Pair of 3-patterns and a 4-pattern There all together, 23, different equivalence classes For the equivalence class of patterns, {{[1, 2, 3, 4], [3, 2, 1], [3, 1, 2]}, {[1, 2, 3, 4], [3, 2, 1], [2, 3, 1]}, {[4, 3, 2, 1], [1, 2, 3], [2, 1, 3]}, {[4, 3, 2, 1], [1, 2, 3], [1, 3, 2]}} the member , {[1, 2, 3, 4], [3, 2, 1], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[2, 0]}, {}], [[2, 1], {[1, 0, 0], [0, 1, 0]}, {1}], [[1, 2], {[2, 0, 0], [1, 1, 0], [0, 2, 0], [0, 0, 3]}, {}], [[2, 3, 1], {[0, 0, 0, 3], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 2, 3], {[2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0], [1, 0, 1, 0], [0, 1, 1, 0], [0, 0, 2, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 0, 0, 3], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {2}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+q^2, 3*q+3*q^2+q^3, 3*q^2+2*q^3, q^3, 0, 0, 0, 0, 0, 0, 0, 0, 0] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+2*q+q^2), q^3*(3*q^2+3*q+1), q^7*(3*q+2), q^12, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of permutations avoiding, {[1, 2, 3, 4], [3, 2, 1], [3, 1, 2]}, is given by [1, 2, 4, 7, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [3, 2, 1], [3, 1, 2]}, is given by [1, 1, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of ODD permutations avoiding, {[1, 2, 3, 4], [3, 2, 1], [3, 1, 2]}, is given by [0, 1, 2, 4, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 3, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] ODD:, [0, 1, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.714285714, 2.400000000, 3., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.6998542123, 0.4898979486, 0., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.489796, 0.240000, 0., FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] Skewness: , [Float(undefined), 0., 0., 0.4592955420, 0.4082482904, FAIL[3] FAIL[3] FAIL[3] FAIL[3] Float(undefined), ----------, ----------, ----------, ----------, 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] ----------, ----------, ----------, ----------, ----------] 3/2 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.104250705, FAIL[4] FAIL[4] FAIL[4] FAIL[4] 1.166666667, Float(undefined), --------, --------, --------, --------, 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------, --------, --------] 2 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] end of this data For the equivalence class of patterns, {{[1, 2, 4, 3], [3, 2, 1], [3, 1, 2]}, {[2, 1, 3, 4], [3, 2, 1], [3, 1, 2]}, {[1, 2, 4, 3], [3, 2, 1], [2, 3, 1]}, {[3, 4, 2, 1], [1, 2, 3], [1, 3, 2]}, {[3, 4, 2, 1], [1, 2, 3], [2, 1, 3]}, {[4, 3, 1, 2], [1, 2, 3], [2, 1, 3]}, {[2, 1, 3, 4], [3, 2, 1], [2, 3, 1]}, {[4, 3, 1, 2], [1, 2, 3], [1, 3, 2]}} the member , {[1, 2, 4, 3], [3, 2, 1], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[2, 0]}, {}], [[2, 1], {[1, 0, 0], [0, 1, 0]}, {1}], [[1, 2], {[2, 0, 0], [1, 1, 0], [0, 2, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {2}], [[1, 2, 3], {[2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+q^2, 1+2*q+3*q^2+q^3, 1+2*q+3*q^2+3*q^3+q^4, 1+2*q+3*q^2+3*q^3+3 *q^4+q^5, 1+2*q+3*q^2+3*q^3+3*q^4+3*q^5+q^6, 1+2*q+3*q^2+3*q^3+3*q^4+3*q^5+3*q^ 6+q^7, 1+2*q+3*q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+q^8, 1+2*q+3*q^2+3*q^3+3*q^4+3 *q^5+3*q^6+3*q^7+3*q^8+q^9, 1+2*q+3*q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+3*q^8+3*q ^9+q^10, 1+2*q+3*q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+3*q^8+3*q^9+3*q^10+q^11, 1+2 *q+3*q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+3*q^8+3*q^9+3*q^10+3*q^11+q^12, 1+2*q+3* q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+3*q^8+3*q^9+3*q^10+3*q^11+3*q^12+q^13, 1+2*q+ 3*q^2+3*q^3+3*q^4+3*q^5+3*q^6+3*q^7+3*q^8+3*q^9+3*q^10+3*q^11+3*q^12+3*q^13+q^ 14] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+2*q+q^2), q^3*(q^3+2*q^2+3*q+1), q^6*(q^4+2*q^3+3*q^2+3*q+1), q^ 10*(1+3*q+3*q^2+3*q^3+2*q^4+q^5), q^15*(q^6+2*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^ 21*(q^7+2*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^28*(q^8+2*q^7+3*q^6+3*q^5+3*q^4 +3*q^3+3*q^2+3*q+1), q^36*(q^9+2*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1) , q^45*(q^10+2*q^9+3*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^55*(q^11 +2*q^10+3*q^9+3*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^66*(q^12+2*q^ 11+3*q^10+3*q^9+3*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^78*(q^13+2* q^12+3*q^11+3*q^10+3*q^9+3*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3*q^2+3*q+1), q^91 *(q^14+2*q^13+3*q^12+3*q^11+3*q^10+3*q^9+3*q^8+3*q^7+3*q^6+3*q^5+3*q^4+3*q^3+3* q^2+3*q+1)] The number of permutations avoiding, {[1, 2, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40] The number of EVEN permutations avoiding, {[1, 2, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [1, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20] The number of ODD permutations avoiding, {[1, 2, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [0, 1, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 4, 5, 6, 8, 10, 11, 12, 14, 16, 17, 18, 20] ODD:, [0, 1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.571428571, 2.100000000, 2.615384615, 3.125000000, 3.631578947, 4.136363636, 4.640000000, 5.142857143, 5.645161290, 6.147058824, 6.648648649, 7.150000000] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.9035079028, 1.135781669, 1.388882315, 1.653594569, 1.925193806, 2.201145457, 2.480000000, 2.760878299, 3.043219720, 3.326651318, 3.610916003, 3.895831105] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.816327, 1.29000, 1.92899, 2.73438, 3.70637, 4.84504, 6.15040, 7.62245, 9.26119, 11.0666, 13.0387, 15.1775] Skewness: , [Float(undefined), 0., 0., -0.2134475891, -0.1965657838, -0.1529074327, -0.1166256765, -0.09011307605, -0.07101039137, -0.05709699448, -0.04675295703, -0.03890881912, -0.03284285493, -0.02806402738, -0.02424023013] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.267426761, 2.246078962, 2.155413809, 2.074032944, 2.012567562, 1.967712709, 1.934728821, 1.910183625, 1.891543436, 1.877134678, 1.865797869, 1.856726372] end of this data For the equivalence class of patterns, {{[2, 1, 4, 3], [3, 2, 1], [3, 1, 2]}, {[3, 4, 1, 2], [1, 2, 3], [1, 3, 2]}, {[3, 4, 1, 2], [1, 2, 3], [2, 1, 3]}, {[2, 1, 4, 3], [3, 2, 1], [2, 3, 1]}} the member , {[2, 1, 4, 3], [3, 2, 1], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[2, 0]}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0], [0, 1, 0]}, {}], [[1, 2], {[2, 0, 0], [1, 1, 0], [0, 2, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+q^2, 1+3*q+2*q^2+q^3, 1+4*q+3*q^2+2*q^3+q^4, 1+5*q+4*q^2+3*q^3+2 *q^4+q^5, 1+6*q+5*q^2+4*q^3+3*q^4+2*q^5+q^6, 1+7*q+6*q^2+5*q^3+4*q^4+3*q^5+2*q^ 6+q^7, 1+8*q+7*q^2+6*q^3+5*q^4+4*q^5+3*q^6+2*q^7+q^8, 1+9*q+8*q^2+7*q^3+6*q^4+5 *q^5+4*q^6+3*q^7+2*q^8+q^9, 1+10*q+9*q^2+8*q^3+7*q^4+6*q^5+5*q^6+4*q^7+3*q^8+2* q^9+q^10, 1+11*q+10*q^2+9*q^3+8*q^4+7*q^5+6*q^6+5*q^7+4*q^8+3*q^9+2*q^10+q^11, 1+12*q+11*q^2+10*q^3+9*q^4+8*q^5+7*q^6+6*q^7+5*q^8+4*q^9+3*q^10+2*q^11+q^12, 1+ 13*q+12*q^2+11*q^3+10*q^4+9*q^5+8*q^6+7*q^7+6*q^8+5*q^9+4*q^10+3*q^11+2*q^12+q^ 13, 1+14*q+13*q^2+12*q^3+11*q^4+10*q^5+9*q^6+8*q^7+7*q^8+6*q^9+5*q^10+4*q^11+3* q^12+2*q^13+q^14] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+2*q+q^2), q^3*(1+2*q+3*q^2+q^3), q^6*(q^4+4*q^3+3*q^2+2*q+1), q^ 10*(q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^15*(q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^ 21*(q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^28*(q^8+8*q^7+7*q^6+6*q^5+5*q^4 +4*q^3+3*q^2+2*q+1), q^36*(q^9+9*q^8+8*q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1) , q^45*(q^10+10*q^9+9*q^8+8*q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^55*(q^ 11+11*q^10+10*q^9+9*q^8+8*q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^66*(q^12+ 12*q^11+11*q^10+10*q^9+9*q^8+8*q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2*q+1), q^78*( q^13+13*q^12+12*q^11+11*q^10+10*q^9+9*q^8+8*q^7+7*q^6+6*q^5+5*q^4+4*q^3+3*q^2+2 *q+1), q^91*(q^14+14*q^13+13*q^12+12*q^11+11*q^10+10*q^9+9*q^8+8*q^7+7*q^6+6*q^ 5+5*q^4+4*q^3+3*q^2+2*q+1)] The number of permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50] The number of ODD permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [3, 1, 2]}, is given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 3, 5, 9, 12, 13, 17, 25, 30, 31, 37, 49, 56] ODD:, [0, 1, 2, 4, 6, 7, 10, 16, 20, 21, 26, 36, 42, 43, 50] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.428571429, 1.818181818, 2.187500000, 2.545454545, 2.896551724, 3.243243243, 3.586956522, 3.928571429, 4.268656716, 4.607594937, 4.945652174, 5.283018868] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.9035079028, 1.113404429, 1.333170563, 1.558766200, 1.787790538, 2.018902638, 2.251338607, 2.484646733, 2.718549271, 2.952869110, 3.187489575, 3.422331538] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.816327, 1.23967, 1.77734, 2.42975, 3.19620, 4.07597, 5.06853, 6.17347, 7.39051, 8.71944, 10.1601, 11.7124] Skewness: , [Float(undefined), 0., 0., 0.2134475891, 0.3625273747, 0.4457236911, 0.4915627629, 0.5177611505, 0.5334349255, 0.5432380696, 0.5496094208, 0.5538919035, 0.5568607378, 0.5589708799, 0.5605008058] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.267456774, 2.324012564, 2.340723758, 2.350328626, 2.357930745, 2.364253509, 2.369486517, 2.373791366, 2.377350063, 2.380287409, 2.382714737, 2.384737219] end of this data For the equivalence class of patterns, {{[1, 3, 2, 4], [3, 2, 1], [2, 3, 1]}, {[4, 2, 3, 1], [1, 2, 3], [2, 1, 3]}, {[1, 3, 2, 4], [3, 2, 1], [3, 1, 2]}, {[4, 2, 3, 1], [1, 2, 3], [1, 3, 2]}} the member , {[1, 3, 2, 4], [3, 2, 1], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+q^2, 1+2*q+3*q^2+q^3, 1+2*q+3*q^2+4*q^3+q^4, 1+2*q+3*q^2+4*q^3+5 *q^4+q^5, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+q^6, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^ 6+q^7, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+q^8, 1+2*q+3*q^2+4*q^3+5*q^4+6 *q^5+7*q^6+8*q^7+9*q^8+q^9, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+9*q^8+10* q^9+q^10, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+9*q^8+10*q^9+11*q^10+q^11, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+9*q^8+10*q^9+11*q^10+12*q^11+q^12, 1+ 2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+9*q^8+10*q^9+11*q^10+12*q^11+13*q^12+q^ 13, 1+2*q+3*q^2+4*q^3+5*q^4+6*q^5+7*q^6+8*q^7+9*q^8+10*q^9+11*q^10+12*q^11+13*q ^12+14*q^13+q^14] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+2*q+q^2), q^3*(1+3*q+2*q^2+q^3), q^6*(1+4*q+3*q^2+2*q^3+q^4), q^ 10*(1+5*q+4*q^2+3*q^3+2*q^4+q^5), q^15*(1+6*q+5*q^2+4*q^3+3*q^4+2*q^5+q^6), q^ 21*(1+7*q+6*q^2+5*q^3+4*q^4+3*q^5+2*q^6+q^7), q^28*(1+8*q+7*q^2+6*q^3+5*q^4+4*q ^5+3*q^6+2*q^7+q^8), q^36*(1+9*q+8*q^2+7*q^3+6*q^4+5*q^5+4*q^6+3*q^7+2*q^8+q^9) , q^45*(1+10*q+9*q^2+8*q^3+7*q^4+6*q^5+5*q^6+4*q^7+3*q^8+2*q^9+q^10), q^55*(1+ 11*q+10*q^2+9*q^3+8*q^4+7*q^5+6*q^6+5*q^7+4*q^8+3*q^9+2*q^10+q^11), q^66*(1+12* q+11*q^2+10*q^3+9*q^4+8*q^5+7*q^6+6*q^7+5*q^8+4*q^9+3*q^10+2*q^11+q^12), q^78*( 1+13*q+12*q^2+11*q^3+10*q^4+9*q^5+8*q^6+7*q^7+6*q^8+5*q^9+4*q^10+3*q^11+2*q^12+ q^13), q^91*(1+14*q+13*q^2+12*q^3+11*q^4+10*q^5+9*q^6+8*q^7+7*q^8+6*q^9+5*q^10+ 4*q^11+3*q^12+2*q^13+q^14)] The number of permutations avoiding, {[1, 3, 2, 4], [3, 2, 1], [2, 3, 1]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[1, 3, 2, 4], [3, 2, 1], [2, 3, 1]}, is given by [1, 1, 2, 4, 5, 9, 10, 16, 17, 25, 26, 36, 37, 49, 50] The number of ODD permutations avoiding, {[1, 3, 2, 4], [3, 2, 1], [2, 3, 1]}, is given by [0, 1, 2, 3, 6, 7, 12, 13, 20, 21, 30, 31, 42, 43, 56] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 4, 5, 7, 12, 16, 17, 21, 30, 36, 37, 43, 56] ODD:, [0, 1, 2, 3, 6, 9, 10, 13, 20, 25, 26, 31, 42, 49, 50] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.571428571, 2.181818182, 2.812500000, 3.454545455, 4.103448276, 4.756756757, 5.413043478, 6.071428571, 6.731343284, 7.392405063, 8.054347826, 8.716981132] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.9035079028, 1.113404429, 1.333170563, 1.558766200, 1.787790538, 2.018902638, 2.251338607, 2.484646733, 2.718549271, 2.952869110, 3.187489575, 3.422331538] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.816327, 1.23967, 1.77734, 2.42975, 3.19620, 4.07597, 5.06853, 6.17347, 7.39051, 8.71944, 10.1601, 11.7124] Skewness: , [Float(undefined), 0., 0., -0.2134475891, -0.3625201297, -0.4457321317, -0.4915654032, -0.5177506502, -0.5334312799, -0.5432310588, -0.5496094208, -0.5538968807, -0.5568607378, -0.5589677921, -0.5604933215] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.267426761, 2.324012564, 2.340723758, 2.350328626, 2.357950323, 2.364265547, 2.369509873, 2.373783495, 2.377350063, 2.380287409, 2.382714737, 2.384737219] end of this data For the equivalence class of patterns, {{[4, 1, 3, 2], [2, 1, 3], [2, 3, 1]}, {[2, 3, 1, 4], [1, 3, 2], [3, 1, 2]}, {[3, 2, 4, 1], [1, 3, 2], [3, 1, 2]}, {[2, 4, 3, 1], [2, 1, 3], [3, 1, 2]}, {[1, 3, 4, 2], [2, 1, 3], [3, 1, 2]}, {[4, 2, 1, 3], [1, 3, 2], [2, 3, 1]}, {[3, 1, 2, 4], [1, 3, 2], [2, 3, 1]}, {[1, 4, 2, 3], [2, 1, 3], [2, 3, 1]}} the member , {[4, 1, 3, 2], [2, 1, 3], [2, 3, 1]}, has a scheme of depth , 3 here it is: {[[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[], {}, {}], [[1], {[1, 1]}, {}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {}], [[2, 1, 3], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [[3, 2, 1], {[1, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q+q^2+q^3, 1+q+q^2+2*q^3+q^5+q^6, 1+q+q^2+2*q^3+q^4+q^5+q^6+q^7+q^9+ q^10, 1+q+q^2+2*q^3+q^4+2*q^5+q^6+q^7+2*q^9+q^10+q^12+q^14+q^15, 1+q+q^2+2*q^3+ q^4+2*q^5+2*q^6+q^7+2*q^9+q^10+q^11+q^12+q^14+2*q^15+q^18+q^20+q^21, 1+q+q^2+2* q^3+q^4+2*q^5+2*q^6+2*q^7+2*q^9+q^10+q^11+q^12+q^13+q^14+2*q^15+2*q^18+q^20+q^ 21+q^22+q^25+q^27+q^28, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+2*q^9+q^10+q^11 +q^12+q^13+q^14+3*q^15+2*q^18+q^20+2*q^21+q^22+q^25+q^26+q^27+q^28+q^30+q^33+q^ 35+q^36, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+3*q^9+q^10+q^11+q^12+q^13+q^14 +3*q^15+q^17+2*q^18+q^20+2*q^21+q^22+q^24+q^25+q^26+q^27+q^28+2*q^30+q^33+2*q^ 35+q^36+q^39+q^42+q^44+q^45, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+3*q^9+2*q^ 10+q^11+q^12+q^13+q^14+3*q^15+q^17+2*q^18+q^19+q^20+2*q^21+q^22+q^24+q^25+q^26+ 2*q^27+q^28+2*q^30+q^33+q^34+2*q^35+q^36+q^39+q^40+q^42+q^44+2*q^45+q^49+q^52+q ^54+q^55, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+3*q^9+2*q^10+2*q^11+q^12+q^13 +q^14+3*q^15+q^17+2*q^18+q^19+q^20+3*q^21+q^22+q^24+q^25+q^26+2*q^27+q^28+3*q^ 30+q^33+q^34+2*q^35+q^36+q^38+q^39+q^40+q^42+q^44+3*q^45+q^49+q^51+q^52+q^54+q^ 55+q^56+q^60+q^63+q^65+q^66, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+3*q^9+2*q^ 10+2*q^11+2*q^12+q^13+q^14+3*q^15+q^17+2*q^18+q^19+q^20+3*q^21+q^22+q^23+q^24+q ^25+q^26+2*q^27+q^28+3*q^30+2*q^33+q^34+2*q^35+q^36+q^38+q^39+q^40+2*q^42+q^44+ 3*q^45+q^49+q^50+q^51+q^52+q^54+q^55+q^56+q^57+q^60+2*q^63+q^65+q^66+q^68+q^72+ q^75+q^77+q^78, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+3*q^9+2*q^10+2*q^11+2*q ^12+2*q^13+q^14+3*q^15+q^17+2*q^18+q^19+q^20+3*q^21+q^22+q^23+q^24+2*q^25+q^26+ 2*q^27+q^28+3*q^30+2*q^33+q^34+2*q^35+2*q^36+q^38+q^39+q^40+2*q^42+q^44+3*q^45+ q^46+q^49+q^50+q^51+q^52+q^54+2*q^55+q^56+q^57+q^60+3*q^63+q^65+q^66+q^68+q^70+ q^72+q^75+q^76+q^77+q^78+q^81+q^85+q^88+q^90+q^91, 1+q+q^2+2*q^3+q^4+2*q^5+2*q^ 6+2*q^7+q^8+3*q^9+2*q^10+2*q^11+2*q^12+2*q^13+2*q^14+3*q^15+q^17+2*q^18+q^19+q^ 20+3*q^21+q^22+q^23+q^24+2*q^25+q^26+3*q^27+q^28+3*q^30+2*q^33+q^34+2*q^35+2*q^ 36+q^38+2*q^39+q^40+2*q^42+q^44+3*q^45+q^46+q^49+2*q^50+q^51+q^52+q^54+2*q^55+q ^56+q^57+2*q^60+3*q^63+q^65+q^66+q^68+q^69+q^70+q^72+q^75+q^76+2*q^77+q^78+q^81 +q^84+q^85+q^88+2*q^90+q^91+q^95+q^99+q^102+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+q^2+q^3, q^6+q^5+q^4+2*q^3+q+1, q^10+q^9+q^8+2*q^7+q^6+q^5+q^4+q^3 +q+1, q^15+q^14+q^13+2*q^12+q^11+2*q^10+q^9+q^8+2*q^6+q^5+q^3+q+1, q^21+q^20+q^ 19+2*q^18+q^17+2*q^16+2*q^15+q^14+2*q^12+q^11+q^10+q^9+q^7+2*q^6+q^3+q+1, q^28+ q^27+q^26+2*q^25+q^24+2*q^23+2*q^22+2*q^21+2*q^19+q^18+q^17+q^16+q^15+q^14+2*q^ 13+2*q^10+q^8+q^7+q^6+q^3+q+1, 1+q^25+q^22+q^24+3*q^21+q^23+q+2*q^30+2*q^29+q^ 34+2*q^33+q^26+q^3+q^6+q^32+q^8+q^9+q^10+q^11+q^14+2*q^15+q^16+q^28+2*q^18+2*q^ 27+2*q^31+q^35+q^36, 1+q^25+2*q^42+2*q^24+q^21+q^23+q+q^44+3*q^30+q^45+q^37+2*q ^39+q^41+q^34+2*q^38+q^33+q^3+q^6+q^32+q^9+2*q^10+q^12+2*q^15+q^43+q^28+q^17+q^ 18+q^19+q^20+2*q^27+2*q^40+q^31+q^35+3*q^36, 1+2*q^25+q^42+q^22+q^21+q+q^51+q^ 44+q^53+q^30+2*q^52+q^55+q^54+2*q^45+2*q^37+q^29+3*q^46+q^41+2*q^34+2*q^50+q^38 +q^33+q^3+q^6+2*q^10+q^11+q^13+q^15+q^16+q^43+2*q^49+2*q^28+q^19+2*q^20+q^27+3* q^40+q^31+q^35+2*q^48+q^36+q^47, 1+q^42+q^22+q^24+3*q^21+q+3*q^51+q^44+q^53+3*q ^57+q^30+q^52+q^58+2*q^55+2*q^60+q^54+3*q^45+q^46+q^65+2*q^39+2*q^61+q^41+q^64+ q^38+q^33+2*q^56+q^26+q^3+q^6+q^32+q^62+q^10+q^11+q^12+q^14+q^15+q^66+q^49+q^28 +q^17+q^27+q^40+2*q^63+2*q^31+2*q^48+2*q^59+3*q^36+q^47, 1+q^42+q^22+q^24+q^21+ q^23+q+2*q^51+q^44+q^53+3*q^57+q^52+q^58+q^55+2*q^60+q^54+2*q^45+2*q^71+q^74+2* q^67+q^29+q^65+3*q^69+q^39+q^61+q^34+q^50+q^64+q^38+3*q^33+q^56+2*q^68+q^26+q^3 +q^6+q^10+q^12+q^76+q^13+2*q^15+2*q^66+q^70+q^77+q^78+2*q^43+q^28+q^18+q^27+2*q ^72+q^40+2*q^73+3*q^63+2*q^75+3*q^48+q^59+2*q^36, 1+q^25+q^42+q^90+q^21+q^23+q+ q^51+q^53+q^57+q^52+2*q^58+2*q^55+q^91+q^45+q^71+q^74+q^37+q^67+3*q^46+q^65+q^ 69+q^39+3*q^61+q^41+q^34+2*q^79+2*q^64+2*q^56+q^68+q^26+q^3+q^6+q^10+3*q^76+q^ 13+q^14+q^15+q^16+2*q^66+3*q^70+q^77+2*q^78+2*q^49+3*q^28+q^19+q^72+q^40+2*q^73 +q^63+q^31+q^35+2*q^80+q^83+2*q^81+2*q^85+2*q^86+q^87+2*q^88+q^89+2*q^84+3*q^82 +2*q^36+q^47, 1+3*q^42+3*q^90+q^24+q^21+q+q^97+q^51+q^53+q^30+2*q^94+2*q^55+3*q ^60+2*q^91+q^54+2*q^45+q^71+3*q^96+2*q^98+q^37+q^67+q^29+2*q^93+2*q^99+q^65+2*q ^69+q^39+q^61+q^79+2*q^50+2*q^100+q^33+q^56+q^3+q^6+2*q^95+q^101+q^10+q^14+2*q^ 15+2*q^66+2*q^70+q^77+3*q^78+2*q^102+q^49+2*q^28+q^17+q^20+q^27+2*q^72+q^40+q^ 103+2*q^63+3*q^75+q^35+q^104+2*q^80+q^48+q^105+q^83+q^81+q^85+q^86+2*q^87+q^88+ 3*q^84+q^82+q^59+q^36+2*q^92] The number of permutations avoiding, {[4, 1, 3, 2], [2, 1, 3], [2, 3, 1]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[4, 1, 3, 2], [2, 1, 3], [2, 3, 1]}, is given by [1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50] The number of ODD permutations avoiding, {[4, 1, 3, 2], [2, 1, 3], [2, 3, 1]}, is given by [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 3, 5, 9, 12, 13, 17, 25, 30, 31, 37, 49, 56] ODD:, [0, 1, 2, 4, 6, 7, 10, 16, 20, 21, 26, 36, 42, 43, 50] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.857142857, 4.545454545, 6.562500000, 8.909090909, 11.58620690, 14.59459459, 17.93478261, 21.60714286, 25.61194030, 29.94936709, 34.61956522, 39.62264151] The standard deviation for each n is given by [0., 0.5000000000, 1.118033988, 1.958758457, 3.055952057, 4.415438115, 6.037075532, 7.919932260, 10.06320421, 12.46630619, 15.12882605, 18.05047254, 21.23103695, 24.67036701, 28.36834964] The centralized moments are Second: , [0., 0.250000, 1.25000, 3.83673, 9.33884, 19.4961, 36.4463, 62.7253, 101.268, 155.409, 228.881, 325.820, 450.757, 608.627, 804.763] Skewness: , [Float(undefined), 0., 0., 0.2001803610, 0.3380246522, 0.4194736357, 0.4686788659, 0.5000603688, 0.5212043339, 0.5361526120, 0.5471658261, 0.5555507431, 0.5621251753, 0.5673999906, 0.5717178064] Kurtosis: , [Float(undefined), 1.000000000, 1.640000000, 1.918868624, 2.041084432, 2.110519337, 2.155928119, 2.187762201, 2.211071721, 2.228676516, 2.242351994, 2.253174372, 2.261934065, 2.269122923, 2.275109735] end of this data For the equivalence class of patterns, {{[4, 3, 1, 2], [2, 1, 3], [2, 3, 1]}, {[3, 4, 2, 1], [1, 3, 2], [3, 1, 2]}, {[2, 1, 3, 4], [1, 3, 2], [3, 1, 2]}, {[1, 2, 4, 3], [2, 1, 3], [3, 1, 2]}, {[3, 4, 2, 1], [2, 1, 3], [3, 1, 2]}, {[2, 1, 3, 4], [1, 3, 2], [2, 3, 1]}, {[4, 3, 1, 2], [1, 3, 2], [2, 3, 1]}, {[1, 2, 4, 3], [2, 1, 3], [2, 3, 1]}} the member , {[4, 3, 1, 2], [2, 1, 3], [2, 3, 1]}, has a scheme of depth , 3 here it is: {[[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[], {}, {}], [[1], {[1, 1]}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {}], [[3, 1, 2], {[0, 1, 1, 0], [1, 0, 0, 0], [0, 0, 0, 1]}, {2}], [[2, 1, 3], {[0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q+q^2+q^3, 1+q+q^2+2*q^3+q^4+q^6, 1+q+q^2+2*q^3+2*q^4+q^5+q^6+q^7+q^ 10, 1+q+q^2+2*q^3+2*q^4+2*q^5+2*q^6+q^7+q^8+q^10+q^11+q^15, 1+q+q^2+2*q^3+2*q^4 +2*q^5+3*q^6+2*q^7+q^8+q^9+q^10+q^11+q^12+q^15+q^16+q^21, 1+q+q^2+2*q^3+2*q^4+2 *q^5+3*q^6+3*q^7+2*q^8+q^9+2*q^10+q^11+q^12+q^13+q^15+q^16+q^17+q^21+q^22+q^28, 1+q+q^2+2*q^3+2*q^4+2*q^5+3*q^6+3*q^7+3*q^8+2*q^9+2*q^10+2*q^11+q^12+q^13+q^14+ q^15+q^16+q^17+q^18+q^21+q^22+q^23+q^28+q^29+q^36, 1+q+q^2+2*q^3+2*q^4+2*q^5+3* q^6+3*q^7+3*q^8+3*q^9+3*q^10+2*q^11+2*q^12+q^13+q^14+2*q^15+q^16+q^17+q^18+q^19 +q^21+q^22+q^23+q^24+q^28+q^29+q^30+q^36+q^37+q^45, 1+q+q^2+2*q^3+2*q^4+2*q^5+3 *q^6+3*q^7+3*q^8+3*q^9+4*q^10+3*q^11+2*q^12+2*q^13+q^14+2*q^15+2*q^16+q^17+q^18 +q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^28+q^29+q^30+q^31+q^36+q^37+q^38+q^45+q^ 46+q^55, 1+q+q^2+2*q^3+2*q^4+2*q^5+3*q^6+3*q^7+3*q^8+3*q^9+4*q^10+4*q^11+3*q^12 +2*q^13+2*q^14+2*q^15+2*q^16+2*q^17+q^18+q^19+q^20+2*q^21+q^22+q^23+q^24+q^25+q ^26+q^28+q^29+q^30+q^31+q^32+q^36+q^37+q^38+q^39+q^45+q^46+q^47+q^55+q^56+q^66, 1+q+q^2+2*q^3+2*q^4+2*q^5+3*q^6+3*q^7+3*q^8+3*q^9+4*q^10+4*q^11+4*q^12+3*q^13+2 *q^14+3*q^15+2*q^16+2*q^17+2*q^18+q^19+q^20+2*q^21+2*q^22+q^23+q^24+q^25+q^26+q ^27+q^28+q^29+q^30+q^31+q^32+q^33+q^36+q^37+q^38+q^39+q^40+q^45+q^46+q^47+q^48+ q^55+q^56+q^57+q^66+q^67+q^78, 1+q+q^2+2*q^3+2*q^4+2*q^5+3*q^6+3*q^7+3*q^8+3*q^ 9+4*q^10+4*q^11+4*q^12+4*q^13+3*q^14+3*q^15+3*q^16+2*q^17+2*q^18+2*q^19+q^20+2* q^21+2*q^22+2*q^23+q^24+q^25+q^26+q^27+2*q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^ 36+q^37+q^38+q^39+q^40+q^41+q^45+q^46+q^47+q^48+q^49+q^55+q^56+q^57+q^58+q^66+q ^67+q^68+q^78+q^79+q^91, 1+q+q^2+2*q^3+2*q^4+2*q^5+3*q^6+3*q^7+3*q^8+3*q^9+4*q^ 10+4*q^11+4*q^12+4*q^13+4*q^14+4*q^15+3*q^16+3*q^17+2*q^18+2*q^19+2*q^20+2*q^21 +2*q^22+2*q^23+2*q^24+q^25+q^26+q^27+2*q^28+2*q^29+q^30+q^31+q^32+q^33+q^34+q^ 35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^45+q^46+q^47+q^48+q^49+q^50+q^55+q^56+q ^57+q^58+q^59+q^66+q^67+q^68+q^69+q^78+q^79+q^80+q^91+q^92+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+q^2+q^3, q^6+q^5+q^4+2*q^3+q^2+1, q^10+q^9+q^8+2*q^7+2*q^6+q^5+q^4 +q^3+1, q^15+q^14+q^13+2*q^12+2*q^11+2*q^10+2*q^9+q^8+q^7+q^5+q^4+1, q^21+q^20+ q^19+2*q^18+2*q^17+2*q^16+3*q^15+2*q^14+q^13+q^12+q^11+q^10+q^9+q^6+q^5+1, q^28 +q^27+q^26+2*q^25+2*q^24+2*q^23+3*q^22+3*q^21+2*q^20+q^19+2*q^18+q^17+q^16+q^15 +q^13+q^12+q^11+q^7+q^6+1, 1+2*q^25+q^22+q^24+q^21+q^23+3*q^30+3*q^29+q^34+2*q^ 33+2*q^26+2*q^32+q^7+q^8+q^13+q^14+q^15+3*q^28+q^18+q^19+q^20+2*q^27+2*q^31+q^ 35+q^36, 1+2*q^42+q^22+q^24+q^21+q^23+q^44+2*q^30+q^45+3*q^37+q^29+3*q^39+2*q^ 41+2*q^34+3*q^38+2*q^33+q^26+q^32+q^8+q^9+q^15+q^16+q^43+q^28+q^17+q^27+2*q^40+ q^31+3*q^35+3*q^36, 1+q^25+2*q^42+q^24+2*q^51+3*q^44+q^53+q^30+2*q^52+q^55+q^54 +4*q^45+q^37+3*q^46+2*q^39+q^41+q^34+2*q^50+q^38+q^33+q^26+q^32+q^9+q^10+2*q^43 +3*q^49+q^17+q^18+q^19+q^27+2*q^40+q^31+q^35+3*q^48+q^36+3*q^47, 1+q^42+q^21+2* q^51+q^44+2*q^53+3*q^57+q^30+2*q^52+3*q^58+4*q^55+3*q^60+3*q^54+2*q^45+q^37+q^ 29+q^46+q^65+2*q^61+q^41+q^34+2*q^50+q^64+q^38+4*q^56+2*q^62+q^10+q^11+q^66+q^ 43+2*q^49+q^28+q^19+q^20+q^27+q^40+2*q^63+q^35+q^48+3*q^59+q^36+q^47, 1+q^42+q^ 22+q^21+q^23+q^51+q^53+2*q^57+q^30+q^52+q^58+q^55+2*q^60+q^54+q^45+3*q^71+2*q^ 74+4*q^67+q^46+3*q^65+3*q^69+q^39+2*q^61+q^41+q^50+2*q^64+q^38+q^33+2*q^56+4*q^ 68+q^32+2*q^62+q^11+q^12+q^76+4*q^66+3*q^70+q^77+q^78+q^49+3*q^72+q^40+2*q^73+3 *q^63+q^31+2*q^75+q^48+q^59+q^47, 1+q^25+q^42+q^90+q^24+q^23+q^51+q^44+q^53+q^ 57+q^52+q^58+q^55+q^60+q^91+q^54+q^45+q^71+2*q^74+q^67+q^46+q^65+2*q^69+q^61+q^ 34+4*q^79+q^50+q^64+q^33+2*q^68+q^62+q^12+3*q^76+q^13+q^66+2*q^70+3*q^77+4*q^78 +q^43+2*q^72+2*q^73+2*q^63+3*q^75+q^35+4*q^80+3*q^83+4*q^81+3*q^85+2*q^86+2*q^ 87+2*q^88+q^89+3*q^84+3*q^82+q^59+q^36, 1+q^25+4*q^90+3*q^97+q^57+4*q^94+q^58+q ^55+q^60+4*q^91+q^71+q^74+3*q^96+3*q^98+q^37+q^67+q^46+4*q^93+3*q^99+q^65+q^69+ q^39+q^79+q^50+q^64+2*q^100+q^38+q^56+q^68+q^26+4*q^95+2*q^101+2*q^76+q^13+q^14 +q^66+q^70+2*q^77+q^78+2*q^102+q^49+q^27+q^72+q^73+q^103+q^63+q^75+q^104+q^80+q ^48+q^105+2*q^83+2*q^81+2*q^85+2*q^86+2*q^87+3*q^88+3*q^89+2*q^84+2*q^82+q^59+q ^36+4*q^92+q^47] The number of permutations avoiding, {[4, 3, 1, 2], [2, 1, 3], [2, 3, 1]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[4, 3, 1, 2], [2, 1, 3], [2, 3, 1]}, is given by [1, 1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46, 53] The number of ODD permutations avoiding, {[4, 3, 1, 2], [2, 1, 3], [2, 3, 1]}, is given by [0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46, 53] ODD:, [0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 2.714285714, 4.090909091, 5.625000000, 7.318181818, 9.172413793, 11.18918919, 13.36956522, 15.71428571, 18.22388060, 20.89873418, 23.73913043, 26.74528302] The standard deviation for each n is given by [0., 0.5000000000, 1.118033988, 1.829464068, 2.712078889, 3.789376598, 5.066703821, 6.544642269, 8.222703698, 10.10027700, 12.17684319, 14.45200073, 16.92544705, 19.59695493, 22.46635275] The centralized moments are Second: , [0., 0.250000, 1.25000, 3.34694, 7.35537, 14.3594, 25.6715, 42.8323, 67.6129, 102.016, 148.276, 208.860, 286.471, 384.041, 504.737] Skewness: , [Float(undefined), 0., 0., 0.2856801023, 0.5875425893, 0.8147302919, 0.9733226669, 1.084021456, 1.163065346, 1.221108198, 1.264903751, 1.298789019, 1.325560234, 1.347136494, 1.364792478] Kurtosis: , [Float(undefined), 1.000000000, 1.640000000, 2.327038285, 2.853397628, 3.243873695, 3.531029595, 3.743588699, 3.903397884, 4.025870676, 4.121661011, 4.197994193, 4.259797721, 4.310650356, 4.353086989] end of this data For the equivalence class of patterns, {{[3, 2, 1, 4], [1, 2, 3], [1, 3, 2]}, {[2, 3, 4, 1], [3, 2, 1], [3, 1, 2]}, {[3, 2, 1], [4, 1, 2, 3], [2, 3, 1]}, {[1, 4, 3, 2], [1, 2, 3], [2, 1, 3]}} the member , {[3, 2, 1, 4], [1, 2, 3], [1, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0], [0, 0, 1]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 2, 0], [0, 1, 1], [0, 0, 2]}, {}], [[1], {[0, 2]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, q+2*q^2+q^3, 3*q^4+3*q^5+q^6, 2*q^7+6*q^8+4*q^9+q^10, q^11+7*q^12+10*q ^13+5*q^14+q^15, 6*q^17+16*q^18+15*q^19+6*q^20+q^21, 3*q^23+19*q^24+30*q^25+21* q^26+7*q^27+q^28, q^30+16*q^31+45*q^32+50*q^33+28*q^34+8*q^35+q^36, 10*q^39+51* q^40+90*q^41+77*q^42+36*q^43+9*q^44+q^45, 4*q^48+45*q^49+126*q^50+161*q^51+112* q^52+45*q^53+10*q^54+q^55, q^58+30*q^59+141*q^60+266*q^61+266*q^62+156*q^63+55* q^64+11*q^65+q^66, 15*q^70+126*q^71+357*q^72+504*q^73+414*q^74+210*q^75+66*q^76 +12*q^77+q^78, 5*q^82+90*q^83+393*q^84+784*q^85+882*q^86+615*q^87+275*q^88+78*q ^89+13*q^90+q^91, q^95+50*q^96+357*q^97+1016*q^98+1554*q^99+1452*q^100+880*q^ 101+352*q^102+91*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^2, 3*q^2+3*q+1, 2*q^3+6*q^2+4*q+1, q^4+7*q^3+10*q^2+5*q+1, 6*q ^4+16*q^3+15*q^2+6*q+1, 3*q^5+19*q^4+30*q^3+21*q^2+7*q+1, q^6+16*q^5+45*q^4+50* q^3+28*q^2+8*q+1, 10*q^6+51*q^5+90*q^4+77*q^3+36*q^2+9*q+1, 4*q^7+45*q^6+126*q^ 5+161*q^4+112*q^3+45*q^2+10*q+1, q^8+30*q^7+141*q^6+266*q^5+266*q^4+156*q^3+55* q^2+11*q+1, 15*q^8+126*q^7+357*q^6+504*q^5+414*q^4+210*q^3+66*q^2+12*q+1, 5*q^9 +90*q^8+393*q^7+784*q^6+882*q^5+615*q^4+275*q^3+78*q^2+13*q+1, q^10+50*q^9+357* q^8+1016*q^7+1554*q^6+1452*q^5+880*q^4+352*q^3+91*q^2+14*q+1] The number of permutations avoiding, {[3, 2, 1, 4], [1, 2, 3], [1, 3, 2]}, is given by [1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768] The number of EVEN permutations avoiding, {[3, 2, 1, 4], [1, 2, 3], [1, 3, 2]}, is given by [1, 1, 2, 4, 7, 12, 22, 41, 75, 137, 252, 464, 853, 1568, 2884] The number of ODD permutations avoiding, {[3, 2, 1, 4], [1, 2, 3], [1, 3, 2]}, is given by [0, 1, 2, 3, 6, 12, 22, 40, 74, 137, 252, 463, 852, 1568, 2884] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 4, 7, 12, 22, 41, 75, 137, 252, 464, 853, 1568, 2884] ODD:, [0, 1, 2, 3, 6, 12, 22, 40, 74, 137, 252, 463, 852, 1568, 2884] The average number of inversions for each n is given by [0., 0.5000000000, 2.000000000, 4.714285714, 8.307692308, 12.91666667, 18.54545455, 25.16049383, 32.77852349, 41.39781022, 51.01587302, 61.63430421, 73.25278592, 85.87117347, 99.48959778] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.6998542123, 0.8213137115, 0.9090593425, 0.9642365195, 1.035860591, 1.098171987, 1.155182998, 1.211611330, 1.264838199, 1.315829470, 1.365102840, 1.412580675] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.489796, 0.674556, 0.826389, 0.929752, 1.07301, 1.20598, 1.33445, 1.46800, 1.59982, 1.73141, 1.86351, 1.99538] Skewness: , [Float(undefined), 0., 0., 0.4592955420, 0.2119590902, 0.1648614895, 0.2514219729, 0.2090256767, 0.2027526225, 0.2068911501, 0.1971838989, 0.1926448637, 0.1886410540, 0.1836796372, 0.1794877618] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.104250705, 2.539724137, 2.694639620, 2.614366741, 2.739044374, 2.774294525, 2.778874687, 2.813332937, 2.829425648, 2.841835735, 2.855814509, 2.866050862] end of this data For the equivalence class of patterns, {{[1, 4, 3, 2], [1, 2, 3], [2, 3, 1]}, {[3, 2, 1, 4], [1, 2, 3], [2, 3, 1]}, {[3, 2, 1, 4], [1, 2, 3], [3, 1, 2]}, {[2, 3, 4, 1], [3, 2, 1], [2, 1, 3]}, {[3, 2, 1], [4, 1, 2, 3], [2, 1, 3]}, {[1, 4, 3, 2], [1, 2, 3], [3, 1, 2]}, {[2, 3, 4, 1], [3, 2, 1], [1, 3, 2]}, {[3, 2, 1], [4, 1, 2, 3], [1, 3, 2]}} the member , {[1, 4, 3, 2], [1, 2, 3], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 0, 1], [0, 2, 0]}, {1}], [[1], {[0, 3]}, {}], [[2, 1], {[0, 1, 1], [0, 3, 0], [0, 0, 3]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2+q^3, q^2+q^3+2*q^4+q^5+q^6, q^4+2*q^6+q^7+2*q^8+q^9+q^10, q^7+ q^9+q^10+2*q^11+q^12+2*q^13+q^14+q^15, q^11+q^13+2*q^15+q^16+2*q^17+q^18+2*q^19 +q^20+q^21, q^16+q^18+q^20+q^21+2*q^22+q^23+2*q^24+q^25+2*q^26+q^27+q^28, q^22+ q^24+q^26+2*q^28+q^29+2*q^30+q^31+2*q^32+q^33+2*q^34+q^35+q^36, q^29+q^31+q^33+ q^35+q^36+2*q^37+q^38+2*q^39+q^40+2*q^41+q^42+2*q^43+q^44+q^45, q^37+q^39+q^41+ q^43+2*q^45+q^46+2*q^47+q^48+2*q^49+q^50+2*q^51+q^52+2*q^53+q^54+q^55, q^46+q^ 48+q^50+q^52+q^54+q^55+2*q^56+q^57+2*q^58+q^59+2*q^60+q^61+2*q^62+q^63+2*q^64+q ^65+q^66, q^56+q^58+q^60+q^62+q^64+2*q^66+q^67+2*q^68+q^69+2*q^70+q^71+2*q^72+q ^73+2*q^74+q^75+2*q^76+q^77+q^78, q^67+q^69+q^71+q^73+q^75+q^77+q^78+2*q^79+q^ 80+2*q^81+q^82+2*q^83+q^84+2*q^85+q^86+2*q^87+q^88+2*q^89+q^90+q^91, q^79+q^81+ q^83+q^85+q^87+q^89+2*q^91+q^92+2*q^93+q^94+2*q^95+q^96+2*q^97+q^98+2*q^99+q^ 100+2*q^101+q^102+2*q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+2*q^2, q^4+q^3+2*q^2+q+1, q^6+2*q^4+q^3+2*q^2+q+1, q^8+q^6+q^5+2*q ^4+q^3+2*q^2+q+1, q^10+q^8+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1, q^12+q^10+q^8+q^7+2*q ^6+q^5+2*q^4+q^3+2*q^2+q+1, q^14+q^12+q^10+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+ q+1, q^16+q^14+q^12+q^10+q^9+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1, q^18+q^16 +q^14+q^12+2*q^10+q^9+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1, q^20+q^18+q^16+q ^14+q^12+q^11+2*q^10+q^9+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1, q^22+q^20+q^ 18+q^16+q^14+2*q^12+q^11+2*q^10+q^9+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1, q^ 24+q^22+q^20+q^18+q^16+q^14+q^13+2*q^12+q^11+2*q^10+q^9+2*q^8+q^7+2*q^6+q^5+2*q ^4+q^3+2*q^2+q+1, q^26+q^24+q^22+q^20+q^18+q^16+2*q^14+q^13+2*q^12+q^11+2*q^10+ q^9+2*q^8+q^7+2*q^6+q^5+2*q^4+q^3+2*q^2+q+1] The number of permutations avoiding, {[1, 4, 3, 2], [1, 2, 3], [2, 3, 1]}, is given by [1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28] The number of EVEN permutations avoiding, {[1, 4, 3, 2], [1, 2, 3], [2, 3, 1]}, is given by [1, 1, 1, 4, 6, 3, 3, 10, 12, 5, 5, 16, 18, 7, 7] The number of ODD permutations avoiding, {[1, 4, 3, 2], [1, 2, 3], [2, 3, 1]}, is given by [0, 1, 3, 2, 2, 7, 9, 4, 4, 13, 15, 6, 6, 19, 21] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21] ODD:, [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 4.000000000, 7.250000000, 11.50000000, 16.75000000, 23.00000000, 30.25000000, 38.50000000, 47.75000000, 58.00000000, 69.25000000, 81.50000000, 94.75000000] The standard deviation for each n is given by [0., 0.5000000000, 0.8291561975, 1.290994449, 1.785357107, 2.291287848, 2.802528620, 3.316624790, 3.832427430, 4.349329451, 4.866980582, 5.385164807, 5.903741751, 6.422616290, 6.941721688] The centralized moments are Second: , [0., 0.250000, 0.687500, 1.66667, 3.18750, 5.25000, 7.85417, 11.0000, 14.6875, 18.9167, 23.6875, 29.0000, 34.8542, 41.2500, 48.1875] -5 Skewness: , [Float(undefined), 0., 0.4933822002, 0.4647566070 10 , -0.2471170362, -0.3740878120, -0.4472063110, -0.4933822003, -0.5246392109, -0.5469447830, -0.5635390285, -0.5762958772, -0.5863625774, -0.5944901029, -0.6011703043] Kurtosis: , [Float(undefined), 1.000000000, 1.628096529, 2.040003840, 2.259482968, 2.371410431, 2.433601886, 2.471074380, 2.495189933, 2.511522287, 2.523090204, 2.531510107, 2.537834440, 2.542698255, 2.546507526] end of this data For the equivalence class of patterns, {{[3, 4, 2, 1], [1, 2, 3], [3, 1, 2]}, {[4, 3, 1, 2], [1, 2, 3], [2, 3, 1]}, {[2, 1, 3, 4], [3, 2, 1], [1, 3, 2]}, {[1, 2, 4, 3], [3, 2, 1], [2, 1, 3]}} the member , {[3, 4, 2, 1], [1, 2, 3], [3, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2], {[0, 0, 1], [2, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2+q^3, q^2+2*q^3+2*q^4+q^6, 2*q^4+3*q^6+2*q^7+q^10, q^6+2*q^7+2* q^9+2*q^10+2*q^11+q^15, 2*q^9+2*q^11+q^12+2*q^13+2*q^15+2*q^16+q^21, q^12+2*q^ 13+4*q^16+2*q^18+2*q^21+2*q^22+q^28, 2*q^16+2*q^18+q^20+2*q^21+2*q^22+2*q^24+2* q^28+2*q^29+q^36, q^20+2*q^21+2*q^24+2*q^25+2*q^27+2*q^29+2*q^31+2*q^36+2*q^37+ q^45, 2*q^25+2*q^27+q^30+4*q^31+2*q^34+2*q^37+2*q^39+2*q^45+2*q^46+q^55, q^30+2 *q^31+2*q^34+2*q^36+2*q^38+2*q^39+2*q^42+2*q^46+2*q^48+2*q^55+2*q^56+q^66, 2*q^ 36+2*q^38+3*q^42+2*q^43+2*q^46+2*q^48+2*q^51+2*q^56+2*q^58+2*q^66+2*q^67+q^78, q^42+2*q^43+2*q^46+2*q^49+4*q^51+2*q^55+2*q^58+2*q^61+2*q^67+2*q^69+2*q^78+2*q^ 79+q^91, 2*q^49+2*q^51+2*q^55+q^56+2*q^57+2*q^60+2*q^61+2*q^65+2*q^69+2*q^72+2* q^79+2*q^81+2*q^91+2*q^92+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+2*q^2, q^4+2*q^3+2*q^2+1, 2*q^6+3*q^4+2*q^3+1, q^9+2*q^8+2*q^6+2*q ^5+2*q^4+1, 2*q^12+2*q^10+q^9+2*q^8+2*q^6+2*q^5+1, q^16+2*q^15+4*q^12+2*q^10+2* q^7+2*q^6+1, 2*q^20+2*q^18+q^16+2*q^15+2*q^14+2*q^12+2*q^8+2*q^7+1, q^25+2*q^24 +2*q^21+2*q^20+2*q^18+2*q^16+2*q^14+2*q^9+2*q^8+1, 2*q^30+2*q^28+q^25+4*q^24+2* q^21+2*q^18+2*q^16+2*q^10+2*q^9+1, q^36+2*q^35+2*q^32+2*q^30+2*q^28+2*q^27+2*q^ 24+2*q^20+2*q^18+2*q^11+2*q^10+1, 2*q^42+2*q^40+3*q^36+2*q^35+2*q^32+2*q^30+2*q ^27+2*q^22+2*q^20+2*q^12+2*q^11+1, q^49+2*q^48+2*q^45+2*q^42+4*q^40+2*q^36+2*q^ 33+2*q^30+2*q^24+2*q^22+2*q^13+2*q^12+1, 2*q^56+2*q^54+2*q^50+q^49+2*q^48+2*q^ 45+2*q^44+2*q^40+2*q^36+2*q^33+2*q^26+2*q^24+2*q^14+2*q^13+1] The number of permutations avoiding, {[3, 4, 2, 1], [1, 2, 3], [3, 1, 2]}, is given by [1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28] The number of EVEN permutations avoiding, {[3, 4, 2, 1], [1, 2, 3], [3, 1, 2]}, is given by [1, 1, 1, 4, 6, 3, 3, 10, 12, 5, 5, 16, 18, 7, 7] The number of ODD permutations avoiding, {[3, 4, 2, 1], [1, 2, 3], [3, 1, 2]}, is given by [0, 1, 3, 2, 2, 7, 9, 4, 4, 13, 15, 6, 6, 19, 21] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21] ODD:, [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.666666667, 6.250000000, 9.500000000, 13.41666667, 18.00000000, 23.25000000, 29.16666667, 35.75000000, 43.00000000, 50.91666667, 59.50000000, 68.75000000] The standard deviation for each n is given by [0., 0.5000000000, 0.8291561975, 1.247219129, 1.785357107, 2.459674775, 3.277660073, 4.242640686, 5.356071322, 6.618576551, 8.030410950, 9.591663046, 11.30234735, 13.16244658, 15.17193132] The centralized moments are Second: , [0., 0.250000, 0.687500, 1.55556, 3.18750, 6.05000, 10.7431, 18.0000, 28.6875, 43.8056, 64.4875, 92.0000, 127.743, 173.250, 230.188] Skewness: , [Float(undefined), 0., 0.4933822002, 0.6490657699, 0.6754292170, 0.6652764232, 0.6507839080, 0.6397627169, 0.6327188238, 0.6286326246, 0.6264826296, 0.6255432421, 0.6253330443, 0.6255498310, 0.6260049821] Kurtosis: , [Float(undefined), 1.000000000, 1.628096529, 2.602027478, 3.071507882, 3.151751930, 3.062916065, 2.928570988, 2.798112407, 2.686402533, 2.595128716, 2.521739130, 2.462901400, 2.415566925, 2.377208601] end of this data For the equivalence class of patterns, {{[3, 2, 4, 1], [1, 2, 3], [3, 1, 2]}, {[4, 1, 3, 2], [1, 2, 3], [2, 3, 1]}, {[4, 2, 1, 3], [1, 2, 3], [2, 3, 1]}, {[2, 4, 3, 1], [1, 2, 3], [3, 1, 2]}, {[2, 3, 1, 4], [3, 2, 1], [1, 3, 2]}, {[1, 4, 2, 3], [3, 2, 1], [2, 1, 3]}, {[3, 1, 2, 4], [3, 2, 1], [1, 3, 2]}, {[1, 3, 4, 2], [3, 2, 1], [2, 1, 3]}} the member , {[3, 2, 4, 1], [1, 2, 3], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {}], [[1, 2], {[0, 0, 1]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2+q^3, q^2+2*q^3+q^4+q^5+q^6, 2*q^4+2*q^6+q^7+q^8+q^9+q^10, q^6+ 2*q^7+2*q^10+q^11+q^12+q^13+q^14+q^15, 2*q^9+2*q^11+2*q^15+q^16+q^17+q^18+q^19+ q^20+q^21, q^12+2*q^13+2*q^16+2*q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28, 2*q^16 +2*q^18+2*q^22+2*q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q^36, q^20+2*q^21+2*q^ 24+2*q^29+2*q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45, 2*q^25+2*q^27+2* q^31+2*q^37+2*q^45+q^46+q^47+q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55, q^30+2*q^ 31+2*q^34+2*q^39+2*q^46+2*q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62+q^63+q^64+q^ 65+q^66, 2*q^36+2*q^38+2*q^42+2*q^48+2*q^56+2*q^66+q^67+q^68+q^69+q^70+q^71+q^ 72+q^73+q^74+q^75+q^76+q^77+q^78, q^42+2*q^43+2*q^46+2*q^51+2*q^58+2*q^67+2*q^ 78+q^79+q^80+q^81+q^82+q^83+q^84+q^85+q^86+q^87+q^88+q^89+q^90+q^91, 2*q^49+2*q ^51+2*q^55+2*q^61+2*q^69+2*q^79+2*q^91+q^92+q^93+q^94+q^95+q^96+q^97+q^98+q^99+ q^100+q^101+q^102+q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+2*q^2, q^4+2*q^3+q^2+q+1, 2*q^6+2*q^4+q^3+q^2+q+1, q^9+2*q^8+2*q^5 +q^4+q^3+q^2+q+1, 2*q^12+2*q^10+2*q^6+q^5+q^4+q^3+q^2+q+1, q^16+2*q^15+2*q^12+2 *q^7+q^6+q^5+q^4+q^3+q^2+q+1, 2*q^20+2*q^18+2*q^14+2*q^8+q^7+q^6+q^5+q^4+q^3+q^ 2+q+1, q^25+2*q^24+2*q^21+2*q^16+2*q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1, 2*q^30+ 2*q^28+2*q^24+2*q^18+2*q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1, q^36+2*q^35+2* q^32+2*q^27+2*q^20+2*q^11+q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1, 2*q^42+2*q^ 40+2*q^36+2*q^30+2*q^22+2*q^12+q^11+q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1, q ^49+2*q^48+2*q^45+2*q^40+2*q^33+2*q^24+2*q^13+q^12+q^11+q^10+q^9+q^8+q^7+q^6+q^ 5+q^4+q^3+q^2+q+1, 2*q^56+2*q^54+2*q^50+2*q^44+2*q^36+2*q^26+2*q^14+q^13+q^12+q ^11+q^10+q^9+q^8+q^7+q^6+q^5+q^4+q^3+q^2+q+1] The number of permutations avoiding, {[3, 2, 4, 1], [1, 2, 3], [3, 1, 2]}, is given by [1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28] The number of EVEN permutations avoiding, {[3, 2, 4, 1], [1, 2, 3], [3, 1, 2]}, is given by [1, 1, 1, 3, 6, 5, 3, 7, 12, 9, 5, 11, 18, 13, 7] The number of ODD permutations avoiding, {[3, 2, 4, 1], [1, 2, 3], [3, 1, 2]}, is given by [0, 1, 3, 3, 2, 5, 9, 7, 4, 9, 15, 11, 6, 13, 21] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21] ODD:, [0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.833333333, 6.750000000, 10.50000000, 15.08333333, 20.50000000, 26.75000000, 33.83333333, 41.75000000, 50.50000000, 60.08333333, 70.50000000, 81.75000000] The standard deviation for each n is given by [0., 0.5000000000, 0.8291561975, 1.343709625, 2.046338193, 2.941088234, 4.030267430, 5.315072905, 6.796138610, 8.473815882, 10.34830904, 12.41974235, 14.68819442, 17.15371680, 19.81634426] The centralized moments are Second: , [0., 0.250000, 0.687500, 1.80556, 4.18750, 8.65000, 16.2431, 28.2500, 46.1875, 71.8056, 107.088, 154.250, 215.743, 294.250, 392.688] Skewness: , [Float(undefined), 0., 0.4933822002, 0.3053189585, 0.07658966573, -0.09433809029, -0.2135199957, -0.2968415563, -0.3563083558, -0.3997895555, -0.4323449287, -0.4572633007, -0.4766991855, -0.4921266503, -0.5045578052] Kurtosis: , [Float(undefined), 1.000000000, 1.628096529, 1.848255611, 1.815341680, 1.756984864, 1.714053661, 1.685945650, 1.667653286, 1.655470683, 1.647093125, 1.641192680, 1.636883256, 1.633656698, 1.631191277] end of this data For the equivalence class of patterns, {{[2, 3, 4, 1], [1, 3, 2], [3, 1, 2]}, {[3, 2, 1, 4], [1, 3, 2], [3, 1, 2]}, {[1, 4, 3, 2], [2, 1, 3], [3, 1, 2]}, {[2, 3, 4, 1], [2, 1, 3], [3, 1, 2]}, {[3, 2, 1, 4], [1, 3, 2], [2, 3, 1]}, {[4, 1, 2, 3], [1, 3, 2], [2, 3, 1]}, {[1, 4, 3, 2], [2, 1, 3], [2, 3, 1]}, {[4, 1, 2, 3], [2, 1, 3], [2, 3, 1]}} the member , {[2, 3, 4, 1], [1, 3, 2], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q+q^2+q^3, 1+q+q^2+q^3+q^4+q^5+q^6, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+ q^9+q^10, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15, 1+ q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+ q^19+q^20+q^21, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^ 15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28, 1+q+q^2+q^ 3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^ 20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q ^36, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^ 17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q ^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45, 1+q+q^2+q^3+q^ 4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^20+q^ 21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q^36+q ^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+q^48+q^49+q^50+q^51+q^52+ q^53+q^54+q^55, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^ 15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q ^31+q^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+ q^47+q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62 +q^63+q^64+q^65+q^66, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q ^14+q^15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+ q^30+q^31+q^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45 +q^46+q^47+q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^ 61+q^62+q^63+q^64+q^65+q^66+q^67+q^68+q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q ^77+q^78, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^ 16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q ^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+ q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62+q^63 +q^64+q^65+q^66+q^67+q^68+q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q^77+q^78+q^ 79+q^80+q^81+q^82+q^83+q^84+q^85+q^86+q^87+q^88+q^89+q^90+q^91, 1+q+q^2+q^3+q^4 +q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^20+q^ 21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q^36+q ^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+q^48+q^49+q^50+q^51+q^52+ q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62+q^63+q^64+q^65+q^66+q^67+q^68 +q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q^77+q^78+q^79+q^80+q^81+q^82+q^83+q^ 84+q^85+q^86+q^87+q^88+q^89+q^90+q^91+q^92+q^93+q^94+q^95+q^96+q^97+q^98+q^99+q ^100+q^101+q^102+q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+q^2+q^3, 1+q+q^2+q^3+q^4+q^5+q^6, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+ q^9+q^10, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15, 1+ q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+ q^19+q^20+q^21, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^ 15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28, 1+q+q^2+q^ 3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^ 20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q ^36, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^ 17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q ^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45, 1+q+q^2+q^3+q^ 4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^20+q^ 21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q^36+q ^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+q^48+q^49+q^50+q^51+q^52+ q^53+q^54+q^55, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^ 15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q ^31+q^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+ q^47+q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62 +q^63+q^64+q^65+q^66, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q ^14+q^15+q^16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+ q^30+q^31+q^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45 +q^46+q^47+q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^ 61+q^62+q^63+q^64+q^65+q^66+q^67+q^68+q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q ^77+q^78, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^ 16+q^17+q^18+q^19+q^20+q^21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q ^32+q^33+q^34+q^35+q^36+q^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+ q^48+q^49+q^50+q^51+q^52+q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62+q^63 +q^64+q^65+q^66+q^67+q^68+q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q^77+q^78+q^ 79+q^80+q^81+q^82+q^83+q^84+q^85+q^86+q^87+q^88+q^89+q^90+q^91, 1+q+q^2+q^3+q^4 +q^5+q^6+q^7+q^8+q^9+q^10+q^11+q^12+q^13+q^14+q^15+q^16+q^17+q^18+q^19+q^20+q^ 21+q^22+q^23+q^24+q^25+q^26+q^27+q^28+q^29+q^30+q^31+q^32+q^33+q^34+q^35+q^36+q ^37+q^38+q^39+q^40+q^41+q^42+q^43+q^44+q^45+q^46+q^47+q^48+q^49+q^50+q^51+q^52+ q^53+q^54+q^55+q^56+q^57+q^58+q^59+q^60+q^61+q^62+q^63+q^64+q^65+q^66+q^67+q^68 +q^69+q^70+q^71+q^72+q^73+q^74+q^75+q^76+q^77+q^78+q^79+q^80+q^81+q^82+q^83+q^ 84+q^85+q^86+q^87+q^88+q^89+q^90+q^91+q^92+q^93+q^94+q^95+q^96+q^97+q^98+q^99+q ^100+q^101+q^102+q^103+q^104+q^105] The number of permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [3, 1, 2]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [3, 1, 2]}, is given by [1, 1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46, 53] The number of ODD permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [3, 1, 2]}, is given by [0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46, 53] ODD:, [0, 1, 2, 3, 5, 8, 11, 14, 18, 23, 28, 33, 39, 46, 53] 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, 1.118033988, 2., 3.162277660, 4.609772228, 6.344288770, 8.366600265, 10.67707825, 13.27591804, 16.16322988, 19.33907961, 22.80350850, 26.55654345, 30.59820256] The centralized moments are Second: , [0., 0.250000, 1.25000, 4.00000, 10.0000, 21.2500, 40.2500, 70.0000, 114.000, 176.250, 261.250, 374.000, 520.000, 705.250, 936.250] Skewness: , [Float(undefined), 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.] Kurtosis: , [Float(undefined), 1.000000000, 1.640000000, 1.750000000, 1.780000000, 1.790582699, 1.795029513, 1.797142857, 1.798245614, 1.798863236, 1.799226208, 1.799465241, 1.799615385, 1.799715282, 1.799788592] end of this data For the equivalence class of patterns, {{[2, 3, 4, 1], [1, 3, 2], [2, 1, 3]}, {[4, 1, 2, 3], [1, 3, 2], [2, 1, 3]}, {[1, 4, 3, 2], [2, 3, 1], [3, 1, 2]}, {[3, 2, 1, 4], [2, 3, 1], [3, 1, 2]}} the member , {[2, 3, 4, 1], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[2, 1], {[0, 0, 1], [1, 2, 0]}, {1}], [[1], {[1, 2]}, {}], [[1, 2], {[0, 1, 0], [1, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 1+q^3+q^4+3*q^5+q^6, 1+q^4+q^6+q^7+3*q^8+4*q^9+q^10, 1+q^ 5+q^8+q^9+2*q^11+2*q^12+6*q^13+5*q^14+q^15, 1+q^6+q^10+q^11+2*q^14+q^15+q^16+3* q^17+5*q^18+10*q^19+6*q^20+q^21, 1+q^7+q^12+q^13+2*q^17+q^18+q^20+3*q^21+q^22+3 *q^23+5*q^24+11*q^25+15*q^26+7*q^27+q^28, 1+q^8+q^14+q^15+2*q^20+q^21+q^24+3*q^ 25+q^26+3*q^28+4*q^29+2*q^30+6*q^31+10*q^32+21*q^33+21*q^34+8*q^35+q^36, 1+q^9+ q^16+q^17+2*q^23+q^24+q^28+3*q^29+q^30+3*q^33+4*q^34+q^35+q^36+6*q^37+5*q^38+5* q^39+11*q^40+21*q^41+36*q^42+28*q^43+9*q^44+q^45, 1+q^10+q^18+q^19+2*q^26+q^27+ q^32+3*q^33+q^34+3*q^38+4*q^39+q^40+q^42+6*q^43+5*q^44+q^45+4*q^46+10*q^47+7*q^ 48+11*q^49+21*q^50+42*q^51+57*q^52+36*q^53+10*q^54+q^55, 1+q^11+q^20+q^21+2*q^ 29+q^30+q^36+3*q^37+q^38+3*q^43+4*q^44+q^45+q^48+6*q^49+5*q^50+q^51+4*q^53+10*q ^54+6*q^55+2*q^56+10*q^57+15*q^58+12*q^59+22*q^60+42*q^61+78*q^62+85*q^63+45*q^ 64+11*q^65+q^66, 1+q^12+q^22+q^23+2*q^32+q^33+q^40+3*q^41+q^42+3*q^48+4*q^49+q^ 50+q^54+6*q^55+5*q^56+q^57+4*q^60+10*q^61+6*q^62+q^63+q^64+10*q^65+15*q^66+7*q^ 67+6*q^68+20*q^69+22*q^70+23*q^71+43*q^72+84*q^73+135*q^74+121*q^75+55*q^76+12* q^77+q^78, 1+q^13+q^24+q^25+2*q^35+q^36+q^44+3*q^45+q^46+3*q^53+4*q^54+q^55+q^ 60+6*q^61+5*q^62+q^63+4*q^67+10*q^68+6*q^69+q^70+q^72+10*q^73+15*q^74+7*q^75+q^ 76+5*q^77+20*q^78+21*q^79+9*q^80+16*q^81+35*q^82+34*q^83+45*q^84+85*q^85+162*q^ 86+220*q^87+166*q^88+66*q^89+13*q^90+q^91, 1+q^14+q^26+q^27+2*q^38+q^39+q^48+3* q^49+q^50+3*q^58+4*q^59+q^60+q^66+6*q^67+5*q^68+q^69+4*q^74+10*q^75+6*q^76+q^77 +q^80+10*q^81+15*q^82+7*q^83+q^84+5*q^86+20*q^87+21*q^88+8*q^89+2*q^90+15*q^91+ 35*q^92+28*q^93+15*q^94+36*q^95+57*q^96+57*q^97+88*q^98+169*q^99+297*q^100+341* q^101+221*q^102+78*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, q^6+q^3+q^2+3*q+1, q^10+q^6+q^4+q^3+3*q^2+4*q+1, q^15+q^10+ q^7+q^6+2*q^4+2*q^3+6*q^2+5*q+1, q^21+q^15+q^11+q^10+2*q^7+q^6+q^5+3*q^4+5*q^3+ 10*q^2+6*q+1, q^28+q^21+q^16+q^15+2*q^11+q^10+q^8+3*q^7+q^6+3*q^5+5*q^4+11*q^3+ 15*q^2+7*q+1, q^36+q^28+q^22+q^21+2*q^16+q^15+q^12+3*q^11+q^10+3*q^8+4*q^7+2*q^ 6+6*q^5+10*q^4+21*q^3+21*q^2+8*q+1, q^45+q^36+q^29+q^28+2*q^22+q^21+q^17+3*q^16 +q^15+3*q^12+4*q^11+q^10+q^9+6*q^8+5*q^7+5*q^6+11*q^5+21*q^4+36*q^3+28*q^2+9*q+ 1, 1+3*q^22+q^21+q^23+10*q+q^55+q^45+q^37+2*q^29+36*q^2+57*q^3+42*q^4+21*q^5+11 *q^6+7*q^7+10*q^8+4*q^9+q^10+5*q^11+6*q^12+q^13+q^15+4*q^16+q^28+3*q^17+q^36, 1 +4*q^22+q^21+3*q^23+11*q+q^30+q^55+q^45+2*q^37+3*q^29+q^46+45*q^2+85*q^3+78*q^4 +42*q^5+22*q^6+12*q^7+15*q^8+10*q^9+2*q^10+6*q^11+10*q^12+4*q^13+q^15+5*q^16+q^ 66+q^28+6*q^17+q^18+q^36, 1+5*q^22+q^24+q^21+6*q^23+12*q+3*q^30+q^55+q^45+3*q^ 37+4*q^29+2*q^46+q^38+q^56+55*q^2+121*q^3+135*q^4+84*q^5+43*q^6+23*q^7+22*q^8+ 20*q^9+6*q^10+7*q^11+15*q^12+10*q^13+q^14+q^15+6*q^16+q^66+q^78+q^28+10*q^17+4* q^18+q^36, 1+6*q^22+4*q^24+q^21+10*q^23+13*q+6*q^30+q^55+q^91+q^45+4*q^37+q^67+ 5*q^29+3*q^46+3*q^38+2*q^56+66*q^2+166*q^3+220*q^4+162*q^5+85*q^6+45*q^7+34*q^8 +35*q^9+16*q^10+9*q^11+21*q^12+20*q^13+5*q^14+q^15+7*q^16+q^66+q^78+q^28+15*q^ 17+10*q^18+q^19+q^31+q^36+q^47, 1+q^25+7*q^22+10*q^24+q^21+15*q^23+14*q+q^57+10 *q^30+q^55+q^91+q^45+5*q^37+2*q^67+6*q^29+4*q^46+q^39+q^79+6*q^38+3*q^56+78*q^2 +221*q^3+341*q^4+297*q^5+169*q^6+88*q^7+57*q^8+57*q^9+36*q^10+15*q^11+28*q^12+ 35*q^13+15*q^14+2*q^15+8*q^16+q^66+q^78+q^28+21*q^17+20*q^18+5*q^19+4*q^31+q^ 105+q^36+3*q^47] The number of permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596] The number of EVEN permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 3, 7, 9, 17, 26, 44, 71, 116, 188, 305, 493, 799] The number of ODD permutations avoiding, {[2, 3, 4, 1], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 4, 5, 11, 16, 28, 44, 72, 116, 188, 304, 493, 797] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 3, 7, 11, 16, 26, 44, 72, 116, 188, 305, 493, 797] ODD:, [0, 1, 3, 4, 5, 9, 17, 28, 44, 71, 116, 188, 304, 493, 799] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 4.000000000, 7.250000000, 11.55000000, 16.90909091, 23.33333333, 30.81818182, 39.35664336, 48.93965517, 59.55851064, 71.20525452, 83.87322515, 97.55701754] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.851640199, 2.680951322, 3.542245051, 4.378761665, 5.163977793, 5.874725244, 6.501646479, 7.041488862, 7.497289005, 7.875399039, 8.184309008, 8.433278283] The centralized moments are Second: , [0., 0.250000, 1.18750, 3.42857, 7.18750, 12.5475, 19.1736, 26.6667, 34.5124, 42.2714, 49.5826, 56.2093, 62.0219, 66.9829, 71.1202] Skewness: , [Float(undefined), 0., -0.6520236645, -1.215141890, -1.616884844, -1.969105663, -2.303639139, -2.632275458, -2.960465612, -3.288519849, -3.614986702, -3.936929886, -4.250633677, -4.552116783, -4.837518549] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 3.354205922, 4.794933233, 6.410896614, 8.291748219, 10.47695818, 13.01916303, 15.94524500, 19.27238872, 22.99621320, 27.09219964, 31.51463654, 36.19612519] end of this data For the equivalence class of patterns, { {[1, 2, 3, 4], [1, 3, 2], [2, 1, 3]}, {[4, 3, 2, 1], [2, 3, 1], [3, 1, 2]}} the member , {[1, 2, 3, 4], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[0, 3]}, {}], [[2, 1], {[0, 0, 1], [0, 3, 0]}, {1}], [[1, 2], {[0, 1, 0], [0, 0, 2]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 2*q^3+q^4+3*q^5+q^6, 2*q^6+3*q^7+3*q^8+4*q^9+q^10, q^9+6* q^11+5*q^12+6*q^13+5*q^14+q^15, 3*q^15+3*q^16+12*q^17+9*q^18+10*q^19+6*q^20+q^ 21, 3*q^21+6*q^22+12*q^23+21*q^24+16*q^25+15*q^26+7*q^27+q^28, q^27+12*q^29+14* q^30+30*q^31+35*q^32+27*q^33+21*q^34+8*q^35+q^36, 4*q^36+6*q^37+30*q^38+35*q^39 +61*q^40+57*q^41+43*q^42+28*q^43+9*q^44+q^45, 4*q^45+10*q^46+30*q^47+65*q^48+81 *q^49+111*q^50+91*q^51+65*q^52+36*q^53+10*q^54+q^55, q^54+20*q^56+30*q^57+90*q^ 58+135*q^59+169*q^60+189*q^61+142*q^62+94*q^63+45*q^64+11*q^65+q^66, 5*q^66+10* q^67+60*q^68+95*q^69+216*q^70+273*q^71+323*q^72+308*q^73+216*q^74+131*q^75+55*q ^76+12*q^77+q^78, 5*q^78+15*q^79+60*q^80+155*q^81+266*q^82+462*q^83+533*q^84+ 577*q^85+486*q^86+320*q^87+177*q^88+66*q^89+13*q^90+q^91, q^90+30*q^92+55*q^93+ 210*q^94+385*q^95+651*q^96+924*q^97+998*q^98+979*q^99+747*q^100+462*q^101+233*q ^102+78*q^103+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, 2*q^3+q^2+3*q+1, 2*q^4+3*q^3+3*q^2+4*q+1, q^6+6*q^4+5*q^3+6 *q^2+5*q+1, 3*q^6+3*q^5+12*q^4+9*q^3+10*q^2+6*q+1, 3*q^7+6*q^6+12*q^5+21*q^4+16 *q^3+15*q^2+7*q+1, q^9+12*q^7+14*q^6+30*q^5+35*q^4+27*q^3+21*q^2+8*q+1, 4*q^9+6 *q^8+30*q^7+35*q^6+61*q^5+57*q^4+43*q^3+28*q^2+9*q+1, 4*q^10+10*q^9+30*q^8+65*q ^7+81*q^6+111*q^5+91*q^4+65*q^3+36*q^2+10*q+1, q^12+20*q^10+30*q^9+90*q^8+135*q ^7+169*q^6+189*q^5+142*q^4+94*q^3+45*q^2+11*q+1, 5*q^12+10*q^11+60*q^10+95*q^9+ 216*q^8+273*q^7+323*q^6+308*q^5+216*q^4+131*q^3+55*q^2+12*q+1, 5*q^13+15*q^12+ 60*q^11+155*q^10+266*q^9+462*q^8+533*q^7+577*q^6+486*q^5+320*q^4+177*q^3+66*q^2 +13*q+1, q^15+30*q^13+55*q^12+210*q^11+385*q^10+651*q^9+924*q^8+998*q^7+979*q^6 +747*q^5+462*q^4+233*q^3+78*q^2+14*q+1] The number of permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 2, 6, 10, 18, 43, 71, 147, 261, 468, 876, 1540, 2884] The number of ODD permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 5, 7, 14, 26, 38, 78, 127, 243, 459, 829, 1596, 2884] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 2, 6, 14, 26, 43, 71, 127, 243, 468, 876, 1596, 2884] ODD:, [0, 1, 3, 5, 7, 10, 18, 38, 78, 147, 261, 459, 829, 1540, 2884] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 4.428571429, 7.923076923, 12.41666667, 17.95454545, 24.46913580, 31.98657718, 40.50729927, 50.02579365, 60.54476807, 72.06392962, 84.58290816, 98.10194175] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.049781318, 1.206491318, 1.381926996, 1.460923516, 1.572027233, 1.674869832, 1.763876580, 1.853067321, 1.937505855, 2.017813874, 2.095491689, 2.170284526] The centralized moments are Second: , [0., 0.250000, 1.18750, 1.10204, 1.45562, 1.90972, 2.13430, 2.47127, 2.80519, 3.11126, 3.43386, 3.75393, 4.07157, 4.39109, 4.71013] Skewness: , [Float(undefined), 0., -0.6520236645, -0.1814416936, -0.1150785265, -0.3004177094, -0.1398726377, -0.1337484775, -0.1416588755, -0.1118373961, -0.1050576143, -0.09849775275, -0.08986488225, -0.08447538118, -0.07955143698] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 1.738669105, 1.942959467, 2.710506550, 2.421520820, 2.508131659, 2.654347264, 2.629324478, 2.670566234, 2.706501978, 2.721291149, 2.742345253, 2.759765793] end of this data For the equivalence class of patterns, {{[3, 4, 2, 1], [1, 3, 2], [2, 1, 3]}, {[4, 3, 1, 2], [1, 3, 2], [2, 1, 3]}, {[1, 2, 4, 3], [2, 3, 1], [3, 1, 2]}, {[2, 1, 3, 4], [2, 3, 1], [3, 1, 2]}} the member , {[3, 4, 2, 1], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 1+2*q^3+q^4+2*q^5+q^6, 1+2*q^4+2*q^6+2*q^7+q^8+2*q^9+q^10 , 1+2*q^5+2*q^8+3*q^9+2*q^11+2*q^12+q^13+2*q^14+q^15, 1+2*q^6+2*q^10+2*q^11+2*q ^12+2*q^14+3*q^15+2*q^17+2*q^18+q^19+2*q^20+q^21, 1+2*q^7+2*q^12+2*q^13+2*q^15+ q^16+2*q^17+2*q^18+2*q^19+2*q^21+3*q^22+2*q^24+2*q^25+q^26+2*q^27+q^28, 1+2*q^8 +2*q^14+2*q^15+2*q^18+4*q^20+2*q^21+2*q^23+q^24+2*q^25+2*q^26+2*q^27+2*q^29+3*q ^30+2*q^32+2*q^33+q^34+2*q^35+q^36, 1+2*q^9+2*q^16+2*q^17+2*q^21+2*q^23+4*q^24+ q^25+2*q^27+4*q^29+2*q^30+2*q^32+q^33+2*q^34+2*q^35+2*q^36+2*q^38+3*q^39+2*q^41 +2*q^42+q^43+2*q^44+q^45, 1+2*q^10+2*q^18+2*q^19+2*q^24+2*q^26+2*q^27+2*q^28+2* q^30+2*q^31+2*q^33+4*q^34+q^35+2*q^37+4*q^39+2*q^40+2*q^42+q^43+2*q^44+2*q^45+2 *q^46+2*q^48+3*q^49+2*q^51+2*q^52+q^53+2*q^54+q^55, 1+2*q^11+2*q^20+2*q^21+2*q^ 27+2*q^29+2*q^30+2*q^32+4*q^35+q^36+2*q^37+2*q^38+2*q^39+2*q^41+2*q^42+2*q^44+4 *q^45+q^46+2*q^48+4*q^50+2*q^51+2*q^53+q^54+2*q^55+2*q^56+2*q^57+2*q^59+3*q^60+ 2*q^62+2*q^63+q^64+2*q^65+q^66, 1+2*q^12+2*q^22+2*q^23+2*q^30+2*q^32+2*q^33+2*q ^36+2*q^39+2*q^40+2*q^41+4*q^42+2*q^44+4*q^47+q^48+2*q^49+2*q^50+2*q^51+2*q^53+ 2*q^54+2*q^56+4*q^57+q^58+2*q^60+4*q^62+2*q^63+2*q^65+q^66+2*q^67+2*q^68+2*q^69 +2*q^71+3*q^72+2*q^74+2*q^75+q^76+2*q^77+q^78, 1+2*q^13+2*q^24+2*q^25+2*q^33+2* q^35+2*q^36+2*q^40+2*q^43+4*q^45+2*q^46+2*q^48+3*q^49+2*q^52+2*q^53+2*q^54+4*q^ 55+2*q^57+4*q^60+q^61+2*q^62+2*q^63+2*q^64+2*q^66+2*q^67+2*q^69+4*q^70+q^71+2*q ^73+4*q^75+2*q^76+2*q^78+q^79+2*q^80+2*q^81+2*q^82+2*q^84+3*q^85+2*q^87+2*q^88+ q^89+2*q^90+q^91, 1+2*q^14+2*q^26+2*q^27+2*q^36+2*q^38+2*q^39+2*q^44+2*q^47+2*q ^49+4*q^50+4*q^54+2*q^56+2*q^57+4*q^59+2*q^60+2*q^62+3*q^63+2*q^66+2*q^67+2*q^ 68+4*q^69+2*q^71+4*q^74+q^75+2*q^76+2*q^77+2*q^78+2*q^80+2*q^81+2*q^83+4*q^84+q ^85+2*q^87+4*q^89+2*q^90+2*q^92+q^93+2*q^94+2*q^95+2*q^96+2*q^98+3*q^99+2*q^101 +2*q^102+q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, q^6+2*q^3+q^2+2*q+1, q^10+2*q^6+2*q^4+2*q^3+q^2+2*q+1, q^15 +2*q^10+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^21+2*q^15+2*q^11+2*q^10+2*q^9+2*q^ 7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^28+2*q^21+2*q^16+2*q^15+2*q^13+q^12+2*q^11+2*q ^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^36+2*q^28+2*q^22+2*q^21+2*q^18+4 *q^16+2*q^15+2*q^13+q^12+2*q^11+2*q^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^45+2*q^36+2*q^29+2*q^28+2*q^24+2*q^22+4*q^21+q^20+2*q^18+4*q^16+2*q^15+2*q^13 +q^12+2*q^11+2*q^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, 1+2*q^25+2*q^22+2* q^24+4*q^21+2*q+q^55+2*q^45+2*q^37+2*q^29+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q ^10+2*q^11+q^12+2*q^13+2*q^15+4*q^16+2*q^28+2*q^18+q^20+2*q^27+2*q^31+2*q^36, 1 +2*q^25+2*q^22+2*q^24+4*q^21+2*q+q^30+2*q^55+2*q^45+2*q^37+2*q^29+2*q^46+2*q^39 +2*q^34+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+2*q^15+4*q^ 16+q^66+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^36, 1+2*q^25+2*q^42+2*q^22+2*q^24+ 4*q^21+2*q+q^30+2*q^55+2*q^45+2*q^37+2*q^29+2*q^46+2*q^39+2*q^34+2*q^38+2*q^56+ q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+2*q^15+4*q^16+2*q^ 66+q^78+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^48+4*q^36, 1+2*q^25+3*q^42+2*q^22+ 2*q^24+4*q^21+2*q+2*q^51+q^30+2*q^58+2*q^55+q^91+2*q^45+2*q^37+2*q^67+2*q^29+4* q^46+2*q^39+2*q^34+2*q^38+2*q^56+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^ 11+q^12+2*q^13+2*q^15+4*q^16+2*q^66+2*q^78+2*q^43+2*q^28+2*q^18+q^20+2*q^27+4*q ^31+2*q^48+4*q^36, 1+2*q^25+3*q^42+2*q^22+2*q^24+4*q^21+2*q+4*q^51+q^30+2*q^58+ 4*q^55+2*q^91+2*q^45+2*q^37+2*q^67+2*q^29+4*q^46+2*q^69+2*q^39+2*q^61+2*q^34+2* q^79+2*q^38+2*q^56+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+ 2*q^15+4*q^16+2*q^66+2*q^78+2*q^43+2*q^49+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^ 48+q^105+4*q^36] The number of permutations avoiding, {[3, 4, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[3, 4, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 3, 7, 7, 13, 13, 21, 21, 31, 31, 43, 43, 57] The number of ODD permutations avoiding, {[3, 4, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 36, 36, 49, 49] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 3, 7, 9, 9, 13, 21, 25, 25, 31, 43, 49, 49] ODD:, [0, 1, 3, 4, 4, 7, 13, 16, 16, 21, 31, 36, 36, 43, 57] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.714285714, 6.363636364, 9.687500000, 13.68181818, 18.34482759, 23.67567568, 29.67391304, 36.33928571, 43.67164179, 51.67088608, 60.33695652, 69.66981132] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.829464068, 2.739367122, 3.835993189, 5.129117977, 6.624280635, 8.324675318, 10.23220480, 12.34804320, 14.67293992, 17.20738841, 19.95172281, 22.90617520] The centralized moments are Second: , [0., 0.250000, 1.18750, 3.34694, 7.50413, 14.7148, 26.3079, 43.8811, 69.3002, 104.698, 152.474, 215.295, 296.094, 398.071, 524.693] Skewness: , [Float(undefined), 0., -0.6520236645, -0.8342005971, -0.8605037324, -0.8302107402, -0.7841469197, -0.7380571523, -0.6971033027, -0.6623464599, -0.6333645786, -0.6093175947, -0.5893441536, -0.5726826445, -0.5587046560] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 2.851721033, 3.198645144, 3.308416481, 3.303137544, 3.250254189, 3.182414078, 3.114169307, 3.051342976, 2.995710836, 2.947349341, 2.905625938, 2.869720759] end of this data For the equivalence class of patterns, { {[4, 2, 3, 1], [1, 3, 2], [2, 1, 3]}, {[1, 3, 2, 4], [2, 3, 1], [3, 1, 2]}} the member , {[4, 2, 3, 1], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 1+2*q^3+q^4+2*q^5+q^6, 1+2*q^4+2*q^6+2*q^7+q^8+2*q^9+q^10 , 1+2*q^5+2*q^8+3*q^9+2*q^11+2*q^12+q^13+2*q^14+q^15, 1+2*q^6+2*q^10+2*q^11+2*q ^12+2*q^14+3*q^15+2*q^17+2*q^18+q^19+2*q^20+q^21, 1+2*q^7+2*q^12+2*q^13+2*q^15+ q^16+2*q^17+2*q^18+2*q^19+2*q^21+3*q^22+2*q^24+2*q^25+q^26+2*q^27+q^28, 1+2*q^8 +2*q^14+2*q^15+2*q^18+4*q^20+2*q^21+2*q^23+q^24+2*q^25+2*q^26+2*q^27+2*q^29+3*q ^30+2*q^32+2*q^33+q^34+2*q^35+q^36, 1+2*q^9+2*q^16+2*q^17+2*q^21+2*q^23+4*q^24+ q^25+2*q^27+4*q^29+2*q^30+2*q^32+q^33+2*q^34+2*q^35+2*q^36+2*q^38+3*q^39+2*q^41 +2*q^42+q^43+2*q^44+q^45, 1+2*q^10+2*q^18+2*q^19+2*q^24+2*q^26+2*q^27+2*q^28+2* q^30+2*q^31+2*q^33+4*q^34+q^35+2*q^37+4*q^39+2*q^40+2*q^42+q^43+2*q^44+2*q^45+2 *q^46+2*q^48+3*q^49+2*q^51+2*q^52+q^53+2*q^54+q^55, 1+2*q^11+2*q^20+2*q^21+2*q^ 27+2*q^29+2*q^30+2*q^32+4*q^35+q^36+2*q^37+2*q^38+2*q^39+2*q^41+2*q^42+2*q^44+4 *q^45+q^46+2*q^48+4*q^50+2*q^51+2*q^53+q^54+2*q^55+2*q^56+2*q^57+2*q^59+3*q^60+ 2*q^62+2*q^63+q^64+2*q^65+q^66, 1+2*q^12+2*q^22+2*q^23+2*q^30+2*q^32+2*q^33+2*q ^36+2*q^39+2*q^40+2*q^41+4*q^42+2*q^44+4*q^47+q^48+2*q^49+2*q^50+2*q^51+2*q^53+ 2*q^54+2*q^56+4*q^57+q^58+2*q^60+4*q^62+2*q^63+2*q^65+q^66+2*q^67+2*q^68+2*q^69 +2*q^71+3*q^72+2*q^74+2*q^75+q^76+2*q^77+q^78, 1+2*q^13+2*q^24+2*q^25+2*q^33+2* q^35+2*q^36+2*q^40+2*q^43+4*q^45+2*q^46+2*q^48+3*q^49+2*q^52+2*q^53+2*q^54+4*q^ 55+2*q^57+4*q^60+q^61+2*q^62+2*q^63+2*q^64+2*q^66+2*q^67+2*q^69+4*q^70+q^71+2*q ^73+4*q^75+2*q^76+2*q^78+q^79+2*q^80+2*q^81+2*q^82+2*q^84+3*q^85+2*q^87+2*q^88+ q^89+2*q^90+q^91, 1+2*q^14+2*q^26+2*q^27+2*q^36+2*q^38+2*q^39+2*q^44+2*q^47+2*q ^49+4*q^50+4*q^54+2*q^56+2*q^57+4*q^59+2*q^60+2*q^62+3*q^63+2*q^66+2*q^67+2*q^ 68+4*q^69+2*q^71+4*q^74+q^75+2*q^76+2*q^77+2*q^78+2*q^80+2*q^81+2*q^83+4*q^84+q ^85+2*q^87+4*q^89+2*q^90+2*q^92+q^93+2*q^94+2*q^95+2*q^96+2*q^98+3*q^99+2*q^101 +2*q^102+q^103+2*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, q^6+2*q^3+q^2+2*q+1, q^10+2*q^6+2*q^4+2*q^3+q^2+2*q+1, q^15 +2*q^10+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^21+2*q^15+2*q^11+2*q^10+2*q^9+2*q^ 7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^28+2*q^21+2*q^16+2*q^15+2*q^13+q^12+2*q^11+2*q ^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^36+2*q^28+2*q^22+2*q^21+2*q^18+4 *q^16+2*q^15+2*q^13+q^12+2*q^11+2*q^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, q^45+2*q^36+2*q^29+2*q^28+2*q^24+2*q^22+4*q^21+q^20+2*q^18+4*q^16+2*q^15+2*q^13 +q^12+2*q^11+2*q^10+2*q^9+2*q^7+3*q^6+2*q^4+2*q^3+q^2+2*q+1, 1+2*q^25+2*q^22+2* q^24+4*q^21+2*q+q^55+2*q^45+2*q^37+2*q^29+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q ^10+2*q^11+q^12+2*q^13+2*q^15+4*q^16+2*q^28+2*q^18+q^20+2*q^27+2*q^31+2*q^36, 1 +2*q^25+2*q^22+2*q^24+4*q^21+2*q+q^30+2*q^55+2*q^45+2*q^37+2*q^29+2*q^46+2*q^39 +2*q^34+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+2*q^15+4*q^ 16+q^66+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^36, 1+2*q^25+2*q^42+2*q^22+2*q^24+ 4*q^21+2*q+q^30+2*q^55+2*q^45+2*q^37+2*q^29+2*q^46+2*q^39+2*q^34+2*q^38+2*q^56+ q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+2*q^15+4*q^16+2*q^ 66+q^78+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^48+4*q^36, 1+2*q^25+3*q^42+2*q^22+ 2*q^24+4*q^21+2*q+2*q^51+q^30+2*q^58+2*q^55+q^91+2*q^45+2*q^37+2*q^67+2*q^29+4* q^46+2*q^39+2*q^34+2*q^38+2*q^56+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^ 11+q^12+2*q^13+2*q^15+4*q^16+2*q^66+2*q^78+2*q^43+2*q^28+2*q^18+q^20+2*q^27+4*q ^31+2*q^48+4*q^36, 1+2*q^25+3*q^42+2*q^22+2*q^24+4*q^21+2*q+4*q^51+q^30+2*q^58+ 4*q^55+2*q^91+2*q^45+2*q^37+2*q^67+2*q^29+4*q^46+2*q^69+2*q^39+2*q^61+2*q^34+2* q^79+2*q^38+2*q^56+q^2+2*q^3+2*q^4+3*q^6+2*q^7+2*q^9+2*q^10+2*q^11+q^12+2*q^13+ 2*q^15+4*q^16+2*q^66+2*q^78+2*q^43+2*q^49+2*q^28+2*q^18+q^20+2*q^27+4*q^31+2*q^ 48+q^105+4*q^36] The number of permutations avoiding, {[4, 2, 3, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[4, 2, 3, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 3, 7, 7, 13, 13, 21, 21, 31, 31, 43, 43, 57] The number of ODD permutations avoiding, {[4, 2, 3, 1], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25, 36, 36, 49, 49] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 3, 7, 9, 9, 13, 21, 25, 25, 31, 43, 49, 49] ODD:, [0, 1, 3, 4, 4, 7, 13, 16, 16, 21, 31, 36, 36, 43, 57] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.714285714, 6.363636364, 9.687500000, 13.68181818, 18.34482759, 23.67567568, 29.67391304, 36.33928571, 43.67164179, 51.67088608, 60.33695652, 69.66981132] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.829464068, 2.739367122, 3.835993189, 5.129117977, 6.624280635, 8.324675318, 10.23220480, 12.34804320, 14.67293992, 17.20738841, 19.95172281, 22.90617520] The centralized moments are Second: , [0., 0.250000, 1.18750, 3.34694, 7.50413, 14.7148, 26.3079, 43.8811, 69.3002, 104.698, 152.474, 215.295, 296.094, 398.071, 524.693] Skewness: , [Float(undefined), 0., -0.6520236645, -0.8342005971, -0.8605037324, -0.8302107402, -0.7841469197, -0.7380571523, -0.6971033027, -0.6623464599, -0.6333645786, -0.6093175947, -0.5893441536, -0.5726826445, -0.5587046560] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 2.851721033, 3.198645144, 3.308416481, 3.303137544, 3.250254189, 3.182414078, 3.114169307, 3.051342976, 2.995710836, 2.947349341, 2.905625938, 2.869720759] end of this data For the equivalence class of patterns, { {[4, 3, 2, 1], [1, 3, 2], [2, 1, 3]}, {[1, 2, 3, 4], [2, 3, 1], [3, 1, 2]}} the member , {[4, 3, 2, 1], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [[2, 1], {[0, 0, 1]}, {}], [[3, 2, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 1+2*q^3+q^4+3*q^5, 1+2*q^4+2*q^6+3*q^7+3*q^8, 1+2*q^5+2*q ^8+4*q^9+6*q^11+q^12, 1+2*q^6+2*q^10+3*q^11+2*q^12+6*q^14+3*q^15+3*q^16, 1+2*q^ 7+2*q^12+3*q^13+2*q^15+q^16+6*q^17+6*q^19+3*q^20+3*q^21, 1+2*q^8+2*q^14+3*q^15+ 2*q^18+8*q^20+6*q^23+6*q^24+6*q^26+q^27, 1+2*q^9+2*q^16+3*q^17+2*q^21+6*q^23+2* q^24+q^25+6*q^27+3*q^28+6*q^29+6*q^31+3*q^32+3*q^33, 1+2*q^10+2*q^18+3*q^19+2*q ^24+6*q^26+2*q^28+2*q^30+6*q^31+3*q^32+6*q^34+3*q^35+6*q^36+6*q^38+3*q^39+3*q^ 40, 1+2*q^11+2*q^20+3*q^21+2*q^27+6*q^29+2*q^32+8*q^35+4*q^36+6*q^39+12*q^41+6* q^44+6*q^45+6*q^47+q^48, 1+2*q^12+2*q^22+3*q^23+2*q^30+6*q^32+2*q^36+6*q^39+5*q ^40+2*q^42+6*q^44+6*q^46+6*q^47+3*q^48+6*q^50+3*q^51+6*q^52+6*q^54+3*q^55+3*q^ 56, 1+2*q^13+2*q^24+3*q^25+2*q^33+6*q^35+2*q^40+6*q^43+3*q^44+2*q^45+2*q^48+7*q ^49+6*q^51+6*q^53+6*q^55+6*q^56+3*q^57+6*q^59+3*q^60+6*q^61+6*q^63+3*q^64+3*q^ 65, 1+2*q^14+2*q^26+3*q^27+2*q^36+6*q^38+2*q^44+6*q^47+3*q^48+2*q^50+8*q^54+8*q ^56+6*q^59+12*q^62+6*q^63+6*q^66+12*q^68+6*q^71+6*q^72+6*q^74+q^75] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, q*(q^5+2*q^2+q+3), q^2*(q^8+2*q^4+2*q^2+3*q+3), q^3*(q^12+2 *q^7+2*q^4+4*q^3+6*q+1), q^5*(q^16+2*q^10+2*q^6+3*q^5+2*q^4+6*q^2+3*q+3), q^7*( q^21+2*q^14+2*q^9+3*q^8+2*q^6+q^5+6*q^4+6*q^2+3*q+3), q^9*(q^27+2*q^19+2*q^13+3 *q^12+2*q^9+8*q^7+6*q^4+6*q^3+6*q+1), q^12*(q^33+2*q^24+2*q^17+3*q^16+2*q^12+6* q^10+2*q^9+q^8+6*q^6+3*q^5+6*q^4+6*q^2+3*q+3), q^15*(q^40+2*q^30+2*q^22+3*q^21+ 2*q^16+6*q^14+2*q^12+2*q^10+6*q^9+3*q^8+6*q^6+3*q^5+6*q^4+6*q^2+3*q+3), q^18*(q ^48+2*q^37+2*q^28+3*q^27+2*q^21+6*q^19+2*q^16+8*q^13+4*q^12+6*q^9+12*q^7+6*q^4+ 6*q^3+6*q+1), q^22*(q^56+2*q^44+2*q^34+3*q^33+2*q^26+6*q^24+2*q^20+6*q^17+5*q^ 16+2*q^14+6*q^12+6*q^10+6*q^9+3*q^8+6*q^6+3*q^5+6*q^4+6*q^2+3*q+3), q^26*(q^65+ 2*q^52+2*q^41+3*q^40+2*q^32+6*q^30+2*q^25+6*q^22+3*q^21+2*q^20+2*q^17+7*q^16+6* q^14+6*q^12+6*q^10+6*q^9+3*q^8+6*q^6+3*q^5+6*q^4+6*q^2+3*q+3), q^30*(q^75+2*q^ 61+2*q^49+3*q^48+2*q^39+6*q^37+2*q^31+6*q^28+3*q^27+2*q^25+8*q^21+8*q^19+6*q^16 +12*q^13+6*q^12+6*q^9+12*q^7+6*q^4+6*q^3+6*q+1)] The number of permutations avoiding, {[4, 3, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[4, 3, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 2, 8, 4, 16, 7, 27, 11, 41, 16, 58, 22, 78] The number of ODD permutations avoiding, {[4, 3, 2, 1], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 5, 3, 12, 6, 22, 10, 35, 15, 51, 21, 70, 28] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 2, 8, 12, 6, 7, 27, 35, 15, 16, 58, 70, 28] ODD:, [0, 1, 3, 5, 3, 4, 16, 22, 10, 11, 41, 51, 21, 22, 78] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.571428571, 5.909090909, 8.750000000, 12.09090909, 15.93103448, 20.27027027, 25.10869565, 30.44642857, 36.28358209, 42.62025316, 49.45652174, 56.79245283] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.678191446, 2.314167646, 3.031088914, 3.848365544, 4.777324562, 5.824684884, 6.994495540, 8.289234937, 9.710443630, 11.25909342, 12.93580558, 14.74098188] The centralized moments are Second: , [0., 0.250000, 1.18750, 2.81633, 5.35537, 9.18750, 14.8099, 22.8228, 33.9270, 48.9230, 68.7114, 94.2927, 126.767, 167.335, 217.297] Skewness: , [Float(undefined), 0., -0.6520236645, -1.121439784, -1.385846254, -1.511544974, -1.552698414, -1.547492754, -1.519824854, -1.483334708, -1.445201487, -1.408864533, -1.375805289, -1.346411824, -1.320621162] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 3.159707711, 4.144109149, 4.830082281, 5.214533519, 5.374287602, 5.391900224, 5.330191596, 5.230006049, 5.115557366, 5.000245089, 4.890824940, 4.790377950] end of this data For the equivalence class of patterns, { {[2, 1, 4, 3], [3, 2, 1], [1, 2, 3]}, {[3, 4, 1, 2], [3, 2, 1], [1, 2, 3]}} the member , {[2, 1, 4, 3], [3, 2, 1], [1, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[0, 0, 1], [0, 2, 0], [2, 1, 0], [3, 0, 0]}, {1}], [[1], {[3, 0], [0, 3]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 0, 2]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+2*q^2, 2*q^3+q^4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] with the reverse patterns and complement patterns having distributions [1, 1+q, 2*q*(1+q), q^2*(1+2*q), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [1, 2, 3]}, is given by [1, 2, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of EVEN permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [1, 2, 3]}, is given by [1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of ODD permutations avoiding, {[2, 1, 4, 3], [3, 2, 1], [1, 2, 3]}, is given by [0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ODD:, [0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.333333333, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The standard deviation for each n is given by [0., 0.5000000000, 0.5000000000, 0.4714045206, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The centralized moments are Second: , [0., 0.250000, 0.250000, 0.222222, FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] FAIL[3] FAIL[3] Skewness: , [Float(undefined), 0., 0., 0.7070975887, ----------, ----------, 3/2 3/2 FAIL[2] FAIL[2] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] ----------, ----------, ----------, ----------, ----------, ----------, 3/2 3/2 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[3] FAIL[3] FAIL[3] ----------, ----------, ----------] 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[4] Kurtosis: , [Float(undefined), 1.000000000, 1.000000000, 1.499718000, --------, 2 FAIL[2] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------, --------, --------, --------, --------, 2 2 2 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------] 2 2 2 FAIL[2] FAIL[2] FAIL[2] end of this data For the equivalence class of patterns, {{[4, 3, 2, 1], [1, 2, 3], [2, 3, 1]}, {[4, 3, 2, 1], [1, 2, 3], [3, 1, 2]}, {[1, 2, 3, 4], [3, 2, 1], [1, 3, 2]}, {[1, 2, 3, 4], [3, 2, 1], [2, 1, 3]}} the member , {[4, 3, 2, 1], [1, 2, 3], [2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 1], [2, 1, 0], [3, 0, 0], [0, 3, 0], [2, 0, 1]}, {}], [[1, 2], {[1, 0, 0], [0, 0, 1], [0, 3, 0]}, {2}], [[2, 1, 3], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[3, 1, 2], {[0, 2, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 0, 3, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1], [1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2+q^3, q^2+2*q^3+2*q^4+q^5, 2*q^4+q^6, q^6, 0, 0, 0, 0, 0, 0, 0, 0, 0] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+2*q^2, q*(1+2*q+2*q^2+q^3), q^4*(2*q^2+1), q^9, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of permutations avoiding, {[4, 3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [1, 2, 4, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of EVEN permutations avoiding, {[4, 3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [1, 1, 1, 3, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of ODD permutations avoiding, {[4, 3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [0, 1, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ODD:, [0, 1, 1, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.500000000, 4.666666667, 6., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The standard deviation for each n is given by [0., 0.5000000000, 0.8291561975, 0.9574271080, 0.9428090414, 0., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The centralized moments are Second: , [0., 0.250000, 0.687500, 0.916667, 0.888889, 0., FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] Skewness: , [Float(undefined), 0., 0.4933822002, 0., 0.7071154877, FAIL[3] FAIL[3] FAIL[3] FAIL[3] Float(undefined), ----------, ----------, ----------, ----------, 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] ----------, ----------, ----------, ----------, ----------] 3/2 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] Kurtosis: , [Float(undefined), 1.000000000, 1.628096529, 2.057889412, FAIL[4] FAIL[4] FAIL[4] FAIL[4] 1.500056344, Float(undefined), --------, --------, --------, --------, 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------, --------, --------] 2 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] end of this data For the equivalence class of patterns, { {[3, 4, 1, 2], [1, 3, 2], [2, 1, 3]}, {[2, 1, 4, 3], [2, 3, 1], [3, 1, 2]}} the member , {[3, 4, 1, 2], [1, 3, 2], [2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q^2+q^3, 1+2*q^3+3*q^5+q^6, 1+2*q^4+3*q^7+4*q^9+q^10, 1+2*q^5+3*q^ 9+4*q^12+5*q^14+q^15, 1+2*q^6+3*q^11+4*q^15+5*q^18+6*q^20+q^21, 1+2*q^7+3*q^13+ 4*q^18+5*q^22+6*q^25+7*q^27+q^28, 1+2*q^8+3*q^15+4*q^21+5*q^26+6*q^30+7*q^33+8* q^35+q^36, 1+2*q^9+3*q^17+4*q^24+5*q^30+6*q^35+7*q^39+8*q^42+9*q^44+q^45, 1+2*q ^10+3*q^19+4*q^27+5*q^34+6*q^40+7*q^45+8*q^49+9*q^52+10*q^54+q^55, 1+2*q^11+3*q ^21+4*q^30+5*q^38+6*q^45+7*q^51+8*q^56+9*q^60+10*q^63+11*q^65+q^66, 1+2*q^12+3* q^23+4*q^33+5*q^42+6*q^50+7*q^57+8*q^63+9*q^68+10*q^72+11*q^75+12*q^77+q^78, 1+ 2*q^13+3*q^25+4*q^36+5*q^46+6*q^55+7*q^63+8*q^70+9*q^76+10*q^81+11*q^85+12*q^88 +13*q^90+q^91, 1+2*q^14+3*q^27+4*q^39+5*q^50+6*q^60+7*q^69+8*q^77+9*q^84+10*q^ 90+11*q^95+12*q^99+13*q^102+14*q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+2*q+q^3, 1+3*q+2*q^3+q^6, 1+4*q+3*q^3+2*q^6+q^10, 1+5*q+4*q^3+3*q^6+ 2*q^10+q^15, 1+6*q+5*q^3+4*q^6+3*q^10+2*q^15+q^21, 1+7*q+6*q^3+5*q^6+4*q^10+3*q ^15+2*q^21+q^28, 1+8*q+7*q^3+6*q^6+5*q^10+4*q^15+3*q^21+2*q^28+q^36, 1+9*q+8*q^ 3+7*q^6+6*q^10+5*q^15+4*q^21+3*q^28+2*q^36+q^45, 1+10*q+9*q^3+8*q^6+7*q^10+6*q^ 15+5*q^21+4*q^28+3*q^36+2*q^45+q^55, 1+11*q+10*q^3+9*q^6+8*q^10+7*q^15+6*q^21+5 *q^28+4*q^36+3*q^45+2*q^55+q^66, 1+12*q+11*q^3+10*q^6+9*q^10+8*q^15+7*q^21+6*q^ 28+5*q^36+4*q^45+3*q^55+2*q^66+q^78, 1+13*q+12*q^3+11*q^6+10*q^10+9*q^15+8*q^21 +7*q^28+6*q^36+5*q^45+4*q^55+3*q^66+2*q^78+q^91, q^105+2*q^91+3*q^78+4*q^66+5*q ^55+6*q^45+7*q^36+8*q^28+9*q^21+10*q^15+11*q^10+12*q^6+13*q^3+14*q+1] The number of permutations avoiding, {[3, 4, 1, 2], [1, 3, 2], [2, 1, 3]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[3, 4, 1, 2], [1, 3, 2], [2, 1, 3]}, is given by [1, 1, 3, 2, 4, 10, 14, 11, 15, 27, 33, 28, 34, 52, 60] The number of ODD permutations avoiding, {[3, 4, 1, 2], [1, 3, 2], [2, 1, 3]}, is given by [0, 1, 1, 5, 7, 6, 8, 18, 22, 19, 23, 39, 45, 40, 46] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 2, 4, 6, 8, 11, 15, 19, 23, 28, 34, 40, 46] ODD:, [0, 1, 3, 5, 7, 10, 14, 18, 22, 27, 33, 39, 45, 52, 60] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 3.857142857, 6.818181818, 10.62500000, 15.27272727, 20.75862069, 27.08108108, 34.23913043, 42.23214286, 51.05970149, 60.72151899, 71.21739130, 82.54716981] The standard deviation for each n is given by [0., 0.5000000000, 1.089724736, 1.884415138, 2.886274158, 4.090767041, 5.495302276, 7.098592248, 8.899979893, 10.89911202, 13.09578868, 15.48989136, 18.08134693, 20.87010869, 23.85614595] The centralized moments are Second: , [0., 0.250000, 1.18750, 3.55102, 8.33058, 16.7344, 30.1983, 50.3900, 79.2096, 118.791, 171.500, 239.937, 326.935, 435.561, 569.116] Skewness: , [Float(undefined), 0., -0.6520236645, -0.9463064582, -1.088709447, -1.170388534, -1.223824304, -1.261938767, -1.290762988, -1.313459396, -1.331871522, -1.347145195, -1.360041123, -1.371080815, -1.380666811] Kurtosis: , [Float(undefined), 1.000000000, 2.096930748, 2.816718777, 3.249217128, 3.525590967, 3.715551403, 3.853983383, 3.959477779, 4.042653278, 4.110056184, 4.165763127, 4.212665065, 4.252688872, 4.287248900] end of this data For the equivalence class of patterns, {{[1, 3, 4, 2], [3, 2, 1], [3, 1, 2]}, {[1, 4, 2, 3], [3, 2, 1], [2, 3, 1]}, {[4, 2, 1, 3], [1, 2, 3], [1, 3, 2]}, {[2, 4, 3, 1], [1, 2, 3], [2, 1, 3]}, {[4, 1, 3, 2], [1, 2, 3], [2, 1, 3]}, {[2, 3, 1, 4], [3, 2, 1], [3, 1, 2]}, {[3, 1, 2, 4], [3, 2, 1], [2, 3, 1]}, {[3, 2, 4, 1], [1, 2, 3], [1, 3, 2]}} the member , {[1, 3, 4, 2], [3, 2, 1], [3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[2, 0]}, {}], [[2, 1], {[1, 0, 0], [0, 1, 0]}, {1}], [[1, 2], {[2, 0, 0], [1, 1, 0], [0, 2, 0]}, {}], [[1, 2, 3], {[2, 0, 0, 0], [1, 0, 1, 0], [0, 0, 2, 0], [0, 1, 0, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q+q^2, 1+3*q+2*q^2+q^3, 1+4*q+4*q^2+2*q^3+q^4, 1+5*q+7*q^2+4*q^3+2 *q^4+q^5, 1+6*q+11*q^2+8*q^3+4*q^4+2*q^5+q^6, 1+7*q+16*q^2+15*q^3+8*q^4+4*q^5+2 *q^6+q^7, 1+8*q+22*q^2+26*q^3+16*q^4+8*q^5+4*q^6+2*q^7+q^8, 1+9*q+29*q^2+42*q^3 +31*q^4+16*q^5+8*q^6+4*q^7+2*q^8+q^9, 1+10*q+37*q^2+64*q^3+57*q^4+32*q^5+16*q^6 +8*q^7+4*q^8+2*q^9+q^10, 1+11*q+46*q^2+93*q^3+99*q^4+63*q^5+32*q^6+16*q^7+8*q^8 +4*q^9+2*q^10+q^11, 1+12*q+56*q^2+130*q^3+163*q^4+120*q^5+64*q^6+32*q^7+16*q^8+ 8*q^9+4*q^10+2*q^11+q^12, 1+13*q+67*q^2+176*q^3+256*q^4+219*q^5+127*q^6+64*q^7+ 32*q^8+16*q^9+8*q^10+4*q^11+2*q^12+q^13, 1+14*q+79*q^2+232*q^3+386*q^4+382*q^5+ 247*q^6+128*q^7+64*q^8+32*q^9+16*q^10+8*q^11+4*q^12+2*q^13+q^14] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+2*q+q^2), q^3*(q^3+3*q^2+2*q+1), q^6*(q^4+4*q^3+4*q^2+2*q+1), q^ 10*(q^5+5*q^4+7*q^3+4*q^2+2*q+1), q^15*(q^6+6*q^5+11*q^4+8*q^3+4*q^2+2*q+1), q^ 21*(q^7+7*q^6+16*q^5+15*q^4+8*q^3+4*q^2+2*q+1), q^28*(q^8+8*q^7+22*q^6+26*q^5+ 16*q^4+8*q^3+4*q^2+2*q+1), q^36*(q^9+9*q^8+29*q^7+42*q^6+31*q^5+16*q^4+8*q^3+4* q^2+2*q+1), q^45*(q^10+10*q^9+37*q^8+64*q^7+57*q^6+32*q^5+16*q^4+8*q^3+4*q^2+2* q+1), q^55*(q^11+11*q^10+46*q^9+93*q^8+99*q^7+63*q^6+32*q^5+16*q^4+8*q^3+4*q^2+ 2*q+1), q^66*(q^12+12*q^11+56*q^10+130*q^9+163*q^8+120*q^7+64*q^6+32*q^5+16*q^4 +8*q^3+4*q^2+2*q+1), q^78*(q^13+13*q^12+67*q^11+176*q^10+256*q^9+219*q^8+127*q^ 7+64*q^6+32*q^5+16*q^4+8*q^3+4*q^2+2*q+1), q^91*(q^14+14*q^13+79*q^12+232*q^11+ 386*q^10+382*q^9+247*q^8+128*q^7+64*q^6+32*q^5+16*q^4+8*q^3+4*q^2+2*q+1)] The number of permutations avoiding, {[1, 3, 4, 2], [3, 2, 1], [3, 1, 2]}, is given by [1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596] The number of EVEN permutations avoiding, {[1, 3, 4, 2], [3, 2, 1], [3, 1, 2]}, is given by [1, 1, 2, 3, 6, 10, 17, 27, 44, 71, 116, 188, 305, 493, 798] The number of ODD permutations avoiding, {[1, 3, 4, 2], [3, 2, 1], [3, 1, 2]}, is given by [0, 1, 2, 4, 6, 10, 16, 27, 44, 72, 116, 188, 304, 493, 798] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 3, 6, 10, 16, 27, 44, 72, 116, 188, 305, 493, 798] ODD:, [0, 1, 2, 4, 6, 10, 17, 27, 44, 71, 116, 188, 304, 493, 798] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.428571429, 1.833333333, 2.200000000, 2.545454545, 2.870370370, 3.181818182, 3.482517483, 3.775862069, 4.063829787, 4.348111658, 4.629817444, 4.909774436] The standard deviation for each n is given by [0., 0.5000000000, 0.7071067810, 0.9035079028, 1.067187373, 1.208304597, 1.327812260, 1.427848632, 1.511662650, 1.581594958, 1.640469902, 1.690451977, 1.733567147, 1.771387237, 1.805208199] The centralized moments are Second: , [0., 0.250000, 0.500000, 0.816327, 1.13889, 1.46000, 1.76309, 2.03875, 2.28512, 2.50144, 2.69114, 2.85763, 3.00526, 3.13781, 3.25878] Skewness: , [Float(undefined), 0., 0., 0.2134475891, 0.3352038637, 0.4625523327, 0.5622762720, 0.6484347911, 0.7164404529, 0.7691374740, 0.8068891248, 0.8314281408, 0.8443624492, 0.8475418564, 0.8428005980] Kurtosis: , [Float(undefined), 1.000000000, 2.000000000, 2.267456774, 2.503796186, 2.765622068, 3.026360404, 3.300203259, 3.563808408, 3.812902271, 4.035342959, 4.226168504, 4.381278404, 4.500260802, 4.584577961] end of this data For the equivalence class of patterns, {{[2, 1, 4, 3], [1, 2, 3], [2, 3, 1]}, {[2, 1, 4, 3], [1, 2, 3], [3, 1, 2]}, {[3, 4, 1, 2], [3, 2, 1], [1, 3, 2]}, {[3, 4, 1, 2], [3, 2, 1], [2, 1, 3]}} the member , {[2, 1, 4, 3], [1, 2, 3], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[2, 1], {[0, 1, 1], [0, 0, 2], [1, 2, 0]}, {1}], [[1, 2], {[1, 0, 0], [0, 0, 1]}, {2}], [[1], {[1, 2]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2+q^3, 2*q^3+2*q^4+q^5+q^6, 2*q^6+2*q^7+2*q^8+q^9+q^10, 2*q^10+2 *q^11+2*q^12+2*q^13+q^14+q^15, 2*q^15+2*q^16+2*q^17+2*q^18+2*q^19+q^20+q^21, 2* q^21+2*q^22+2*q^23+2*q^24+2*q^25+2*q^26+q^27+q^28, 2*q^28+2*q^29+2*q^30+2*q^31+ 2*q^32+2*q^33+2*q^34+q^35+q^36, 2*q^36+2*q^37+2*q^38+2*q^39+2*q^40+2*q^41+2*q^ 42+2*q^43+q^44+q^45, 2*q^45+2*q^46+2*q^47+2*q^48+2*q^49+2*q^50+2*q^51+2*q^52+2* q^53+q^54+q^55, 2*q^55+2*q^56+2*q^57+2*q^58+2*q^59+2*q^60+2*q^61+2*q^62+2*q^63+ 2*q^64+q^65+q^66, 2*q^66+2*q^67+2*q^68+2*q^69+2*q^70+2*q^71+2*q^72+2*q^73+2*q^ 74+2*q^75+2*q^76+q^77+q^78, 2*q^78+2*q^79+2*q^80+2*q^81+2*q^82+2*q^83+2*q^84+2* q^85+2*q^86+2*q^87+2*q^88+2*q^89+q^90+q^91, 2*q^91+2*q^92+2*q^93+2*q^94+2*q^95+ 2*q^96+2*q^97+2*q^98+2*q^99+2*q^100+2*q^101+2*q^102+2*q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+2*q^2, 2*q^3+2*q^2+q+1, 2*q^4+2*q^3+2*q^2+q+1, 2*q^5+2*q^4+2*q^3+2 *q^2+q+1, 2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^ 2+q+1, 2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q^9+2*q^8+2*q^7+2*q^6+2 *q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q^10+2*q^9+2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2* q^2+q+1, 2*q^11+2*q^10+2*q^9+2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q ^12+2*q^11+2*q^10+2*q^9+2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q^13+2 *q^12+2*q^11+2*q^10+2*q^9+2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+1, 2*q^14 +2*q^13+2*q^12+2*q^11+2*q^10+2*q^9+2*q^8+2*q^7+2*q^6+2*q^5+2*q^4+2*q^3+2*q^2+q+ 1] The number of permutations avoiding, {[2, 1, 4, 3], [1, 2, 3], [2, 3, 1]}, is given by [1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28] The number of EVEN permutations avoiding, {[2, 1, 4, 3], [1, 2, 3], [2, 3, 1]}, is given by [1, 1, 1, 3, 5, 5, 5, 7, 9, 9, 9, 11, 13, 13, 13] The number of ODD permutations avoiding, {[2, 1, 4, 3], [1, 2, 3], [2, 3, 1]}, is given by [0, 1, 3, 3, 3, 5, 7, 7, 7, 9, 11, 11, 11, 13, 15] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15] ODD:, [0, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13] The average number of inversions for each n is given by [0., 0.5000000000, 1.750000000, 4.166666667, 7.625000000, 12.10000000, 17.58333333, 24.07142857, 31.56250000, 40.05555556, 49.55000000, 60.04545455, 71.54166667, 84.03846154, 97.53571429] The standard deviation for each n is given by [0., 0.5000000000, 0.8291561975, 1.067187373, 1.316956719, 1.577973384, 1.846543317, 2.120117440, 2.397101114, 2.676486549, 2.957617284, 3.240051525, 3.523482731, 3.807692308, 4.092520553] The centralized moments are Second: , [0., 0.250000, 0.687500, 1.13889, 1.73438, 2.49000, 3.40972, 4.49490, 5.74609, 7.16358, 8.74750, 10.4979, 12.4149, 14.4985, 16.7487] Skewness: , [Float(undefined), 0., 0.4933822002, 0.4875722481, 0.3847981384, 0.2931929061, 0.2251932017, 0.1762171759, 0.1407003366, 0.1144669182, 0.09470945410, 0.07951506101, 0.06763041545, 0.05817506002, 0.05054100091] Kurtosis: , [Float(undefined), 1.000000000, 1.628096529, 1.989891941, 2.064543636, 2.046692795, 2.009635060, 1.973982937, 1.944202384, 1.920344788, 1.901390027, 1.886260098, 1.874109411, 1.864252513, 1.856163360] end of this data For the equivalence class of patterns, {{[1, 2, 3, 4], [1, 3, 2], [3, 1, 2]}, {[4, 3, 2, 1], [1, 3, 2], [3, 1, 2]}, {[1, 2, 3, 4], [2, 1, 3], [3, 1, 2]}, {[4, 3, 2, 1], [2, 1, 3], [3, 1, 2]}, {[1, 2, 3, 4], [1, 3, 2], [2, 3, 1]}, {[1, 2, 3, 4], [2, 1, 3], [2, 3, 1]}, {[4, 3, 2, 1], [1, 3, 2], [2, 3, 1]}, {[4, 3, 2, 1], [2, 1, 3], [2, 3, 1]}} the member , {[1, 2, 3, 4], [1, 3, 2], [3, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[2, 1], {[0, 1, 0], [0, 0, 3]}, {1}], [[1], {[0, 3]}, {}], [[1, 2], {[0, 1, 0], [0, 0, 2]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q+q^2+q^3, q+q^2+2*q^3+q^4+q^5+q^6, q^3+q^4+2*q^5+2*q^6+2*q^7+q^8+q^ 9+q^10, q^6+q^7+2*q^8+2*q^9+3*q^10+2*q^11+2*q^12+q^13+q^14+q^15, q^10+q^11+2*q^ 12+2*q^13+3*q^14+3*q^15+3*q^16+2*q^17+2*q^18+q^19+q^20+q^21, q^15+q^16+2*q^17+2 *q^18+3*q^19+3*q^20+4*q^21+3*q^22+3*q^23+2*q^24+2*q^25+q^26+q^27+q^28, q^21+q^ 22+2*q^23+2*q^24+3*q^25+3*q^26+4*q^27+4*q^28+4*q^29+3*q^30+3*q^31+2*q^32+2*q^33 +q^34+q^35+q^36, q^28+q^29+2*q^30+2*q^31+3*q^32+3*q^33+4*q^34+4*q^35+5*q^36+4*q ^37+4*q^38+3*q^39+3*q^40+2*q^41+2*q^42+q^43+q^44+q^45, q^36+q^37+2*q^38+2*q^39+ 3*q^40+3*q^41+4*q^42+4*q^43+5*q^44+5*q^45+5*q^46+4*q^47+4*q^48+3*q^49+3*q^50+2* q^51+2*q^52+q^53+q^54+q^55, q^45+q^46+2*q^47+2*q^48+3*q^49+3*q^50+4*q^51+4*q^52 +5*q^53+5*q^54+6*q^55+5*q^56+5*q^57+4*q^58+4*q^59+3*q^60+3*q^61+2*q^62+2*q^63+q ^64+q^65+q^66, q^55+q^56+2*q^57+2*q^58+3*q^59+3*q^60+4*q^61+4*q^62+5*q^63+5*q^ 64+6*q^65+6*q^66+6*q^67+5*q^68+5*q^69+4*q^70+4*q^71+3*q^72+3*q^73+2*q^74+2*q^75 +q^76+q^77+q^78, q^66+q^67+2*q^68+2*q^69+3*q^70+3*q^71+4*q^72+4*q^73+5*q^74+5*q ^75+6*q^76+6*q^77+7*q^78+6*q^79+6*q^80+5*q^81+5*q^82+4*q^83+4*q^84+3*q^85+3*q^ 86+2*q^87+2*q^88+q^89+q^90+q^91, q^78+q^79+2*q^80+2*q^81+3*q^82+3*q^83+4*q^84+4 *q^85+5*q^86+5*q^87+6*q^88+6*q^89+7*q^90+7*q^91+7*q^92+6*q^93+6*q^94+5*q^95+5*q ^96+4*q^97+4*q^98+3*q^99+3*q^100+2*q^101+2*q^102+q^103+q^104+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 1+q+q^2+q^3, 1+q+q^2+2*q^3+q^4+q^5, 1+q+q^2+2*q^3+2*q^4+2*q^5+q^6+q^7, 1+q+q^2+2*q^3+2*q^4+3*q^5+2*q^6+2*q^7+q^8+q^9, 1+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+ 3*q^7+2*q^8+2*q^9+q^10+q^11, 1+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+3*q^8+3*q^9+ 2*q^10+2*q^11+q^12+q^13, 1+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+4*q^9+3*q^ 10+3*q^11+2*q^12+2*q^13+q^14+q^15, 1+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+ 5*q^9+4*q^10+4*q^11+3*q^12+3*q^13+2*q^14+2*q^15+q^16+q^17, 1+q+q^2+2*q^3+2*q^4+ 3*q^5+3*q^6+4*q^7+4*q^8+5*q^9+5*q^10+5*q^11+4*q^12+4*q^13+3*q^14+3*q^15+2*q^16+ 2*q^17+q^18+q^19, 1+q^21+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+5*q^9+5*q^10 +6*q^11+5*q^12+5*q^13+4*q^14+4*q^15+3*q^16+3*q^17+2*q^18+2*q^19+q^20, 1+q^22+2* q^21+q^23+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+5*q^9+5*q^10+6*q^11+6*q^12+ 6*q^13+5*q^14+5*q^15+4*q^16+4*q^17+3*q^18+3*q^19+2*q^20, 1+q^25+2*q^22+q^24+3*q ^21+2*q^23+q+q^2+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+5*q^9+5*q^10+6*q^11+6*q^12 +7*q^13+6*q^14+6*q^15+5*q^16+5*q^17+4*q^18+4*q^19+3*q^20, 1+2*q^25+3*q^22+2*q^ 24+4*q^21+3*q^23+q+q^2+q^26+2*q^3+2*q^4+3*q^5+3*q^6+4*q^7+4*q^8+5*q^9+5*q^10+6* q^11+6*q^12+7*q^13+7*q^14+7*q^15+6*q^16+6*q^17+5*q^18+5*q^19+4*q^20+q^27] The number of permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [3, 1, 2]}, is given by [1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106] The number of EVEN permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [3, 1, 2]}, is given by [1, 1, 2, 3, 5, 9, 12, 13, 17, 25, 30, 31, 37, 49, 56] The number of ODD permutations avoiding, {[1, 2, 3, 4], [1, 3, 2], [3, 1, 2]}, is given by [0, 1, 2, 4, 6, 7, 10, 16, 20, 21, 26, 36, 42, 43, 50] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 3, 5, 7, 10, 13, 17, 21, 26, 31, 37, 43, 50] ODD:, [0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.428571429, 6.363636364, 10.31250000, 15.27272727, 21.24137931, 28.21621622, 36.19565217, 45.17857143, 55.16417910, 66.15189873, 78.14130435, 91.13207547] The standard deviation for each n is given by [0., 0.5000000000, 1.118033988, 1.590789818, 2.012358511, 2.416576866, 2.815247739, 3.212638538, 3.610309076, 4.008792321, 4.408219364, 4.808565167, 5.209748913, 5.611675567, 6.014252682] The centralized moments are Second: , [0., 0.250000, 1.25000, 2.53061, 4.04959, 5.83984, 7.92562, 10.3210, 13.0343, 16.0704, 19.4324, 23.1223, 27.1415, 31.4909, 36.1712] Skewness: , [Float(undefined), 0., 0., 0.1303560433, 0.1659543264, 0.1634801384, 0.1484946789, 0.1308750997, 0.1141766217, 0.09946313424, 0.08686669838, 0.07620356368, 0.06719710231, 0.05957671010, 0.05310591299] Kurtosis: , [Float(undefined), 1.000000000, 1.640000000, 1.984123337, 2.199817419, 2.320054057, 2.383983468, 2.417073317, 2.433437853, 2.440809559, 2.443317281, 2.443248073, 2.441815090, 2.439746540, 2.437395257] end of this data For the equivalence class of patterns, { {[2, 4, 1, 3], [3, 2, 1], [1, 2, 3]}, {[3, 1, 4, 2], [3, 2, 1], [1, 2, 3]}} the member , {[2, 4, 1, 3], [3, 2, 1], [1, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 1, 2], [0, 0, 3]}, {1}], [[1], {[3, 0], [0, 3]}, {}], [[1, 2], {[0, 0, 1], [1, 1, 0], [0, 2, 0], [3, 0, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+2*q^2, q^2+q^3+q^4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] with the reverse patterns and complement patterns having distributions [1, 1+q, 2*q*(1+q), q^2*(1+q+q^2), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of permutations avoiding, {[2, 4, 1, 3], [3, 2, 1], [1, 2, 3]}, is given by [1, 2, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of EVEN permutations avoiding, {[2, 4, 1, 3], [3, 2, 1], [1, 2, 3]}, is given by [1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of ODD permutations avoiding, {[2, 4, 1, 3], [3, 2, 1], [1, 2, 3]}, is given by [0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ODD:, [0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The average number of inversions for each n is given by [0., 0.5000000000, 1.500000000, 3.000000000, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The standard deviation for each n is given by [0., 0.5000000000, 0.5000000000, 0.8164965809, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The centralized moments are Second: , [0., 0.250000, 0.250000, 0.666667, FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] FAIL[3] FAIL[3] FAIL[3] Skewness: , [Float(undefined), 0., 0., 0., ----------, ----------, ----------, 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] FAIL[3] ----------, ----------, ----------, ----------, ----------, ----------, 3/2 3/2 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[3] FAIL[3] ----------, ----------] 3/2 3/2 FAIL[2] FAIL[2] FAIL[4] Kurtosis: , [Float(undefined), 1.000000000, 1.000000000, 1.500006000, --------, 2 FAIL[2] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------, --------, --------, --------, --------, 2 2 2 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[2] FAIL[4] FAIL[4] FAIL[4] --------, --------, --------] 2 2 2 FAIL[2] FAIL[2] FAIL[2] end of this data Out of a total of , 23, cases 23, were successful and , 0, failed Success Rate: , 1. Here are the failures {}