There all together, 56, different equivalence classes For the equivalence class of patterns, {{[1, 3, 4, 2], [2, 3, 4, 1]}, {[1, 4, 2, 3], [4, 1, 2, 3]}, {[1, 4, 3, 2], [2, 4, 3, 1]}, {[1, 4, 3, 2], [4, 1, 3, 2]}, {[2, 3, 1, 4], [2, 3, 4, 1]}, {[3, 1, 2, 4], [4, 1, 2, 3]}, {[3, 2, 1, 4], [3, 2, 4, 1]}, {[3, 2, 1, 4], [4, 2, 1, 3]}} the member , {[1, 4, 2, 3], [4, 1, 2, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {}, {2}], [[3, 2, 1], {}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {1}], [[2, 3, 1], {}, {1}], [[2, 1, 3], {}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 4, 3], [3, 2, 1, 4]}, {[1, 4, 3, 2], [2, 1, 3, 4]}, {[2, 3, 4, 1], [4, 3, 1, 2]}, {[3, 4, 2, 1], [4, 1, 2, 3]}} the member , {[1, 2, 4, 3], [3, 2, 1, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[2, 4, 1, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {5}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[3, 1, 2], {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[3, 2, 4, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {2}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[1, 2], {}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {3}], [[3, 2, 5, 4, 1], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 0, 1]}, {3}], [[1], {}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}], [[3, 5, 2, 4, 1], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 5, 1, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}], [[3, 5, 1, 4, 2], {[0, 0, 0, 0, 0, 1]}, {2}], [[3, 4, 1, 2], {}, {2}], [[2, 1], {}, {}], [[2, 1, 5, 4, 3], {[0, 0, 0, 0, 0, 1]}, {3}], [[2, 4, 1, 3], {}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1]}, {1}], [[2, 5, 1, 4, 3], {[0, 0, 0, 0, 0, 1]}, {2}], [[2, 1, 4, 3], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1]}, {1}], [[1, 3, 2], {}, {}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1]}, {2}], [[3, 1, 4, 2], {}, {3}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {2}], [[3, 1, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[2, 3, 1], {}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {5}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 86, 338, 1318, 5110, 19770, 76466] For the equivalence class of patterns, {{[1, 2, 4, 3], [2, 3, 4, 1]}, {[1, 2, 4, 3], [4, 1, 2, 3]}, {[1, 4, 3, 2], [3, 4, 2, 1]}, {[1, 4, 3, 2], [4, 3, 1, 2]}, {[2, 1, 3, 4], [2, 3, 4, 1]}, {[2, 1, 3, 4], [4, 1, 2, 3]}, {[3, 2, 1, 4], [3, 4, 2, 1]}, {[3, 2, 1, 4], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [4, 1, 2, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[2, 1, 3], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 4, 3, 1], {[0, 0, 0, 2, 0]}, {}], [[3, 1, 5, 4, 2], { [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}], [[2, 1, 5, 4, 3], {[0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 0, 0]}, {4}], [[1, 3, 2], {[0, 0, 2, 0]}, {}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0]}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}], [[3, 5, 4, 2, 1], {[0, 0, 0, 0, 2, 0]}, {4}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}], [[3, 4, 2, 1], {}, {3}], [[2, 5, 4, 1, 3], %2, {2}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 2, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[3, 2, 5, 4, 1], {[0, 0, 0, 0, 2, 0]}, {1}], [[2, 3, 1], {}, {}], [[3, 4, 1, 2], %1, {1}], [[2, 5, 3, 1, 4], %2, {2}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1, 5], %2, {2}], [[3, 1, 4, 2], {[0, 0, 0, 2, 0], [0, 0, 1, 0, 0]}, {}], [[3, 5, 4, 1, 2], %3, {2}], [[4, 1, 5, 3, 2], {[0, 0, 0, 0, 2, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}], [[1, 4, 3, 2], {[0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0]}, {3}], [[4, 1, 3, 2], %1, {3}], [[2, 1, 4, 3, 5], %2, {3}], [[3, 1, 5, 2, 4], %3, {3}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 1, 4, 2, 5], %3, {3}], [[3, 1, 2, 4], %1, {3}], [[1, 4, 2, 3], %1, {2}], [[2, 1, 5, 3, 4], %2, {3}], [[1, 3, 2, 4], %1, {2}], [[3, 2, 4, 1], {}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} %2 := {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]} %3 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 88, 365, 1540, 6568, 28269, 122752] For the equivalence class of patterns, {{[1, 2, 4, 3], [3, 4, 2, 1]}, {[1, 2, 4, 3], [4, 3, 1, 2]}, {[2, 1, 3, 4], [3, 4, 2, 1]}, {[2, 1, 3, 4], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[3, 1, 2], {}, {}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0]}, {1}], [[2, 1], {}, {}], [[3, 1, 4, 2], {}, {1}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}], [[1, 3, 2], {}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {2}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0]}, {2}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 3, 1], {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {1}], [[2, 1, 4, 3], {}, {1}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 86, 330, 1206, 4174, 13726, 43134, 130302, 380414, 1078270, 2978814, 8046590, 21311486, 55468030, 142147582, 359268350, 896794622, 2213543934, 5408292862, 13091995646, 31424774142, 74845257726, 176991240190, 415789744126, 970830381054, 2253985415166, 5205567471614] This enumerating sequence seems to have the rational generating function, 2 3 4 5 6 7 1 - 10 x + 41 x - 86 x + 96 x - 48 x + 4 x + 4 x ------------------------------------------------------ 2 3 4 5 6 1 - 11 x + 50 x - 120 x + 160 x - 112 x + 32 x For the equivalence class of patterns, {{[1, 2, 4, 3], [3, 2, 4, 1]}, {[1, 2, 4, 3], [4, 2, 1, 3]}, {[1, 3, 4, 2], [4, 3, 1, 2]}, {[1, 4, 2, 3], [3, 4, 2, 1]}, {[2, 1, 3, 4], [2, 4, 3, 1]}, {[2, 1, 3, 4], [4, 1, 3, 2]}, {[2, 3, 1, 4], [4, 3, 1, 2]}, {[3, 1, 2, 4], [3, 4, 2, 1]}} the member , {[1, 2, 4, 3], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[3, 1, 2], {}, {}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {}, {}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {3}], [[1], {}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}], [[2, 1], {}, {}], [[2, 1, 4, 3], {}, {}], [[1, 3, 2], {}, {}], [[2, 4, 1, 3], {}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {}, {1}], [[2, 3, 1], {}, {}], [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [ [3, 1, 4, 2, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [[3, 1, 2, 4], %1, {3}], [[1, 3, 2, 4], %1, {2}], [[2, 1, 5, 4, 3], {[0, 0, 0, 0, 1, 0]}, {3}], [[3, 2, 5, 4, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[3, 1, 5, 4, 2], {[0, 0, 0, 0, 1, 0]}, {3}], [[3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}], [ [4, 2, 5, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[4, 1, 5, 3, 2], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [[4, 2, 3, 1], %1, {1}], [[4, 1, 5, 2, 3], {[0, 0, 1, 0, 0, 0]}, {4}], [[3, 1, 4, 2], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 0]}, {3}], [[3, 4, 2, 1], %1, {3}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0]}, {2}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0]}, {1}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 86, 337, 1299, 4910, 18228, 66640] For the equivalence class of patterns, {{[1, 2, 4, 3], [2, 3, 1, 4]}, {[1, 2, 4, 3], [3, 1, 2, 4]}, {[1, 3, 4, 2], [2, 1, 3, 4]}, {[1, 4, 2, 3], [2, 1, 3, 4]}, {[2, 4, 3, 1], [4, 3, 1, 2]}, {[3, 2, 4, 1], [4, 3, 1, 2]}, {[3, 4, 2, 1], [4, 1, 3, 2]}, {[3, 4, 2, 1], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [2, 1, 3, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [0, 0, 1, 1, 0]}, {2}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 0, 1]}, {2}], [[4, 2, 3, 1], {[0, 0, 1, 0, 1], [0, 0, 1, 1, 0]}, {1}], [[1, 4, 3, 2], {[0, 1, 1, 1, 0], [0, 0, 0, 1, 1], [0, 1, 1, 0, 1]}, {2}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}], [[3, 2, 1], {}, {1}], [[4, 1, 3, 2], {[0, 1, 1, 1, 0], [0, 1, 1, 0, 1]}, {1}], [[1, 3, 2], {[0, 1, 1, 1]}, {}], [[1, 4, 2, 3], {[0, 0, 1, 1, 1], [0, 1, 0, 0, 0]}, {3}], [[2, 3, 1], {[0, 0, 1, 1]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 88, 367, 1571, 6861, 30468, 137229, 625573, 2881230, 13388094, 62688448, 295504025, 1401195334, 6678877732, 31984089193, 153809536017, 742462191363, 3596290278723, 17473993136316, 85147347832182, 415997039428899, 2037323575386383, 10000024336253853, 49186129273614768, 242393790200039756, 1196687427997471342, 5917920133042928976] For the equivalence class of patterns, {{[1, 2, 4, 3], [2, 4, 3, 1]}, {[1, 2, 4, 3], [4, 1, 3, 2]}, {[1, 3, 4, 2], [3, 4, 2, 1]}, {[1, 4, 2, 3], [4, 3, 1, 2]}, {[2, 1, 3, 4], [3, 2, 4, 1]}, {[2, 1, 3, 4], [4, 2, 1, 3]}, {[2, 3, 1, 4], [3, 4, 2, 1]}, {[3, 1, 2, 4], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[2, 1, 3], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 0]}, {4}], [[2, 1], {}, {}], [[2, 1, 4, 3], {}, {}], [[1, 3, 2], {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[2, 4, 3, 1, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {2}], [[2, 1, 4, 3, 5], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[3, 2, 4, 1], {}, {1}], [[5, 3, 4, 1, 2], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {2, 3}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0]}, {}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {1}], [[2, 1, 5, 4, 3], {}, {4}], [[2, 4, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 2, 5, 4, 1], {}, {1}], [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}], [[2, 5, 4, 1, 3], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}], [[3, 5, 4, 1, 2], {[0, 1, 0, 0, 0, 0]}, {3}], [[2, 5, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}], [ [4, 2, 3, 1, 5], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[5, 2, 4, 1, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[5, 2, 3, 1, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}], [[1, 4, 3, 2], {}, {3}], [[2, 4, 3, 1], {}, {}], [[3, 5, 4, 2, 1], {}, {4}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}], [[5, 3, 4, 2, 1], {[0, 0, 0, 1, 0, 0]}, {4}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 88, 363, 1507, 6241, 25721, 105485] For the equivalence class of patterns, {{[1, 3, 4, 2], [3, 2, 1, 4]}, {[1, 4, 2, 3], [3, 2, 1, 4]}, {[1, 4, 3, 2], [2, 3, 1, 4]}, {[1, 4, 3, 2], [3, 1, 2, 4]}, {[2, 3, 4, 1], [4, 1, 3, 2]}, {[2, 3, 4, 1], [4, 2, 1, 3]}, {[2, 4, 3, 1], [4, 1, 2, 3]}, {[3, 2, 4, 1], [4, 1, 2, 3]}} the member , {[1, 3, 4, 2], [3, 2, 1, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[3, 2, 4, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[1, 2], {}, {}], [[3, 2, 5, 4, 1], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[1], {}, {}], [[3, 5, 2, 4, 1], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1], {}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1]}, {2}], [[2, 3, 1], {}, {}], [[3, 1, 2], {[0, 1, 0, 1]}, {}], [[2, 1, 5, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}], [ [4, 1, 5, 2, 3], {[0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {4}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 1], [0, 1, 0, 0, 0]}, {2}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0]}, {2}], [[2, 1, 4, 3, 5], %1, {1}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 4, 1, 3, 5], %1, {1}], [[4, 1, 5, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {2}], [[4, 2, 5, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 4, 2, 5], %1, {1}], [[3, 4, 1, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 0, 0, 1], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}], [ [3, 5, 1, 2, 4], {[0, 0, 1, 0, 0, 1], [0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0]}, {3}], [ [4, 5, 1, 3, 2], {[0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[4, 5, 2, 3, 1], {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 1], [0, 1, 0, 0, 0]}, {3}], [[2, 5, 1, 3, 4], {[0, 0, 0, 1, 0, 1], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 1, 0, 0, 1], [0, 1, 0, 1, 0]}, {}], [[2, 5, 1, 4, 3], {[0, 1, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [ [3, 1, 5, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {3}], [ [3, 5, 1, 4, 2], {[0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[2, 4, 1, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}], [[1, 3, 2], {[0, 1, 0, 1]}, {}], [[1, 4, 3, 2], {[0, 1, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[2, 1, 4, 3], {[0, 1, 0, 0, 1], [0, 0, 1, 0, 1]}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[4, 1, 3, 2], {[0, 1, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}]} %1 := {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 86, 336, 1290, 4870, 18164, 67234] For the equivalence class of patterns, {{[1, 4, 3, 2], [2, 3, 4, 1]}, {[1, 4, 3, 2], [4, 1, 2, 3]}, {[2, 3, 4, 1], [3, 2, 1, 4]}, {[3, 2, 1, 4], [4, 1, 2, 3]}} the member , {[1, 4, 3, 2], [4, 1, 2, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 3, 0]}, {}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {2}], [[2, 3, 1], {[0, 0, 2, 0], [0, 3, 0, 0], [0, 2, 1, 0]}, {1}], [[1, 2], {[0, 3, 0]}, {}], [[2, 1, 3], {[0, 2, 0, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {1}], [[3, 1, 2], {[0, 2, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2, 3], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {2}], [[1, 3, 2], {[0, 0, 2, 0], [0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486, 357913942, 1431655766, 5726623062, 22906492246, 91625968982, 366503875926, 1466015503702, 5864062014806, 23456248059222, 93824992236886, 375299968947542, 1501199875790166, 6004799503160662, 24019198012642646, 96076792050570582] This enumerating sequence seems to have the 2 1 - 4 x + x rational generating function, -------------- 2 1 - 5 x + 4 x For the equivalence class of patterns, {{[1, 3, 4, 2], [3, 2, 4, 1]}, {[1, 3, 4, 2], [4, 1, 3, 2]}, {[1, 4, 2, 3], [2, 4, 3, 1]}, {[1, 4, 2, 3], [4, 2, 1, 3]}, {[2, 3, 1, 4], [2, 4, 3, 1]}, {[2, 3, 1, 4], [4, 2, 1, 3]}, {[3, 1, 2, 4], [3, 2, 4, 1]}, {[3, 1, 2, 4], [4, 1, 3, 2]}} the member , {[1, 3, 4, 2], [4, 1, 3, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[2, 1, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 1, 4, 3], {}, {}], [[1, 3, 2], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[3, 2, 4, 1], {}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [ [2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[2, 1, 5, 4, 3], {}, {4}], [[3, 2, 5, 4, 1], {}, {1}], [[3, 1, 5, 4, 2], {[0, 1, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {}, {3}], [[1, 3, 2, 4], %1, {1}], [[3, 1, 4, 2], {[0, 1, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[1, 4, 2, 3], %1, {3}], [[2, 1, 3, 4], %1, {1}], [[2, 4, 1, 3], %1, {3}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[2, 1, 5, 3, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[2, 4, 3, 1], {}, {3}]} %1 := {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310] For the equivalence class of patterns, {{[1, 3, 2, 4], [1, 3, 4, 2]}, {[1, 3, 2, 4], [1, 4, 2, 3]}, {[1, 3, 2, 4], [2, 3, 1, 4]}, {[1, 3, 2, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [4, 2, 3, 1]}, {[3, 2, 4, 1], [4, 2, 3, 1]}, {[4, 1, 3, 2], [4, 2, 3, 1]}, {[4, 2, 1, 3], [4, 2, 3, 1]}} the member , {[1, 3, 2, 4], [1, 3, 4, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {1}], [[1, 2], {[0, 1, 1]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[2, 1, 4, 3], [3, 4, 1, 2]}} the member , {[2, 1, 4, 3], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 3, 2], {}, {}], [[1, 3, 2, 4], {[0, 0, 0, 1, 0]}, {1}], [[1, 4, 3, 2], {}, {2}], [[1, 2, 3], {}, {1}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0]}, {1}], [[2, 3, 1], {[0, 1, 0, 0]}, {}], [[3, 2, 1], {}, {1}], [[4, 1, 3, 2], {}, {1}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {}, {2}], [[3, 1, 2, 4], {[0, 0, 0, 1, 0]}, {2}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0]}, {2}], [[2, 1, 3], {[0, 0, 1, 0]}, {}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[1, 4, 2, 3], {}, {1}], [[2, 4, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 86, 340, 1340, 5254, 20518, 79932, 311028, 1209916, 4707964, 18330728, 71429176, 278586182, 1087537414, 4249391468, 16618640836, 65048019092, 254814326164, 998953992728, 3919041821896, 15385395144092, 60438585676636, 237563884988120, 934311596780040, 3676495517376184, 14474185732012088, 57011153530262480] This enumerating sequence seems to have the 2 12 (1 + 2 n) 2 (16 + 13 n) N (19 + 9 n) N 3 annihilating operator , - ------------ + --------------- - ------------- + N 3 + n 3 + n 3 + n For the equivalence class of patterns, {{[1, 3, 4, 2], [3, 1, 4, 2]}, {[1, 4, 2, 3], [2, 4, 1, 3]}, {[2, 3, 1, 4], [2, 4, 1, 3]}, {[2, 4, 1, 3], [2, 4, 3, 1]}, {[2, 4, 1, 3], [4, 2, 1, 3]}, {[3, 1, 2, 4], [3, 1, 4, 2]}, {[3, 1, 4, 2], [3, 2, 4, 1]}, {[3, 1, 4, 2], [4, 1, 3, 2]}} the member , {[1, 4, 2, 3], [2, 4, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 1], {}, {1}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 3, 4, 2], [2, 4, 3, 1]}, {[1, 4, 2, 3], [4, 1, 3, 2]}, {[2, 3, 1, 4], [3, 2, 4, 1]}, {[3, 1, 2, 4], [4, 2, 1, 3]}} the member , {[1, 4, 2, 3], [4, 1, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {}, {2}], [[3, 2, 1], {}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 1], {}, {}], [[2, 1, 3], {[0, 2, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 3, 1], {[0, 0, 2, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368, 853829272, 3733693872, 16333556838, 71476391800, 312865382004, 1369760107576, 5998008630244, 26268304208032, 115055864102504, 503997820344464, 2207927106851580, 9673223726469136, 42382192892577128, 185702341264971696, 813710332253257688] This enumerating sequence seems to have the annihilating operator , 2 3 8 (1 + 2 n) 12 (2 + 3 n) N 4 (7 + 8 n) N 2 (7 + 5 n) N 4 ----------- - -------------- + -------------- - -------------- + N 2 + n 2 + n 2 + n 2 + n For the equivalence class of patterns, {{[1, 3, 4, 2], [4, 2, 1, 3]}, {[1, 4, 2, 3], [3, 2, 4, 1]}, {[2, 3, 1, 4], [4, 1, 3, 2]}, {[2, 4, 3, 1], [3, 1, 2, 4]}} the member , {[1, 3, 4, 2], [4, 2, 1, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[3, 1, 2], {[0, 1, 1, 0]}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[2, 3, 1], {}, {}], [[1, 3, 2], {[0, 1, 1, 0]}, {}], [[1, 4, 3, 2], {[0, 1, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 1, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 1, 5, 4, 2], {[0, 1, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [ [2, 1, 5, 3, 4], {[0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}], [ [2, 1, 5, 4, 3], {[0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [ [3, 5, 1, 4, 2], {[0, 1, 1, 0, 0, 0], [0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [ [2, 5, 1, 4, 3], {[0, 1, 0, 1, 0, 0], [0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[3, 5, 2, 4, 1], %1, {1}], [[2, 5, 1, 3, 4], {[0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}], [ [3, 1, 5, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [4, 1, 5, 2, 3], {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {1}], [ [4, 1, 5, 3, 2], {[0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [ [4, 5, 1, 3, 2], {[0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [ [4, 5, 1, 2, 3], {[0, 0, 1, 1, 0, 0], [0, 0, 1, 0, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}], [ [3, 5, 1, 2, 4], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0], [0, 1, 0, 0, 0, 0]}, {3}], [[4, 5, 2, 3, 1], %1, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0]}, {2}], [[2, 1, 4, 3, 5], %2, {1}], [[2, 4, 1, 3, 5], %2, {1}], [[3, 1, 4, 2, 5], %2, {1}], [[3, 4, 1, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {}], [[3, 2, 5, 4, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[4, 2, 5, 3, 1], %1, {1}], [[2, 1, 4, 3], {[0, 1, 0, 1, 0], [0, 0, 1, 1, 0]}, {}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0]}, {2}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 1, 0, 1, 0], [0, 0, 1, 1, 0]}, {}], [[3, 4, 1, 2], {[0, 1, 1, 0, 0], [0, 1, 0, 1, 0]}, {}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 1, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[1, 4, 2, 3], {[0, 0, 1, 1, 0], [0, 1, 0, 0, 0]}, {4}]} %1 := {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]} %2 := {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 86, 338, 1318, 5106, 19718, 76066] For the equivalence class of patterns, {{[1, 3, 4, 2], [1, 4, 3, 2]}, {[1, 4, 2, 3], [1, 4, 3, 2]}, {[2, 3, 1, 4], [3, 2, 1, 4]}, {[2, 3, 4, 1], [2, 4, 3, 1]}, {[2, 3, 4, 1], [3, 2, 4, 1]}, {[3, 1, 2, 4], [3, 2, 1, 4]}, {[4, 1, 2, 3], [4, 1, 3, 2]}, {[4, 1, 2, 3], [4, 2, 1, 3]}} the member , {[1, 4, 2, 3], [1, 4, 3, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {1}], [[1, 2], {[0, 2, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 3, 4], [1, 2, 4, 3]}, {[1, 2, 3, 4], [2, 1, 3, 4]}, {[3, 4, 2, 1], [4, 3, 2, 1]}, {[4, 3, 1, 2], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [1, 2, 4, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {1}], [[1, 2], {[0, 0, 2]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 4, 3], [2, 1, 4, 3]}, {[2, 1, 3, 4], [2, 1, 4, 3]}, {[3, 4, 1, 2], [3, 4, 2, 1]}, {[3, 4, 1, 2], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[3, 1, 2], {}, {1}], [[3, 2, 1], {}, {1}], [[2, 1, 3], {[0, 0, 1, 0]}, {1}], [[1, 3, 2], {}, {2}], [[2, 3, 1], {}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 3, 4], [1, 3, 2, 4]}, {[4, 2, 3, 1], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [1, 3, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[2, 1], {}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1]}, {1, 2}], [[1, 2, 3], {[0, 0, 0, 1]}, {3}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {4}], [[3, 4, 1, 2], {}, {1, 2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 396, 1837, 8864, 44074, 224352, 1163724, 6129840, 32703074, 176351644, 959658200, 5262988330, 29057961666, 161374413196, 900792925199, 5050924332096, 28434661250454, 160644331001476, 910455895039056, 5174722258676440, 29486753617569684, 168411404890025412, 963888354223500966, 5527261077912111036, 31750323052401948174, 182673854776113608008] For the equivalence class of patterns, {{[1, 2, 3, 4], [1, 3, 4, 2]}, {[1, 2, 3, 4], [1, 4, 2, 3]}, {[1, 2, 3, 4], [2, 3, 1, 4]}, {[1, 2, 3, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [4, 3, 2, 1]}, {[3, 2, 4, 1], [4, 3, 2, 1]}, {[4, 1, 3, 2], [4, 3, 2, 1]}, {[4, 2, 1, 3], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [1, 3, 4, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[1, 3, 2], {[0, 0, 1, 1]}, {2}], [[2, 1], {}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1]}, {1, 2}], [[2, 4, 1, 3], {[0, 0, 0, 1, 1]}, {1, 2}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {4}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 89, 380, 1678, 7584, 34875, 162560, 766124, 3644066, 17469863, 84324840, 409471090, 1998933556, 9804748548, 48298256084, 238840150970, 1185256302910, 5900843531665, 29464355189120, 147522603762870, 740471407808372, 3725334547101464, 18782663124890072, 94889671255981134, 480277198610107582, 2435135955040907671, 12367005533005155312] For the equivalence class of patterns, {{[1, 2, 3, 4], [1, 4, 3, 2]}, {[1, 2, 3, 4], [3, 2, 1, 4]}, {[2, 3, 4, 1], [4, 3, 2, 1]}, {[4, 1, 2, 3], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [1, 4, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 2, 0], [0, 1, 0, 0], [0, 0, 1, 2], [0, 0, 0, 3]}, {2}], [[2, 3, 1, 4], {[0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0], [0, 0, 0, 2, 0], [0, 0, 0, 0, 1]}, {1, 2}], [[3, 4, 1, 2], { [0, 0, 2, 1, 0], [0, 3, 0, 0, 0], [0, 0, 3, 0, 0], [0, 0, 2, 0, 1], [0, 0, 1, 1, 1], [0, 0, 1, 0, 2], [0, 0, 0, 1, 2], [0, 0, 0, 0, 3], [0, 0, 0, 2, 0]}, {1, 2}], [[2, 1], {}, {1}], [[2, 3, 1], {[0, 0, 0, 3], [0, 0, 3, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 3, 0], [0, 0, 0, 0, 3]}, {3}], [[2, 4, 1, 3], { [0, 3, 0, 0, 0], [0, 0, 0, 1, 2], [0, 0, 0, 0, 3], [0, 0, 0, 2, 0], [0, 0, 1, 0, 0]}, {1, 2}], [[1, 2], {[0, 3, 0], [0, 0, 3]}, {}], [[1, 2, 3], {[0, 0, 2, 0], [0, 0, 0, 1], [0, 3, 0, 0], [0, 2, 1, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486, 357913942, 1431655766, 5726623062, 22906492246, 91625968982, 366503875926, 1466015503702, 5864062014806, 23456248059222, 93824992236886, 375299968947542, 1501199875790166, 6004799503160662, 24019198012642646, 96076792050570582] This enumerating sequence seems to have the 2 1 - 4 x + x rational generating function, -------------- 2 1 - 5 x + 4 x For the equivalence class of patterns, {{[1, 2, 3, 4], [2, 1, 4, 3]}, {[3, 4, 1, 2], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1, 3], {[0, 0, 0, 2], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 2]}, {2}], [[3, 1, 2], {[0, 0, 0, 2]}, {1}], [[2, 1], {[0, 0, 3]}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {3}], [[3, 2, 1], {[0, 0, 2, 1], [0, 0, 1, 2], [0, 0, 0, 3], [0, 0, 3, 0]}, {1}], [[2, 3, 1], {[0, 0, 0, 2], [0, 0, 2, 1], [0, 0, 3, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 86, 342, 1366, 5462, 21846, 87382, 349526, 1398102, 5592406, 22369622, 89478486, 357913942, 1431655766, 5726623062, 22906492246, 91625968982, 366503875926, 1466015503702, 5864062014806, 23456248059222, 93824992236886, 375299968947542, 1501199875790166, 6004799503160662, 24019198012642646, 96076792050570582] This enumerating sequence seems to have the 2 1 - 4 x + x rational generating function, -------------- 2 1 - 5 x + 4 x For the equivalence class of patterns, {{[1, 3, 4, 2], [4, 1, 2, 3]}, {[1, 4, 2, 3], [2, 3, 4, 1]}, {[1, 4, 3, 2], [3, 2, 4, 1]}, {[1, 4, 3, 2], [4, 2, 1, 3]}, {[2, 3, 1, 4], [4, 1, 2, 3]}, {[2, 3, 4, 1], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 2, 1, 4]}, {[3, 2, 1, 4], [4, 1, 3, 2]}} the member , {[1, 3, 4, 2], [4, 1, 2, 3]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[2, 1, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[3, 1, 2], {[0, 0, 1, 0]}, {}], [[2, 1, 4, 3], {}, {}], [[1, 3, 2], {}, {}], [[3, 4, 2, 1], {}, {3}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0]}, {1}], [[2, 3, 1], {}, {}], [[3, 4, 1, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[4, 1, 5, 2, 3], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 4, 1], {}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 1, 0]}, {1}], [[2, 1, 5, 4, 3], {[0, 0, 0, 1, 1, 0]}, {4}], [[2, 4, 3, 1], {[0, 0, 1, 1, 0]}, {3}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0]}, {1}], [[2, 1, 4, 3, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [ [3, 1, 4, 2, 5], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[3, 1, 5, 4, 2], {[0, 0, 0, 1, 1, 0], [0, 0, 1, 0, 0, 0]}, {4}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 1, 0]}, {3}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 1, 0, 0]}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 2, 5, 4, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 1, 0]}, {1}], [ [2, 1, 5, 3, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [ [3, 1, 5, 2, 4], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [[4, 1, 5, 3, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {4}], [[4, 2, 5, 3, 1], {[0, 0, 1, 0, 1, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 87, 352, 1434, 5861, 24019, 98677] For the equivalence class of patterns, {{[1, 3, 4, 2], [2, 1, 4, 3]}, {[1, 4, 2, 3], [2, 1, 4, 3]}, {[2, 1, 4, 3], [2, 3, 1, 4]}, {[2, 1, 4, 3], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 4, 1, 2]}, {[3, 2, 4, 1], [3, 4, 1, 2]}, {[3, 4, 1, 2], [4, 1, 3, 2]}, {[3, 4, 1, 2], [4, 2, 1, 3]}} the member , {[1, 3, 4, 2], [2, 1, 4, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 3, 2], {}, {}], [[1, 4, 2, 3], {[0, 1, 0, 0, 0]}, {3}], [[1, 3, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 4, 3, 1], {}, {2}], [[3, 2, 4, 1], {[0, 0, 0, 1, 0]}, {1}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[4, 2, 3, 1], {}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 3, 2], {}, {2}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0]}, {2}], [[3, 2, 1], {}, {1}], [[4, 1, 3, 2], {}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0]}, {1}], [[2, 1, 3], {[0, 0, 1, 0]}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1], {}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 88, 368, 1584, 6968, 31192, 141656, 651136, 3023840, 14166496, 66876096, 317809216, 1519163456, 7299577216, 35237444736, 170812433536, 831127053696, 4057858988416, 19873611712896, 97609555091456, 480665412429312, 2372681964103168, 11738252228246528, 58192137103545344, 289040850441827328, 1438241007404642304, 7168538977121976320] This enumerating sequence seems to have the annihilating operator , 2 3 4 (1 + 2 n) 12 (3 + 2 n) N 12 (5 + 2 n) N 3 (11 + 3 n) N 4 ----------- - -------------- + --------------- - --------------- + N 5 + n 5 + n 5 + n 5 + n For the equivalence class of patterns, {{[1, 2, 3, 4], [3, 4, 2, 1]}, {[1, 2, 3, 4], [4, 3, 1, 2]}, {[1, 2, 4, 3], [4, 3, 2, 1]}, {[2, 1, 3, 4], [4, 3, 2, 1]}} the member , {[1, 2, 3, 4], [4, 3, 1, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[2, 1, 3], {}, {}], [[3, 1, 2], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0]}, {1}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[3, 1, 4, 2], {}, {1}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0]}, {3}], [[1, 3, 2], {}, {}], [[2, 4, 1, 3], {}, {1}], [[3, 4, 1, 2], {}, {1}], [[2, 3, 1], {}, {}], [[2, 1, 4, 3], {}, {1}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}], [[2, 3, 4, 1], {[0, 0, 0, 0, 1]}, {}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1]}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 1]}, {}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0]}, {3}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {3}], [[4, 1, 2, 3], {[0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[1, 4, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {2}], [[1, 3, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {2}], [ [1, 4, 2, 5, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[1, 3, 2, 5, 4], %1, {1}], [[3, 1, 4, 5, 2], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 3, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 2, 4, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 4, 5, 3], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 1, 3, 5, 4], %1, {1}], [[3, 4, 1, 5, 2], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 3, 1, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 5, 3], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 3, 1, 5, 4], %1, {1}], [[3, 4, 2, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 3, 4, 1, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[3, 4, 5, 1, 2], {[0, 0, 0, 0, 0, 1]}, {1}], [[2, 3, 5, 1, 4], %1, {1}], [[2, 4, 5, 1, 3], {[0, 0, 0, 0, 0, 1]}, {1}], [[3, 4, 5, 2, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[3, 1, 2, 5, 4], %1, {1}], [[4, 1, 2, 5, 3], {[0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[3, 1, 2, 4, 5], {[0, 0, 0, 0, 0, 0]}, {1}], [[4, 1, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [ [4, 2, 3, 5, 1], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[4, 2, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}], [[2, 3, 1, 4], {[0, 0, 0, 0, 1]}, {}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1]}, {}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1]}, {2}]} %1 := {[0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]} Using the scheme, the first, , 11, terms are [1, 1, 2, 6, 22, 86, 321, 1085, 3266, 8797, 21478] For the equivalence class of patterns, {{[1, 2, 4, 3], [1, 3, 2, 4]}, {[1, 3, 2, 4], [2, 1, 3, 4]}, {[3, 4, 2, 1], [4, 2, 3, 1]}, {[4, 2, 3, 1], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [1, 3, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[2, 1], {}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1]}, {1, 2}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1]}, {1, 2}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1, 2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 4, 3], [1, 3, 4, 2]}, {[1, 2, 4, 3], [1, 4, 2, 3]}, {[2, 1, 3, 4], [2, 3, 1, 4]}, {[2, 1, 3, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 4, 2, 1]}, {[3, 2, 4, 1], [3, 4, 2, 1]}, {[4, 1, 3, 2], [4, 3, 1, 2]}, {[4, 2, 1, 3], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [1, 3, 4, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[2, 4, 1, 3], {}, {1, 2}], [[2, 1], {}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {}, {2}], [[3, 4, 1, 2], {}, {1, 2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090, 85249942588971314, 469286147871837366, 2588758890960637798, 14308406109097843626, 79228031819993134650] This enumerating sequence seems to have the -1 + n 3 (1 + 2 n) N 2 annihilating operator , ------ - ------------- + N 2 + n 2 + n For the equivalence class of patterns, {{[1, 2, 4, 3], [1, 4, 3, 2]}, {[2, 1, 3, 4], [3, 2, 1, 4]}, {[2, 3, 4, 1], [3, 4, 2, 1]}, {[4, 1, 2, 3], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [1, 4, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[3, 4, 2, 1], {}, {3}], [[2, 3, 1], {}, {}], [[2, 1], {}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0]}, {1, 2}], [[3, 4, 1, 2], {}, {1, 2}], [[1, 3, 2], {[0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0]}, {1, 2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 89, 382, 1711, 7922, 37663, 182936, 904302, 4535994, 23034564, 118209806, 612165222, 3195359360, 16795435994, 88825567814, 472356139660, 2524292893556, 13549955878141, 73026827854516, 395017112175542, 2143881709415478, 11671226062503926, 63717160152793490, 348759626655606356, 1913558711839152932, 10522719736133406089, 57985227476697094630] For the equivalence class of patterns, {{[1, 2, 4, 3], [2, 1, 3, 4]}, {[3, 4, 2, 1], [4, 3, 1, 2]}} the member , {[1, 2, 4, 3], [2, 1, 3, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {}, {}], [[1, 3, 2], {[0, 0, 0, 2]}, {2}], [[3, 1, 2], {[0, 0, 0, 2]}, {1}], [[2, 1, 3], {[0, 0, 0, 1]}, {3}], [[3, 2, 1], {}, {1}], [[2, 3, 1], {[0, 0, 0, 2]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 6, 22, 87, 354, 1459, 6056, 25252, 105632, 442916, 1860498, 7826120, 32956964, 138911074, 585926818, 2472923499, 10442263142, 44112331275, 186413949540, 788000866243, 3331853294090, 14090947775581, 59604161832772, 252164535493017, 1066972399657514, 4515192187721619, 19109373797471836, 80883153826692957, 342378241152493302] Out of a total of , 56, cases 29, were successful and , 27, failed Success Rate: , 0.518 Here are the failures {{{[1, 3, 2, 4], [2, 4, 1, 3]}, {[1, 3, 2, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [4, 2, 3, 1]}, {[3, 1, 4, 2], [4, 2, 3, 1]}}, {{[1, 3, 2, 4], [2, 1, 4, 3]}, {[3, 4, 1, 2], [4, 2, 3, 1]}}, {{[1, 2, 3, 4], [4, 3, 2, 1]}}, {{[1, 2, 4, 3], [4, 2, 3, 1]}, {[1, 3, 2, 4], [3, 4, 2, 1]}, {[1, 3, 2, 4], [4, 3, 1, 2]}, {[2, 1, 3, 4], [4, 2, 3, 1]}}, {{[1, 4, 3, 2], [3, 2, 1, 4]}, {[2, 3, 4, 1], [4, 1, 2, 3]}}, { {[1, 2, 4, 3], [2, 4, 1, 3]}, {[1, 2, 4, 3], [3, 1, 4, 2]}, {[2, 1, 3, 4], [2, 4, 1, 3]}, {[2, 1, 3, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 4, 2, 1]}, {[2, 4, 1, 3], [4, 3, 1, 2]}, {[3, 1, 4, 2], [3, 4, 2, 1]}, {[3, 1, 4, 2], [4, 3, 1, 2]}}, { {[1, 4, 3, 2], [2, 4, 1, 3]}, {[1, 4, 3, 2], [3, 1, 4, 2]}, {[2, 3, 4, 1], [2, 4, 1, 3]}, {[2, 3, 4, 1], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 2, 1, 4]}, {[2, 4, 1, 3], [4, 1, 2, 3]}, {[3, 1, 4, 2], [3, 2, 1, 4]}, {[3, 1, 4, 2], [4, 1, 2, 3]}}, { {[1, 3, 4, 2], [1, 4, 2, 3]}, {[2, 3, 1, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 2, 4, 1]}, {[4, 1, 3, 2], [4, 2, 1, 3]}}, {{[1, 3, 2, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [4, 2, 3, 1]}}, { {[1, 3, 2, 4], [2, 4, 3, 1]}, {[1, 3, 2, 4], [3, 2, 4, 1]}, {[1, 3, 2, 4], [4, 1, 3, 2]}, {[1, 3, 2, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [4, 2, 3, 1]}, {[1, 4, 2, 3], [4, 2, 3, 1]}, {[2, 3, 1, 4], [4, 2, 3, 1]}, {[3, 1, 2, 4], [4, 2, 3, 1]}}, { {[1, 3, 2, 4], [1, 4, 3, 2]}, {[1, 3, 2, 4], [3, 2, 1, 4]}, {[2, 3, 4, 1], [4, 2, 3, 1]}, {[4, 1, 2, 3], [4, 2, 3, 1]}}, { {[1, 4, 3, 2], [2, 1, 4, 3]}, {[2, 1, 4, 3], [3, 2, 1, 4]}, {[2, 3, 4, 1], [3, 4, 1, 2]}, {[3, 4, 1, 2], [4, 1, 2, 3]}}, { {[1, 3, 4, 2], [2, 4, 1, 3]}, {[1, 4, 2, 3], [3, 1, 4, 2]}, {[2, 3, 1, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 1, 2, 4]}, {[2, 4, 1, 3], [3, 2, 4, 1]}, {[2, 4, 1, 3], [4, 1, 3, 2]}, {[2, 4, 3, 1], [3, 1, 4, 2]}, {[3, 1, 4, 2], [4, 2, 1, 3]}}, { {[1, 2, 4, 3], [3, 4, 1, 2]}, {[2, 1, 3, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [3, 4, 2, 1]}, {[2, 1, 4, 3], [4, 3, 1, 2]}}, {{[1, 3, 2, 4], [4, 2, 3, 1]}}, {{[2, 1, 4, 3], [2, 4, 1, 3]}, {[2, 1, 4, 3], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 4, 1, 2]}, {[3, 1, 4, 2], [3, 4, 1, 2]}}, {{[1, 4, 3, 2], [3, 4, 1, 2]}, {[2, 1, 4, 3], [2, 3, 4, 1]}, {[2, 1, 4, 3], [4, 1, 2, 3]}, {[3, 2, 1, 4], [3, 4, 1, 2]}}, {{[1, 3, 2, 4], [2, 3, 4, 1]}, {[1, 3, 2, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [4, 2, 3, 1]}, {[3, 2, 1, 4], [4, 2, 3, 1]}}, {{[1, 3, 4, 2], [3, 1, 2, 4]}, {[1, 4, 2, 3], [2, 3, 1, 4]}, {[2, 4, 3, 1], [4, 2, 1, 3]}, {[3, 2, 4, 1], [4, 1, 3, 2]}}, {{[1, 2, 3, 4], [2, 3, 4, 1]}, {[1, 2, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [4, 3, 2, 1]}, {[3, 2, 1, 4], [4, 3, 2, 1]}}, {{[2, 4, 1, 3], [3, 1, 4, 2]}}, { {[1, 2, 3, 4], [2, 4, 1, 3]}, {[1, 2, 3, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [4, 3, 2, 1]}, {[3, 1, 4, 2], [4, 3, 2, 1]}}, { {[1, 2, 3, 4], [2, 4, 3, 1]}, {[1, 2, 3, 4], [3, 2, 4, 1]}, {[1, 2, 3, 4], [4, 1, 3, 2]}, {[1, 2, 3, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [4, 3, 2, 1]}, {[1, 4, 2, 3], [4, 3, 2, 1]}, {[2, 3, 1, 4], [4, 3, 2, 1]}, {[3, 1, 2, 4], [4, 3, 2, 1]}}, {{[1, 2, 3, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [4, 3, 2, 1]}}, { {[1, 3, 4, 2], [3, 4, 1, 2]}, {[1, 4, 2, 3], [3, 4, 1, 2]}, {[2, 1, 4, 3], [2, 4, 3, 1]}, {[2, 1, 4, 3], [3, 2, 4, 1]}, {[2, 1, 4, 3], [4, 1, 3, 2]}, {[2, 1, 4, 3], [4, 2, 1, 3]}, {[2, 3, 1, 4], [3, 4, 1, 2]}, {[3, 1, 2, 4], [3, 4, 1, 2]}}, {{[1, 2, 3, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [4, 3, 2, 1]}}, { {[1, 3, 4, 2], [2, 3, 1, 4]}, {[1, 4, 2, 3], [3, 1, 2, 4]}, {[2, 4, 3, 1], [4, 1, 3, 2]}, {[3, 2, 4, 1], [4, 2, 1, 3]}}} {{{[1, 3, 2, 4], [2, 4, 1, 3]}, {[1, 3, 2, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [4, 2, 3, 1]}, {[3, 1, 4, 2], [4, 2, 3, 1]}}, {{[1, 3, 2, 4], [2, 1, 4, 3]}, {[3, 4, 1, 2], [4, 2, 3, 1]}}, {{[1, 2, 3, 4], [4, 3, 2, 1]}}, {{[1, 2, 4, 3], [4, 2, 3, 1]}, {[1, 3, 2, 4], [3, 4, 2, 1]}, {[1, 3, 2, 4], [4, 3, 1, 2]}, {[2, 1, 3, 4], [4, 2, 3, 1]}}, {{[1, 4, 3, 2], [3, 2, 1, 4]}, {[2, 3, 4, 1], [4, 1, 2, 3]}}, { {[1, 2, 4, 3], [2, 4, 1, 3]}, {[1, 2, 4, 3], [3, 1, 4, 2]}, {[2, 1, 3, 4], [2, 4, 1, 3]}, {[2, 1, 3, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 4, 2, 1]}, {[2, 4, 1, 3], [4, 3, 1, 2]}, {[3, 1, 4, 2], [3, 4, 2, 1]}, {[3, 1, 4, 2], [4, 3, 1, 2]}}, { {[1, 4, 3, 2], [2, 4, 1, 3]}, {[1, 4, 3, 2], [3, 1, 4, 2]}, {[2, 3, 4, 1], [2, 4, 1, 3]}, {[2, 3, 4, 1], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 2, 1, 4]}, {[2, 4, 1, 3], [4, 1, 2, 3]}, {[3, 1, 4, 2], [3, 2, 1, 4]}, {[3, 1, 4, 2], [4, 1, 2, 3]}}, { {[1, 3, 4, 2], [1, 4, 2, 3]}, {[2, 3, 1, 4], [3, 1, 2, 4]}, {[2, 4, 3, 1], [3, 2, 4, 1]}, {[4, 1, 3, 2], [4, 2, 1, 3]}}, {{[1, 3, 2, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [4, 2, 3, 1]}}, { {[1, 3, 2, 4], [2, 4, 3, 1]}, {[1, 3, 2, 4], [3, 2, 4, 1]}, {[1, 3, 2, 4], [4, 1, 3, 2]}, {[1, 3, 2, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [4, 2, 3, 1]}, {[1, 4, 2, 3], [4, 2, 3, 1]}, {[2, 3, 1, 4], [4, 2, 3, 1]}, {[3, 1, 2, 4], [4, 2, 3, 1]}}, { {[1, 3, 2, 4], [1, 4, 3, 2]}, {[1, 3, 2, 4], [3, 2, 1, 4]}, {[2, 3, 4, 1], [4, 2, 3, 1]}, {[4, 1, 2, 3], [4, 2, 3, 1]}}, { {[1, 4, 3, 2], [2, 1, 4, 3]}, {[2, 1, 4, 3], [3, 2, 1, 4]}, {[2, 3, 4, 1], [3, 4, 1, 2]}, {[3, 4, 1, 2], [4, 1, 2, 3]}}, { {[1, 3, 4, 2], [2, 4, 1, 3]}, {[1, 4, 2, 3], [3, 1, 4, 2]}, {[2, 3, 1, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 1, 2, 4]}, {[2, 4, 1, 3], [3, 2, 4, 1]}, {[2, 4, 1, 3], [4, 1, 3, 2]}, {[2, 4, 3, 1], [3, 1, 4, 2]}, {[3, 1, 4, 2], [4, 2, 1, 3]}}, { {[1, 2, 4, 3], [3, 4, 1, 2]}, {[2, 1, 3, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [3, 4, 2, 1]}, {[2, 1, 4, 3], [4, 3, 1, 2]}}, {{[1, 3, 2, 4], [4, 2, 3, 1]}}, {{[2, 1, 4, 3], [2, 4, 1, 3]}, {[2, 1, 4, 3], [3, 1, 4, 2]}, {[2, 4, 1, 3], [3, 4, 1, 2]}, {[3, 1, 4, 2], [3, 4, 1, 2]}}, {{[1, 4, 3, 2], [3, 4, 1, 2]}, {[2, 1, 4, 3], [2, 3, 4, 1]}, {[2, 1, 4, 3], [4, 1, 2, 3]}, {[3, 2, 1, 4], [3, 4, 1, 2]}}, {{[1, 3, 2, 4], [2, 3, 4, 1]}, {[1, 3, 2, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [4, 2, 3, 1]}, {[3, 2, 1, 4], [4, 2, 3, 1]}}, {{[1, 3, 4, 2], [3, 1, 2, 4]}, {[1, 4, 2, 3], [2, 3, 1, 4]}, {[2, 4, 3, 1], [4, 2, 1, 3]}, {[3, 2, 4, 1], [4, 1, 3, 2]}}, {{[1, 2, 3, 4], [2, 3, 4, 1]}, {[1, 2, 3, 4], [4, 1, 2, 3]}, {[1, 4, 3, 2], [4, 3, 2, 1]}, {[3, 2, 1, 4], [4, 3, 2, 1]}}, {{[2, 4, 1, 3], [3, 1, 4, 2]}}, { {[1, 2, 3, 4], [2, 4, 1, 3]}, {[1, 2, 3, 4], [3, 1, 4, 2]}, {[2, 4, 1, 3], [4, 3, 2, 1]}, {[3, 1, 4, 2], [4, 3, 2, 1]}}, { {[1, 2, 3, 4], [2, 4, 3, 1]}, {[1, 2, 3, 4], [3, 2, 4, 1]}, {[1, 2, 3, 4], [4, 1, 3, 2]}, {[1, 2, 3, 4], [4, 2, 1, 3]}, {[1, 3, 4, 2], [4, 3, 2, 1]}, {[1, 4, 2, 3], [4, 3, 2, 1]}, {[2, 3, 1, 4], [4, 3, 2, 1]}, {[3, 1, 2, 4], [4, 3, 2, 1]}}, {{[1, 2, 3, 4], [3, 4, 1, 2]}, {[2, 1, 4, 3], [4, 3, 2, 1]}}, { {[1, 3, 4, 2], [3, 4, 1, 2]}, {[1, 4, 2, 3], [3, 4, 1, 2]}, {[2, 1, 4, 3], [2, 4, 3, 1]}, {[2, 1, 4, 3], [3, 2, 4, 1]}, {[2, 1, 4, 3], [4, 1, 3, 2]}, {[2, 1, 4, 3], [4, 2, 1, 3]}, {[2, 3, 1, 4], [3, 4, 1, 2]}, {[3, 1, 2, 4], [3, 4, 1, 2]}}, {{[1, 2, 3, 4], [4, 2, 3, 1]}, {[1, 3, 2, 4], [4, 3, 2, 1]}}, { {[1, 3, 4, 2], [2, 3, 1, 4]}, {[1, 4, 2, 3], [3, 1, 2, 4]}, {[2, 4, 3, 1], [4, 1, 3, 2]}, {[3, 2, 4, 1], [4, 2, 1, 3]}}} It took, 287345.641, seconds of CPU time .