There are all together, 42, different equivalence classes For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 4, 5, 1]}, {[3, 2, 1], [5, 1, 2, 3, 4]}, {[1, 2, 3], [1, 5, 4, 3, 2]}, {[1, 2, 3], [4, 3, 2, 1, 5]}} the member , {[3, 2, 1], [2, 3, 4, 5, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {}, {}], [[1, 2, 3, 4], {[1, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 3, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[2, 3, 4, 1], {[1, 0, 0, 0, 0]}, {4}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 2, 1], [3, 4, 5, 1, 2]}, {[3, 2, 1], [4, 5, 1, 2, 3]}, {[1, 2, 3], [2, 1, 5, 4, 3]}, {[1, 2, 3], [3, 2, 1, 5, 4]}} the member , {[3, 2, 1], [3, 4, 5, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {[2, 0, 0, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481, 34315188682442] This enumerating sequence seems to have the 2 1 - 3 x + x rational generating function, -------------- 2 1 - 4 x + 3 x For the equivalence class of patterns, {{[3, 1, 2], [2, 1, 3, 4, 5]}, {[2, 3, 1], [2, 1, 3, 4, 5]}, {[1, 3, 2], [4, 5, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 5, 4]}, {[2, 1, 3], [4, 5, 3, 2, 1]}, {[1, 3, 2], [5, 4, 3, 1, 2]}, {[2, 3, 1], [1, 2, 3, 5, 4]}, {[2, 1, 3], [5, 4, 3, 1, 2]}} the member , {[3, 1, 2], [2, 1, 3, 4, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[1, 2], {}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 0, 1]}, {3}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 119, 334, 902, 2351, 5945] For the equivalence class of patterns, {{[3, 1, 2], [2, 1, 3, 5, 4]}, {[2, 3, 1], [2, 1, 3, 5, 4]}, {[1, 3, 2], [4, 5, 3, 1, 2]}, {[2, 1, 3], [4, 5, 3, 1, 2]}} the member , {[3, 1, 2], [2, 1, 3, 5, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[1, 2], {}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 354, 1021, 2901, 8130] For the equivalence class of patterns, {{[3, 1, 2], [2, 1, 5, 4, 3]}, {[1, 3, 2], [4, 5, 1, 2, 3]}, {[3, 1, 2], [3, 2, 1, 5, 4]}, {[2, 1, 3], [3, 4, 5, 1, 2]}, {[1, 3, 2], [3, 4, 5, 1, 2]}, {[2, 1, 3], [4, 5, 1, 2, 3]}, {[2, 3, 1], [2, 1, 5, 4, 3]}, {[2, 3, 1], [3, 2, 1, 5, 4]}} the member , {[3, 1, 2], [2, 1, 5, 4, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 2, 0]}, {}], [[1, 2], {}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[1, 3, 2], [5, 2, 3, 1, 4]}, {[1, 3, 2], [4, 2, 3, 5, 1]}, {[2, 1, 3], [5, 1, 3, 4, 2]}, {[2, 3, 1], [1, 5, 3, 2, 4]}, {[3, 1, 2], [1, 4, 3, 5, 2]}, {[3, 1, 2], [2, 4, 3, 1, 5]}, {[2, 3, 1], [4, 1, 3, 2, 5]}, {[2, 1, 3], [2, 5, 3, 4, 1]}} the member , {[1, 3, 2], [5, 2, 3, 1, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {2}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[3, 2, 1], {}, {2}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[2, 1], {}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[2, 3, 1], [5, 4, 2, 1, 3]}, {[1, 3, 2], [2, 3, 1, 4, 5]}, {[2, 3, 1], [5, 4, 1, 3, 2]}, {[2, 1, 3], [1, 2, 5, 3, 4]}, {[1, 3, 2], [3, 1, 2, 4, 5]}, {[3, 1, 2], [4, 3, 5, 2, 1]}, {[2, 1, 3], [1, 2, 4, 5, 3]}, {[3, 1, 2], [3, 5, 4, 2, 1]}} the member , {[2, 3, 1], [5, 4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 2, 1], {[1, 1, 0, 0]}, {3}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}], [[2, 1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481, 34315188682442] This enumerating sequence seems to have the 2 1 - 3 x + x rational generating function, -------------- 2 1 - 4 x + 3 x For the equivalence class of patterns, {{[3, 1, 2], [3, 2, 5, 4, 1]}, {[1, 3, 2], [3, 4, 1, 2, 5]}, {[2, 1, 3], [1, 4, 5, 2, 3]}, {[2, 3, 1], [5, 2, 1, 4, 3]}} the member , {[3, 1, 2], [3, 2, 5, 4, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {}], [[2, 1, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}] , [[1, 2], {}, {1}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [1, 0, 1, 0, 0], [1, 0, 0, 1, 0]}, {3}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[2, 1, 3], {[0, 1, 0, 0], [1, 0, 1, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[1, 3, 2], [4, 3, 2, 1, 5]}, {[3, 1, 2], [2, 3, 4, 5, 1]}, {[2, 1, 3], [1, 5, 4, 3, 2]}, {[2, 3, 1], [5, 1, 2, 3, 4]}} the member , {[1, 3, 2], [4, 3, 2, 1, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[4, 3, 1, 2], {[0, 1, 0, 0, 0]}, {3}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 2, 1], {}, {}], [[4, 2, 1, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}], [[2, 1], {}, {}], [[4, 3, 2, 1], {[0, 0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 119, 336, 927, 2527, 6870] For the equivalence class of patterns, {{[3, 1, 2], [1, 2, 5, 4, 3]}, {[3, 1, 2], [3, 2, 1, 4, 5]}, {[2, 1, 3], [3, 4, 5, 2, 1]}, {[2, 1, 3], [5, 4, 1, 2, 3]}, {[1, 3, 2], [3, 4, 5, 2, 1]}, {[1, 3, 2], [5, 4, 1, 2, 3]}, {[2, 3, 1], [1, 2, 5, 4, 3]}, {[2, 3, 1], [3, 2, 1, 4, 5]}} the member , {[3, 1, 2], [1, 2, 5, 4, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1], {}, {}], [[1, 2, 3, 4], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 3, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 3], {[0, 0, 2, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496] For the equivalence class of patterns, {{[2, 3, 1], [5, 4, 3, 1, 2]}, {[2, 1, 3], [1, 2, 3, 5, 4]}, {[3, 1, 2], [4, 5, 3, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4, 5]}} the member , {[2, 3, 1], [5, 4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 2, 1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}], [[2, 1], {}, {}], [[4, 2, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 2, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 3, 2, 1], {[0, 1, 0, 0, 0]}, {3}], [[4, 3, 1, 2], {[1, 0, 0, 0, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 1, 2], [2, 1, 4, 3, 5]}, {[1, 3, 2], [4, 5, 2, 3, 1]}, {[3, 1, 2], [1, 3, 2, 5, 4]}, {[2, 3, 1], [1, 3, 2, 5, 4]}, {[2, 1, 3], [5, 3, 4, 1, 2]}, {[1, 3, 2], [5, 3, 4, 1, 2]}, {[2, 1, 3], [4, 5, 2, 3, 1]}, {[2, 3, 1], [2, 1, 4, 3, 5]}} the member , {[3, 1, 2], [2, 1, 4, 3, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {}], [[2, 1, 3], {[0, 1, 0, 0]}, {}], [[1, 2], {}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 354, 1021, 2901, 8130] For the equivalence class of patterns, {{[3, 1, 2], [1, 2, 4, 5, 3]}, {[3, 1, 2], [2, 3, 1, 4, 5]}, {[1, 3, 2], [4, 3, 5, 2, 1]}, {[1, 3, 2], [5, 4, 2, 1, 3]}, {[2, 3, 1], [1, 2, 5, 3, 4]}, {[2, 1, 3], [5, 4, 1, 3, 2]}, {[2, 1, 3], [3, 5, 4, 2, 1]}, {[2, 3, 1], [3, 1, 2, 4, 5]}} the member , {[3, 1, 2], [1, 2, 4, 5, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 3, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 3, 4], {[0, 0, 1, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496] For the equivalence class of patterns, {{[1, 3, 2], [2, 3, 4, 5, 1]}, {[3, 1, 2], [4, 3, 2, 1, 5]}, {[1, 3, 2], [5, 1, 2, 3, 4]}, {[2, 1, 3], [5, 1, 2, 3, 4]}, {[2, 3, 1], [1, 5, 4, 3, 2]}, {[2, 1, 3], [2, 3, 4, 5, 1]}, {[3, 1, 2], [1, 5, 4, 3, 2]}, {[2, 3, 1], [4, 3, 2, 1, 5]}} the member , {[1, 3, 2], [5, 1, 2, 3, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {2}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 2, 0]}, {2}], [[2, 1], {[0, 3, 0]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481, 34315188682442] This enumerating sequence seems to have the 2 1 - 3 x + x rational generating function, -------------- 2 1 - 4 x + 3 x For the equivalence class of patterns, {{[2, 3, 1], [5, 4, 1, 2, 3]}, {[1, 3, 2], [3, 2, 1, 4, 5]}, {[3, 1, 2], [3, 4, 5, 2, 1]}, {[2, 1, 3], [1, 2, 5, 4, 3]}} the member , {[2, 3, 1], [5, 4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[4, 3, 1, 2], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {4}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 2, 1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}], [[2, 1], {}, {}], [[4, 2, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 2, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 3, 2, 1], {[0, 1, 1, 0, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 356, 1044, 3057, 8948] For the equivalence class of patterns, {{[3, 1, 2], [1, 3, 4, 2, 5]}, {[2, 1, 3], [5, 2, 4, 3, 1]}, {[2, 3, 1], [1, 4, 2, 3, 5]}, {[1, 3, 2], [5, 3, 2, 4, 1]}} the member , {[3, 1, 2], [1, 3, 4, 2, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 3, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0]}, {3}], [[1, 2, 3, 4], {}, {2}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496] For the equivalence class of patterns, {{[2, 3, 1], [5, 4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4, 5]}, {[3, 1, 2], [5, 4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4, 5]}} the member , {[2, 3, 1], [5, 4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[4, 3, 2, 1], {[1, 0, 0, 0, 0]}, {4}], [[3, 2, 1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}], [[2, 1], {}, {}], [[4, 2, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 2, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 3, 1, 2], {[1, 0, 0, 0, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 1, 2], [1, 3, 4, 5, 2]}, {[1, 3, 2], [4, 3, 2, 5, 1]}, {[2, 1, 3], [2, 5, 4, 3, 1]}, {[2, 3, 1], [1, 5, 2, 3, 4]}, {[2, 1, 3], [5, 1, 4, 3, 2]}, {[1, 3, 2], [5, 3, 2, 1, 4]}, {[3, 1, 2], [2, 3, 4, 1, 5]}, {[2, 3, 1], [4, 1, 2, 3, 5]}} the member , {[3, 1, 2], [1, 3, 4, 5, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 3, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 3, 4], {[0, 1, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0]}, {3}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 356, 1044, 3057, 8948] For the equivalence class of patterns, {{[3, 1, 2], [1, 4, 3, 2, 5]}, {[2, 3, 1], [1, 4, 3, 2, 5]}, {[1, 3, 2], [5, 2, 3, 4, 1]}, {[2, 1, 3], [5, 2, 3, 4, 1]}} the member , {[3, 1, 2], [1, 4, 3, 2, 5]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[1, 3, 2, 5, 4], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {2, 3}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0]}, {}], [[1, 3, 2, 4, 5], {[0, 0, 1, 0, 0, 0]}, {4}], [[1, 2, 3], {}, {2}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {2}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {2}], [[2, 4, 3, 5, 1], { [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {4}], [[1, 4, 3, 5, 2], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]}, {4}], [[1, 4, 2, 5, 3], {[0, 0, 0, 0, 0, 0]}, {4}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 121, 355, 1032, 2973, 8496] For the equivalence class of patterns, {{[3, 2, 1], [2, 4, 5, 1, 3]}, {[3, 2, 1], [3, 4, 1, 5, 2]}, {[3, 2, 1], [3, 5, 1, 2, 4]}, {[3, 2, 1], [4, 1, 5, 2, 3]}, {[1, 2, 3], [2, 5, 1, 4, 3]}, {[1, 2, 3], [3, 1, 5, 4, 2]}, {[1, 2, 3], [3, 2, 5, 1, 4]}, {[1, 2, 3], [4, 2, 1, 5, 3]}} the member , {[3, 2, 1], [2, 4, 5, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 3], {[1, 1, 0, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481, 34315188682442] This enumerating sequence seems to have the 2 1 - 3 x + x rational generating function, -------------- 2 1 - 4 x + 3 x For the equivalence class of patterns, {{[1, 3, 2], [5, 3, 4, 2, 1]}, {[3, 1, 2], [1, 2, 4, 3, 5]}, {[1, 3, 2], [5, 4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4, 5]}, {[2, 1, 3], [5, 3, 4, 2, 1]}, {[2, 1, 3], [5, 4, 2, 3, 1]}, {[3, 1, 2], [1, 3, 2, 4, 5]}, {[2, 3, 1], [1, 2, 4, 3, 5]}} the member , {[1, 3, 2], [5, 3, 4, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {2}], [[4, 1, 2, 3], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}], [[3, 2, 1], {}, {2}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[2, 1], {}, {}], [[3, 1, 2, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[4, 2, 3, 1], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 119, 334, 902, 2351, 5945] For the equivalence class of patterns, {{[1, 3, 2], [3, 2, 4, 5, 1]}, {[1, 3, 2], [5, 2, 1, 3, 4]}, {[2, 1, 3], [2, 3, 5, 4, 1]}, {[3, 1, 2], [3, 4, 2, 1, 5]}, {[3, 1, 2], [1, 4, 5, 3, 2]}, {[2, 1, 3], [5, 1, 2, 4, 3]}, {[2, 3, 1], [1, 5, 4, 2, 3]}, {[2, 3, 1], [4, 3, 1, 2, 5]}} the member , {[1, 3, 2], [5, 2, 1, 3, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 2, 1], {[0, 0, 2, 0]}, {2}], [[2, 1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481, 34315188682442] This enumerating sequence seems to have the 2 1 - 3 x + x rational generating function, -------------- 2 1 - 4 x + 3 x For the equivalence class of patterns, {{[1, 3, 2], [3, 4, 2, 5, 1]}, {[2, 3, 1], [1, 5, 2, 4, 3]}, {[2, 1, 3], [5, 1, 4, 2, 3]}, {[1, 3, 2], [5, 3, 1, 2, 4]}, {[2, 1, 3], [2, 4, 5, 3, 1]}, {[3, 1, 2], [1, 3, 5, 4, 2]}, {[3, 1, 2], [3, 2, 4, 1, 5]}, {[2, 3, 1], [4, 2, 1, 3, 5]}} the member , {[1, 3, 2], [3, 4, 2, 5, 1]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1], {}, {}], [[4, 5, 2, 3, 1], {[0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {1, 2}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {}], [[2, 1], {}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {[0, 1, 0]}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {3}], [ [2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[4, 5, 1, 2, 3], {[0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [[4, 5, 1, 3, 2], {[0, 0, 0, 0, 0, 0]}, {3}], [[3, 5, 1, 2, 4], {[0, 0, 0, 0, 0, 0]}, {3}], [[3, 4, 1, 2, 5], { [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 5, 1, 4]}, {[3, 2, 1], [2, 5, 1, 3, 4]}, {[3, 2, 1], [3, 1, 4, 5, 2]}, {[3, 2, 1], [4, 1, 2, 5, 3]}, {[1, 2, 3], [2, 5, 4, 1, 3]}, {[1, 2, 3], [3, 5, 2, 1, 4]}, {[1, 2, 3], [4, 1, 5, 3, 2]}, {[1, 2, 3], [4, 3, 1, 5, 2]}} the member , {[3, 2, 1], [2, 3, 5, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2, 3], {[1, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 122, 364, 1083, 3208, 9462, 27812, 81545, 238696, 698005, 2040025, 5960779, 17415475, 50882771, 148670706, 434413073, 1269406641, 3709491754, 10840243660, 31678942133, 92577687779, 270547814849, 790646863094, 2310581123630, 6752424224926, 19733219149950] This enumerating sequence seems to have the 2 3 -1 + 6 x - 12 x + 8 x rational generating function, - ----------------------------------- 2 3 4 5 1 - 7 x + 17 x - 16 x + 3 x + x For the equivalence class of patterns, {{[3, 2, 1], [1, 2, 4, 5, 3]}, {[3, 2, 1], [1, 2, 5, 3, 4]}, {[3, 2, 1], [2, 3, 1, 4, 5]}, {[3, 2, 1], [3, 1, 2, 4, 5]}, {[1, 2, 3], [3, 5, 4, 2, 1]}, {[1, 2, 3], [4, 3, 5, 2, 1]}, {[1, 2, 3], [5, 4, 1, 3, 2]}, {[1, 2, 3], [5, 4, 2, 1, 3]}} the member , {[3, 2, 1], [1, 2, 5, 3, 4]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[1, 2, 3, 4], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {3}], [[2, 3, 4, 1], {[1, 0, 0, 0, 0], [0, 0, 0, 2, 0]}, {}], [[1, 2, 3], {[0, 0, 2, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {}], [[2, 3, 4, 1, 5], { [1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 0, 0]}, {3}], [[3, 4, 2, 5, 1], {[0, 0, 0, 0, 0, 0]}, {3}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 2, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[2, 4, 1, 5, 3], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 2, 0]}, {3}], [ [2, 3, 1, 4, 5], {[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0], [0, 0, 0, 2, 0, 0]}, {4}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 4, 2, 3], %1, {1}], [[3, 4, 1, 5, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0]}, {1, 2}], [[2, 4, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {2}], [[2, 4, 1, 3], %1, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], %1, {2}], [ [3, 1, 2, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 4, 5, 1, 3], { [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 2, 0]}, {1}], [[3, 4, 5, 1, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0]}, {1}], [[3, 4, 5, 2, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[1, 3, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 2, 0]}, {2}], [ [2, 3, 5, 1, 4], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {1}], [ [2, 3, 1, 5, 4], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0]}, {3}], [[2, 1], {[1, 0, 0]}, {}]} %1 := {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 116, 307, 760, 1779, 3986] For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 4, 5, 2]}, {[3, 2, 1], [1, 5, 2, 3, 4]}, {[3, 2, 1], [2, 3, 4, 1, 5]}, {[3, 2, 1], [4, 1, 2, 3, 5]}, {[1, 2, 3], [2, 5, 4, 3, 1]}, {[1, 2, 3], [4, 3, 2, 5, 1]}, {[1, 2, 3], [5, 1, 4, 3, 2]}, {[1, 2, 3], [5, 3, 2, 1, 4]}} the member , {[3, 2, 1], [1, 3, 4, 5, 2]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[1, 2, 3, 4], {[0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 5, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {5}], [[1, 3, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {4}], [[2, 3, 4, 1], {[1, 0, 0, 0, 0]}, {}], [[3, 4, 2, 5, 1], {[0, 0, 0, 0, 0, 0]}, {3}], [[2, 3, 1, 4, 5], {[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1, 2}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[2, 3, 4, 1, 5], {[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 4, 5, 1, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [ [2, 4, 5, 1, 3], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {4}], [ [2, 4, 1, 5, 3], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0]}, {3}], [ [2, 3, 5, 1, 4], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[2, 3, 1, 5, 4], { [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 5, 2, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 119, 336, 924, 2492, 6636] For the equivalence class of patterns, {{[3, 2, 1], [1, 4, 5, 2, 3]}, {[3, 2, 1], [3, 4, 1, 2, 5]}, {[1, 2, 3], [3, 2, 5, 4, 1]}, {[1, 2, 3], [5, 2, 1, 4, 3]}} the member , {[3, 2, 1], [1, 4, 5, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 2, 0, 0]}, {1}], [[1, 2, 3], {[0, 2, 0, 0]}, {2}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 119, 336, 924, 2492, 6636] For the equivalence class of patterns, {{[1, 3, 2], [2, 3, 4, 1, 5]}, {[1, 3, 2], [4, 1, 2, 3, 5]}, {[2, 3, 1], [5, 3, 2, 1, 4]}, {[3, 1, 2], [2, 5, 4, 3, 1]}, {[2, 1, 3], [1, 3, 4, 5, 2]}, {[2, 1, 3], [1, 5, 2, 3, 4]}, {[3, 1, 2], [4, 3, 2, 5, 1]}, {[2, 3, 1], [5, 1, 4, 3, 2]}} the member , {[1, 3, 2], [2, 3, 4, 1, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0]}, {3}], [[1, 2], {[0, 1, 0]}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {3}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[1, 2, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[1, 3, 2], [4, 3, 1, 2, 5]}, {[1, 3, 2], [3, 4, 2, 1, 5]}, {[2, 1, 3], [1, 4, 5, 3, 2]}, {[2, 1, 3], [1, 5, 4, 2, 3]}, {[3, 1, 2], [3, 2, 4, 5, 1]}, {[3, 1, 2], [2, 3, 5, 4, 1]}, {[2, 3, 1], [5, 1, 2, 4, 3]}, {[2, 3, 1], [5, 2, 1, 3, 4]}} the member , {[1, 3, 2], [4, 3, 1, 2, 5]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[2, 1], {}, {}], [[3, 2, 1], {[0, 1, 0, 1]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 14, 41, 121, 355, 1033, 2986, 8594, 24674, 70757, 202814, 581272, 1666003, 4775323, 13688536, 39240095, 112490042, 322480613, 924475493, 2650255885, 7597664869, 21780717418, 62440153552, 179001040232, 513153190661, 1471087325732, 4217254874996, 12089859016909] This enumerating sequence seems to have the 2 3 -1 + 5 x - 8 x + 4 x rational generating function, - --------------------------- 2 3 4 1 - 6 x + 12 x - 9 x + x For the equivalence class of patterns, {{[3, 1, 2], [2, 3, 1, 5, 4]}, {[1, 3, 2], [4, 3, 5, 1, 2]}, {[3, 1, 2], [2, 1, 4, 5, 3]}, {[1, 3, 2], [4, 5, 2, 1, 3]}, {[2, 1, 3], [3, 5, 4, 1, 2]}, {[2, 1, 3], [4, 5, 1, 3, 2]}, {[2, 3, 1], [2, 1, 5, 3, 4]}, {[2, 3, 1], [3, 1, 2, 5, 4]}} the member , {[3, 1, 2], [2, 3, 1, 5, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {3}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[2, 3, 1, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[3, 4, 2, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {3}], [[1, 2, 3], {}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 4, 2, 5]}, {[3, 2, 1], [1, 4, 2, 3, 5]}, {[1, 2, 3], [5, 2, 4, 3, 1]}, {[1, 2, 3], [5, 3, 2, 4, 1]}} the member , {[3, 2, 1], [1, 3, 4, 2, 5]}, has a scheme of depth , 5 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 2, 4, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[3, 4, 1, 5, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {5}], [[2, 3, 1, 4, 5], {[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0]}, {1, 2}], [[2, 3, 4, 1], {[1, 0, 0, 0, 0]}, {}], [[3, 4, 2, 5, 1], {[0, 0, 0, 0, 0, 0]}, {3}], [[1, 3, 4, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 1]}, {1}], [ [2, 4, 5, 1, 3], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {1}], [[2, 4, 1, 5, 3], { [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1]}, {3}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 4, 5, 1, 2], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0]}, {4}], [ [2, 3, 5, 1, 4], {[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {1}], [[2, 3, 1, 5, 4], { [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 4, 1, 5], {[1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0]}, {2}], [[3, 4, 5, 2, 1], {[0, 0, 0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[1, 2, 3, 4], {[0, 1, 0, 1, 0]}, {2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 118, 321, 816, 1946, 4396] For the equivalence class of patterns, {{[1, 3, 2], [3, 2, 4, 1, 5]}, {[3, 1, 2], [2, 4, 5, 3, 1]}, {[1, 3, 2], [4, 2, 1, 3, 5]}, {[2, 1, 3], [1, 5, 2, 4, 3]}, {[2, 1, 3], [1, 3, 5, 4, 2]}, {[3, 1, 2], [3, 4, 2, 5, 1]}, {[2, 3, 1], [5, 1, 4, 2, 3]}, {[2, 3, 1], [5, 3, 1, 2, 4]}} the member , {[1, 3, 2], [3, 2, 4, 1, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[2, 1], {}, {}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]}, {3}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {3}], [[3, 2, 1], {}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842] For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 5, 2, 4]}, {[3, 2, 1], [1, 4, 2, 5, 3]}, {[3, 2, 1], [2, 4, 1, 3, 5]}, {[3, 2, 1], [3, 1, 4, 2, 5]}, {[1, 2, 3], [3, 5, 2, 4, 1]}, {[1, 2, 3], [4, 2, 5, 3, 1]}, {[1, 2, 3], [5, 2, 4, 1, 3]}, {[1, 2, 3], [5, 3, 1, 4, 2]}} the member , {[3, 2, 1], [1, 3, 5, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2], {}, {}], [[1], {}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 120, 345, 972, 2691, 7348] For the equivalence class of patterns, {{[2, 1, 3], [5, 4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4, 5]}, {[3, 1, 2], [1, 2, 3, 4, 5]}, {[1, 3, 2], [5, 4, 3, 2, 1]}} the member , {[3, 1, 2], [1, 2, 3, 4, 5]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1], {}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 1]}, {1}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 3, 4, 1], {[0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}], [[1, 2, 4, 3], {[0, 0, 0, 1, 0]}, {3}], [[1, 3, 4, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {4}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 14, 41, 116, 302, 715, 1549, 3106] Out of a total of , 42, cases 34, were successful and , 8, failed Success Rate: , 0.810 Here are the failures {{{[3, 2, 1], [2, 1, 3, 5, 4]}, {[1, 2, 3], [4, 5, 3, 1, 2]}}, { {[3, 2, 1], [2, 4, 1, 5, 3]}, {[3, 2, 1], [3, 1, 5, 2, 4]}, {[1, 2, 3], [3, 5, 1, 4, 2]}, {[1, 2, 3], [4, 2, 5, 1, 3]}}, { {[3, 2, 1], [2, 1, 4, 5, 3]}, {[3, 2, 1], [2, 1, 5, 3, 4]}, {[3, 2, 1], [2, 3, 1, 5, 4]}, {[3, 2, 1], [3, 1, 2, 5, 4]}, {[1, 2, 3], [3, 5, 4, 1, 2]}, {[1, 2, 3], [4, 3, 5, 1, 2]}, {[1, 2, 3], [4, 5, 1, 3, 2]}, {[1, 2, 3], [4, 5, 2, 1, 3]}}, { {[3, 2, 1], [1, 2, 3, 5, 4]}, {[3, 2, 1], [2, 1, 3, 4, 5]}, {[1, 2, 3], [4, 5, 3, 2, 1]}, {[1, 2, 3], [5, 4, 3, 1, 2]}}, { {[3, 2, 1], [1, 2, 4, 3, 5]}, {[3, 2, 1], [1, 3, 2, 4, 5]}, {[1, 2, 3], [5, 3, 4, 2, 1]}, {[1, 2, 3], [5, 4, 2, 3, 1]}}, {{[3, 2, 1], [1, 2, 3, 4, 5]}, {[1, 2, 3], [5, 4, 3, 2, 1]}}, { {[3, 1, 2], [2, 4, 3, 5, 1]}, {[2, 1, 3], [1, 5, 3, 4, 2]}, {[1, 3, 2], [4, 2, 3, 1, 5]}, {[2, 3, 1], [5, 1, 3, 2, 4]}}, { {[3, 2, 1], [1, 3, 2, 5, 4]}, {[3, 2, 1], [2, 1, 4, 3, 5]}, {[1, 2, 3], [4, 5, 2, 3, 1]}, {[1, 2, 3], [5, 3, 4, 1, 2]}}} {{{[3, 2, 1], [2, 1, 3, 5, 4]}, {[1, 2, 3], [4, 5, 3, 1, 2]}}, { {[3, 2, 1], [2, 4, 1, 5, 3]}, {[3, 2, 1], [3, 1, 5, 2, 4]}, {[1, 2, 3], [3, 5, 1, 4, 2]}, {[1, 2, 3], [4, 2, 5, 1, 3]}}, { {[3, 2, 1], [2, 1, 4, 5, 3]}, {[3, 2, 1], [2, 1, 5, 3, 4]}, {[3, 2, 1], [2, 3, 1, 5, 4]}, {[3, 2, 1], [3, 1, 2, 5, 4]}, {[1, 2, 3], [3, 5, 4, 1, 2]}, {[1, 2, 3], [4, 3, 5, 1, 2]}, {[1, 2, 3], [4, 5, 1, 3, 2]}, {[1, 2, 3], [4, 5, 2, 1, 3]}}, { {[3, 2, 1], [1, 2, 3, 5, 4]}, {[3, 2, 1], [2, 1, 3, 4, 5]}, {[1, 2, 3], [4, 5, 3, 2, 1]}, {[1, 2, 3], [5, 4, 3, 1, 2]}}, { {[3, 2, 1], [1, 2, 4, 3, 5]}, {[3, 2, 1], [1, 3, 2, 4, 5]}, {[1, 2, 3], [5, 3, 4, 2, 1]}, {[1, 2, 3], [5, 4, 2, 3, 1]}}, {{[3, 2, 1], [1, 2, 3, 4, 5]}, {[1, 2, 3], [5, 4, 3, 2, 1]}}, { {[3, 1, 2], [2, 4, 3, 5, 1]}, {[2, 1, 3], [1, 5, 3, 4, 2]}, {[1, 3, 2], [4, 2, 3, 1, 5]}, {[2, 3, 1], [5, 1, 3, 2, 4]}}, { {[3, 2, 1], [1, 3, 2, 5, 4]}, {[3, 2, 1], [2, 1, 4, 3, 5]}, {[1, 2, 3], [4, 5, 2, 3, 1]}, {[1, 2, 3], [5, 3, 4, 1, 2]}}}