228,

    11184811, 44739243]

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

     is given by

[0, 1, 3, 10, 42, 171, 683, 2726, 10918, 43691, 174763, 699026, 2796178,

    11184811, 44739243]

            For the reverse patterns and complement patterns, we get

EVEN:, [1, 1, 3, 12, 44, 171, 683, 2736, 10928, 43691, 174763, 699076, 2796228,

    11184811, 44739243]

 ODD:, [0, 1, 3, 10, 42, 171, 683, 2726, 10918, 43691, 174763, 699026, 2796178,

    11184811, 44739243]



            The average number of inversions for each n is given by

[0., 0.5000000000, 1.500000000, 3.000000000, 5.000000000, 7.500000000,

    10.50000000, 14.00000000, 18.00000000, 22.50000000, 27.50000000,

    33.00000000, 39.00000000, 45.50000000, 52.50000000]

                 The standard deviation for each n is given by

[0., 0.5000000000, 0.9574271080, 1.537412230, 2.292556205, 3.230655430,

    4.340005938, 5.604014621, 7.007163387, 8.536569080, 10.18189114,

    11.93482357, 13.78859700, 15.73759115, 17.77705406]

                          The centralized moments are

Second: , [0., 0.250000, 0.916667, 2.36364, 5.25581, 10.4371, 18.8357, 31.4050,

    49.1003, 72.8730, 103.671, 142.440, 190.125, 247.672, 316.024]

                                                      -5
Skewness: , [Float(undefined), 0., 0., 0.2751870570 10  , 0.,

                    -5                 -5                 -6
    -0.2965720597 10  , 0.1223284162 10  , 0.5682007509 10  , 0., 0., 0., 0.,

                    -6                 -6                 -6
    -0.3814531097 10  , 0.2565574518 10  , 0.1779999588 10  ]

Kurtosis: , [Float(undefined), 1.000000000, 2.057889412, 2.180463409,

    2.160269925, 2.148952832, 2.162529530, 2.193041635, 2.231886482,

    2.273450488, 2.314611963, 2.353784734, 2.390274920, 2.423876086,

    2.454662376]

                                end of this data



                         Out of a total of , 56, cases

                     29, were successful and , 27, failed

                             Success Rate: , 0.518

                             Here are the failures

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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