Trio of 3-patterns There all together, 5, different equivalence classes For the equivalence class of patterns, {{[3, 2, 1], [1, 2, 3], [2, 3, 1]}, {[3, 2, 1], [1, 2, 3], [3, 1, 2]}, {[3, 2, 1], [1, 2, 3], [1, 3, 2]}, {[3, 2, 1], [1, 2, 3], [2, 1, 3]}} the member , {[3, 2, 1], [1, 2, 3], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[2, 1], [3, 0], [0, 3]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 1, 1], [0, 0, 3]}, {1}], [[1, 2], {[1, 0, 0], [0, 0, 1], [0, 2, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^2, q^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(2*q+1), q^4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of permutations avoiding, {[3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [1, 2, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of EVEN permutations avoiding, {[3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] The number of ODD permutations avoiding, {[3, 2, 1], [1, 2, 3], [2, 3, 1]}, is given by [0, 1, 2, 0, 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, 1, 0, 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, 1.333333333, 2., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The standard deviation for each n is given by [0., 0.5000000000, 0.4714045206, 0., FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL, FAIL] The centralized moments are Second: , [0., 0.250000, 0.222222, 0., FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2], FAIL[2]] FAIL[3] Skewness: , [Float(undefined), 0., 0.7070975887, Float(undefined), ----------, 3/2 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] FAIL[3] ----------, ----------, ----------, ----------] 3/2 3/2 3/2 3/2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] Kurtosis: , [Float(undefined), 1.000000000, 1.499718000, Float(undefined), 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] FAIL[4] --------, --------, --------, --------] 2 2 2 2 FAIL[2] FAIL[2] FAIL[2] FAIL[2] end of this data For the equivalence class of patterns, {{[1, 3, 2], [2, 1, 3], [3, 1, 2]}, {[1, 3, 2], [2, 3, 1], [3, 1, 2]}, {[2, 1, 3], [2, 3, 1], [3, 1, 2]}, {[1, 3, 2], [2, 1, 3], [2, 3, 1]}} the member , {[1, 3, 2], [2, 1, 3], [3, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {[0, 1, 0], [0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q^2+q^3, 1+q^3+q^5+q^6, 1+q^4+q^7+q^9+q^10, 1+q^5+q^9+q^12+q^14+q^15 , 1+q^6+q^11+q^15+q^18+q^20+q^21, 1+q^7+q^13+q^18+q^22+q^25+q^27+q^28, 1+q^8+q^ 15+q^21+q^26+q^30+q^33+q^35+q^36, 1+q^9+q^17+q^24+q^30+q^35+q^39+q^42+q^44+q^45 , 1+q^10+q^19+q^27+q^34+q^40+q^45+q^49+q^52+q^54+q^55, 1+q^11+q^21+q^30+q^38+q^ 45+q^51+q^56+q^60+q^63+q^65+q^66, 1+q^12+q^23+q^33+q^42+q^50+q^57+q^63+q^68+q^ 72+q^75+q^77+q^78, 1+q^13+q^25+q^36+q^46+q^55+q^63+q^70+q^76+q^81+q^85+q^88+q^ 90+q^91, 1+q^14+q^27+q^39+q^50+q^60+q^69+q^77+q^84+q^90+q^95+q^99+q^102+q^104+q ^105] with the reverse patterns and complement patterns having distributions [1, 1+q, q^3+q+1, q^6+q^3+q+1, q^10+q^6+q^3+q+1, q^15+q^10+q^6+q^3+q+1, q^21+q^ 15+q^10+q^6+q^3+q+1, q^28+q^21+q^15+q^10+q^6+q^3+q+1, q^36+q^28+q^21+q^15+q^10+ q^6+q^3+q+1, q^45+q^36+q^28+q^21+q^15+q^10+q^6+q^3+q+1, q^55+q^45+q^36+q^28+q^ 21+q^15+q^10+q^6+q^3+q+1, q^66+q^55+q^45+q^36+q^28+q^21+q^15+q^10+q^6+q^3+q+1, q^78+q^66+q^55+q^45+q^36+q^28+q^21+q^15+q^10+q^6+q^3+q+1, q^91+q^78+q^66+q^55+q ^45+q^36+q^28+q^21+q^15+q^10+q^6+q^3+q+1, q^105+q^91+q^78+q^66+q^55+q^45+q^36+q ^28+q^21+q^15+q^10+q^6+q^3+q+1] The number of permutations avoiding, {[1, 3, 2], [2, 1, 3], [3, 1, 2]}, is given by [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] The number of EVEN permutations avoiding, {[1, 3, 2], [2, 1, 3], [3, 1, 2]}, is given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8] The number of ODD permutations avoiding, {[1, 3, 2], [2, 1, 3], [3, 1, 2]}, is given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7] ODD:, [0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8] The average number of inversions for each n is given by [0., 0.5000000000, 1.666666667, 3.500000000, 6.000000000, 9.166666667, 13.00000000, 17.50000000, 22.66666667, 28.50000000, 35.00000000, 42.16666667, 50.00000000, 58.50000000, 67.66666667] The standard deviation for each n is given by [0., 0.5000000000, 1.247219129, 2.291287848, 3.633180424, 5.273097341, 7.211102550, 9.447221815, 11.98146717, 14.81384488, 17.94435844, 21.37300999, 25.09980080, 29.12473176, 33.44780344] The centralized moments are Second: , [0., 0.250000, 1.55556, 5.25000, 13.2000, 27.8056, 52.0000, 89.2500, 143.556, 219.450, 322.000, 456.806, 630.000, 848.250, 1118.76] Skewness: , [Float(undefined), 0., -0.3817946015, -0.4987837494, -0.5504818825, -0.5779510783, -0.5943212577, -0.6048641417, -0.6120530991, -0.6171832640, -0.6209671516, -0.6238387693, -0.6260721284, -0.6278391529, -0.6292627626] Kurtosis: , [Float(undefined), 1.000000000, 1.499979184, 1.761886621, 1.893939394, 1.968178570, 2.013738905, 2.043622155, 2.064228884, 2.079059992, 2.090062112, 2.098444373, 2.104988662, 2.110174820, 2.114365122] end of this data For the equivalence class of patterns, {{[3, 2, 1], [2, 1, 3], [2, 3, 1]}, {[1, 2, 3], [1, 3, 2], [3, 1, 2]}, {[1, 2, 3], [1, 3, 2], [2, 3, 1]}, {[3, 2, 1], [2, 1, 3], [3, 1, 2]}, {[3, 2, 1], [1, 3, 2], [2, 3, 1]}, {[1, 2, 3], [2, 1, 3], [2, 3, 1]}, {[1, 2, 3], [2, 1, 3], [3, 1, 2]}, {[3, 2, 1], [1, 3, 2], [3, 1, 2]}} the member , {[3, 2, 1], [2, 1, 3], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[], {}, {}], [[2, 1], {[1, 0, 0], [0, 0, 1]}, {2}], [[1], {[1, 1]}, {}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+q+q^2, 1+q+q^2+q^3, 1+q+q^2+q^3+q^4, 1+q+q^2+q^3+q^4+q^5, 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, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8, 1+ q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9, 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, 1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^ 9+q^10+q^11+q^12, 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, 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] with the reverse patterns and complement patterns having distributions [1, 1+q, q*(1+q+q^2), q^3*(1+q+q^2+q^3), q^6*(1+q+q^2+q^3+q^4), q^10*(1+q+q^2+q ^3+q^4+q^5), q^15*(1+q+q^2+q^3+q^4+q^5+q^6), q^21*(1+q+q^2+q^3+q^4+q^5+q^6+q^7) , q^28*(1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8), q^36*(1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8 +q^9), q^45*(1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9+q^10), 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^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^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^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)] The number of permutations avoiding, {[3, 2, 1], [2, 1, 3], [2, 3, 1]}, is given by [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] The number of EVEN permutations avoiding, {[3, 2, 1], [2, 1, 3], [2, 3, 1]}, is given by [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8] The number of ODD permutations avoiding, {[3, 2, 1], [2, 1, 3], [2, 3, 1]}, is given by [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 7, 7] ODD:, [0, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8] The average number of inversions for each n is given by [0., 0.5000000000, 1.000000000, 1.500000000, 2.000000000, 2.500000000, 3.000000000, 3.500000000, 4.000000000, 4.500000000, 5.000000000, 5.500000000, 6.000000000, 6.500000000, 7.000000000] The standard deviation for each n is given by [0., 0.5000000000, 0.8164965809, 1.118033988, 1.414213562, 1.707825129, 2., 2.291287848, 2.581988897, 2.872281324, 3.162277660, 3.452052531, 3.741657387, 4.031128874, 4.320493799] The centralized moments are Second: , [0., 0.250000, 0.666667, 1.25000, 2.00000, 2.91667, 4.00000, 5.25000, 6.66667, 8.25000, 10.0000, 11.9167, 14.0000, 16.2500, 18.6667] -5 Skewness: , [Float(undefined), 0., 0., 0., 0., 0.2007562421 10 , 0., 0., 0., -5 -5 0., 0., 0.2430893825 10 , 0., 0., 0.1239934312 10 ] Kurtosis: , [Float(undefined), 1.000000000, 1.500006000, 1.640000000, 1.700000000, 1.731428532, 1.750000000, 1.761886621, 1.769998980, 1.775750230, 1.780000000, 1.783205633, 1.785714286, 1.787690414, 1.789280281] end of this data For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [3, 1, 2]}, {[1, 2, 3], [1, 3, 2], [2, 1, 3]}} the member , {[3, 2, 1], [2, 3, 1], [3, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[2, 0]}, {}], [[2, 1], {[1, 0, 0], [0, 1, 0]}, {1}], [[1, 2], {[1, 0, 0], [0, 2, 0]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 1+2*q, 1+3*q+q^2, 1+4*q+3*q^2, 1+5*q+6*q^2+q^3, 1+6*q+10*q^2+4*q^3, 1+ 7*q+15*q^2+10*q^3+q^4, 1+8*q+21*q^2+20*q^3+5*q^4, 1+9*q+28*q^2+35*q^3+15*q^4+q^ 5, 1+10*q+36*q^2+56*q^3+35*q^4+6*q^5, 1+11*q+45*q^2+84*q^3+70*q^4+21*q^5+q^6, 1 +12*q+55*q^2+120*q^3+126*q^4+56*q^5+7*q^6, 1+13*q+66*q^2+165*q^3+210*q^4+126*q^ 5+28*q^6+q^7, 1+14*q+78*q^2+220*q^3+330*q^4+252*q^5+84*q^6+8*q^7] with the reverse patterns and complement patterns having distributions [1, 1+q, q^2*(q+2), q^4*(1+3*q+q^2), q^8*(q^2+4*q+3), q^12*(q^3+5*q^2+6*q+1), q ^18*(q^3+6*q^2+10*q+4), q^24*(q^4+7*q^3+15*q^2+10*q+1), q^32*(q^4+8*q^3+21*q^2+ 20*q+5), q^40*(q^5+9*q^4+28*q^3+35*q^2+15*q+1), q^50*(q^5+10*q^4+36*q^3+56*q^2+ 35*q+6), q^60*(q^6+11*q^5+45*q^4+84*q^3+70*q^2+21*q+1), q^72*(q^6+12*q^5+55*q^4 +120*q^3+126*q^2+56*q+7), q^84*(q^7+13*q^6+66*q^5+165*q^4+210*q^3+126*q^2+28*q+ 1), q^98*(q^7+14*q^6+78*q^5+220*q^4+330*q^3+252*q^2+84*q+8)] The number of permutations avoiding, {[3, 2, 1], [2, 3, 1], [3, 1, 2]}, is given by [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987] The number of EVEN permutations avoiding, {[3, 2, 1], [2, 3, 1], [3, 1, 2]}, is given by [1, 1, 1, 2, 4, 7, 11, 17, 27, 44, 72, 117, 189, 305, 493] The number of ODD permutations avoiding, {[3, 2, 1], [2, 3, 1], [3, 1, 2]}, is given by [0, 1, 2, 3, 4, 6, 10, 17, 28, 45, 72, 116, 188, 305, 494] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 2, 2, 4, 6, 10, 17, 27, 45, 72, 117, 189, 305, 494] ODD:, [0, 1, 1, 3, 4, 7, 11, 17, 28, 44, 72, 116, 188, 305, 493] The average number of inversions for each n is given by [0., 0.5000000000, 0.6666666666, 1.000000000, 1.250000000, 1.538461538, 1.809523810, 2.088235294, 2.363636364, 2.640449438, 2.916666667, 3.193133047, 3.469496021, 3.745901639, 4.022289767] The standard deviation for each n is given by [0., 0.5000000000, 0.4714045206, 0.6324555320, 0.6614378278, 0.7457969011, 0.7939681905, 0.8529411765, 0.9017888559, 0.9510159088, 0.9965217283, 1.040611022, 1.082657116, 1.123237967, 1.162356908] The centralized moments are Second: , [0., 0.250000, 0.222222, 0.400000, 0.437500, 0.556213, 0.630385, 0.727509, 0.813223, 0.904431, 0.993056, 1.08287, 1.17215, 1.26166, 1.35107] Skewness: , [Float(undefined), 0., -0.7071166806, 0., -0.3239695482, -0.1316709573, -0.2178997277, -0.1692444588, -0.1844126389, -0.1703234577, -0.1695833097, -0.1632699016, -0.1597351104, -0.1553901679, -0.1517361434] Kurtosis: , [Float(undefined), 1.000000000, 1.499718000, 2.500000000, 2.224483265, 2.699493059, 2.557870491, 2.728780749, 2.704331490, 2.765418860, 2.774397047, 2.801193218, 2.814324003, 2.829840270, 2.841255616] end of this data For the equivalence class of patterns, {{[1, 2, 3], [2, 3, 1], [3, 1, 2]}, {[3, 2, 1], [1, 3, 2], [2, 1, 3]}} the member , {[1, 2, 3], [2, 3, 1], [3, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2], {[1, 0, 0], [0, 0, 1]}, {1}]} Using the scheme, the first, , 15, terms are [1, 1+q, 2*q+q^3, q^2+2*q^3+q^6, 2*q^4+2*q^6+q^10, q^6+2*q^7+2*q^10+q^15, 2*q^9 +2*q^11+2*q^15+q^21, q^12+2*q^13+2*q^16+2*q^21+q^28, 2*q^16+2*q^18+2*q^22+2*q^ 28+q^36, q^20+2*q^21+2*q^24+2*q^29+2*q^36+q^45, 2*q^25+2*q^27+2*q^31+2*q^37+2*q ^45+q^55, q^30+2*q^31+2*q^34+2*q^39+2*q^46+2*q^55+q^66, 2*q^36+2*q^38+2*q^42+2* q^48+2*q^56+2*q^66+q^78, q^42+2*q^43+2*q^46+2*q^51+2*q^58+2*q^67+2*q^78+q^91, 2 *q^49+2*q^51+2*q^55+2*q^61+2*q^69+2*q^79+2*q^91+q^105] with the reverse patterns and complement patterns having distributions [1, 1+q, 2*q^2+1, q^4+2*q^3+1, 2*q^6+2*q^4+1, q^9+2*q^8+2*q^5+1, 2*q^12+2*q^10+ 2*q^6+1, q^16+2*q^15+2*q^12+2*q^7+1, 2*q^20+2*q^18+2*q^14+2*q^8+1, q^25+2*q^24+ 2*q^21+2*q^16+2*q^9+1, 2*q^30+2*q^28+2*q^24+2*q^18+2*q^10+1, q^36+2*q^35+2*q^32 +2*q^27+2*q^20+2*q^11+1, 2*q^42+2*q^40+2*q^36+2*q^30+2*q^22+2*q^12+1, q^49+2*q^ 48+2*q^45+2*q^40+2*q^33+2*q^24+2*q^13+1, 2*q^56+2*q^54+2*q^50+2*q^44+2*q^36+2*q ^26+2*q^14+1] The number of permutations avoiding, {[1, 2, 3], [2, 3, 1], [3, 1, 2]}, is given by [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] The number of EVEN permutations avoiding, {[1, 2, 3], [2, 3, 1], [3, 1, 2]}, is given by [1, 1, 0, 2, 5, 3, 0, 4, 9, 5, 0, 6, 13, 7, 0] The number of ODD permutations avoiding, {[1, 2, 3], [2, 3, 1], [3, 1, 2]}, is given by [0, 1, 3, 2, 0, 3, 7, 4, 0, 5, 11, 6, 0, 7, 15] For the reverse patterns and complement patterns, we get EVEN:, [1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15] ODD:, [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0] The average number of inversions for each n is given by [0., 0.5000000000, 1.666666667, 3.500000000, 6.000000000, 9.166666667, 13.00000000, 17.50000000, 22.66666667, 28.50000000, 35.00000000, 42.16666667, 50.00000000, 58.50000000, 67.66666667] The standard deviation for each n is given by [0., 0.5000000000, 0.9428090414, 1.500000000, 2.190890230, 3.023059526, 4., 5.123475385, 6.394442029, 7.813449942, 9.380831520, 11.09679634, 12.96148140, 14.97497913, 17.13735362] The centralized moments are Second: , [0., 0.250000, 0.888889, 2.25000, 4.80000, 9.13889, 16.0000, 26.2500, 40.8889, 61.0500, 88.0000, 123.139, 168.000, 224.250, 293.689] Skewness: , [Float(undefined), 0., 0.7071154877, 0.8888888889, 0.9128709292, 0.8901522035, 0.8571421875, 0.8253326331, 0.7977955128, 0.7748247062, 0.7558934485, 0.7403057013, 0.7274300779, 0.7167243376, 0.7077612452] Kurtosis: , [Float(undefined), 1.000000000, 1.500056344, 2.185185185, 2.500000000, 2.601170901, 2.607144531, 2.576050794, 2.533551391, 2.490315198, 2.450413223, 2.415068576, 2.384353741, 2.357835918, 2.335019836] end of this data Out of a total of , 5, cases 5, were successful and , 0, failed Success Rate: , 1. Here are the failures {}