There are all together, 23, different equivalence classes For the equivalence class of patterns, {{[2, 1, 3], [2, 3, 1], [1, 4, 3, 2]}, {[2, 1, 3], [3, 1, 2], [2, 3, 4, 1]}, {[2, 1, 3], [3, 1, 2], [1, 4, 3, 2]}, {[2, 1, 3], [2, 3, 1], [4, 1, 2, 3]}, {[1, 3, 2], [2, 3, 1], [3, 2, 1, 4]}, {[1, 3, 2], [2, 3, 1], [4, 1, 2, 3]}, {[1, 3, 2], [3, 1, 2], [2, 3, 4, 1]}, {[1, 3, 2], [3, 1, 2], [3, 2, 1, 4]}} the member , {[2, 1, 3], [2, 3, 1], [1, 4, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {1}], [[1], {[1, 1]}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[2, 1, 3], [2, 3, 1], [4, 3, 1, 2]}, {[2, 1, 3], [3, 1, 2], [1, 2, 4, 3]}, {[2, 1, 3], [3, 1, 2], [3, 4, 2, 1]}, {[1, 3, 2], [2, 3, 1], [2, 1, 3, 4]}, {[1, 3, 2], [2, 3, 1], [4, 3, 1, 2]}, {[1, 3, 2], [3, 1, 2], [2, 1, 3, 4]}, {[1, 3, 2], [3, 1, 2], [3, 4, 2, 1]}, {[2, 1, 3], [2, 3, 1], [1, 2, 4, 3]}} the member , {[2, 1, 3], [2, 3, 1], [4, 3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1], {[1, 1]}, {}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2], [1, 2, 3, 4]}, {[3, 2, 1], [2, 1, 3], [1, 2, 3, 4]}, {[1, 2, 3], [2, 3, 1], [4, 3, 2, 1]}, {[1, 2, 3], [3, 1, 2], [4, 3, 2, 1]}} the member , {[3, 2, 1], [1, 3, 2], [1, 2, 3, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0], [0, 0, 2]}, {1}], [[2, 1], {[1, 0, 0], [0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]}, {1}], [[1], {[0, 3]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2], [2, 3, 1, 4]}, {[3, 2, 1], [1, 3, 2], [3, 1, 2, 4]}, {[3, 2, 1], [2, 1, 3], [1, 3, 4, 2]}, {[3, 2, 1], [2, 1, 3], [1, 4, 2, 3]}, {[1, 2, 3], [2, 3, 1], [4, 1, 3, 2]}, {[1, 2, 3], [2, 3, 1], [4, 2, 1, 3]}, {[1, 2, 3], [3, 1, 2], [2, 4, 3, 1]}, {[1, 2, 3], [3, 1, 2], [3, 2, 4, 1]}} the member , {[3, 2, 1], [1, 3, 2], [2, 3, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58] the sequence seems to be polynomial -4 + 2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [1, 2, 4, 3]}, {[3, 2, 1], [2, 3, 1], [2, 1, 3, 4]}, {[3, 2, 1], [3, 1, 2], [1, 2, 4, 3]}, {[3, 2, 1], [3, 1, 2], [2, 1, 3, 4]}, {[1, 2, 3], [1, 3, 2], [3, 4, 2, 1]}, {[1, 2, 3], [1, 3, 2], [4, 3, 1, 2]}, {[1, 2, 3], [2, 1, 3], [3, 4, 2, 1]}, {[1, 2, 3], [2, 1, 3], [4, 3, 1, 2]}} the member , {[3, 2, 1], [2, 3, 1], [1, 2, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[1, 2], {[1, 0, 0]}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85] the sequence seems to be polynomial -8 + 3 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [4, 1, 2, 3]}, {[3, 2, 1], [3, 1, 2], [2, 3, 4, 1]}, {[1, 2, 3], [1, 3, 2], [3, 2, 1, 4]}, {[1, 2, 3], [2, 1, 3], [1, 4, 3, 2]}} the member , {[3, 2, 1], [2, 3, 1], [4, 1, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 3, 0]}, {1}], [[1], {[3, 0]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080] This enumerating sequence seems to have the 1 rational generating function, - ---------------- 2 3 -1 + x + x + x For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2], [2, 1, 3, 4]}, {[3, 2, 1], [2, 1, 3], [1, 2, 4, 3]}, {[1, 2, 3], [2, 3, 1], [4, 3, 1, 2]}, {[1, 2, 3], [3, 1, 2], [3, 4, 2, 1]}} the member , {[3, 2, 1], [1, 3, 2], [2, 1, 3, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {[1, 0, 0], [0, 0, 2]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58] the sequence seems to be polynomial -4 + 2 n For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 3], [2, 4, 1, 3]}, {[3, 2, 1], [1, 2, 3], [3, 1, 4, 2]}} the member , {[3, 2, 1], [1, 2, 3], [2, 4, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[3, 0], [0, 3]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 1, 2], [0, 0, 3]}, {1}], [[1, 2], {[0, 0, 1], [1, 1, 0], [0, 2, 0], [3, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, { {[2, 3, 1], [3, 1, 2], [1, 3, 2, 4]}, {[1, 3, 2], [2, 1, 3], [4, 2, 3, 1]}} the member , {[2, 3, 1], [3, 1, 2], [1, 3, 2, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, { {[2, 3, 1], [3, 1, 2], [2, 1, 4, 3]}, {[1, 3, 2], [2, 1, 3], [3, 4, 1, 2]}} the member , {[2, 3, 1], [3, 1, 2], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2], [2, 3, 4, 1]}, {[3, 2, 1], [1, 3, 2], [4, 1, 2, 3]}, {[3, 2, 1], [2, 1, 3], [2, 3, 4, 1]}, {[3, 2, 1], [2, 1, 3], [4, 1, 2, 3]}, {[1, 2, 3], [2, 3, 1], [1, 4, 3, 2]}, {[1, 2, 3], [2, 3, 1], [3, 2, 1, 4]}, {[1, 2, 3], [3, 1, 2], [1, 4, 3, 2]}, {[1, 2, 3], [3, 1, 2], [3, 2, 1, 4]}} the member , {[3, 2, 1], [1, 3, 2], [2, 3, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58] the sequence seems to be polynomial -4 + 2 n For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 3], [3, 4, 1, 2]}, {[1, 2, 3], [2, 3, 1], [2, 1, 4, 3]}, {[1, 2, 3], [3, 1, 2], [2, 1, 4, 3]}} the member , {[3, 2, 1], [1, 3, 2], [3, 4, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {1}], [[1, 2], {[0, 1, 0], [2, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58] the sequence seems to be polynomial -4 + 2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [1, 4, 2, 3]}, {[3, 2, 1], [2, 3, 1], [3, 1, 2, 4]}, {[3, 2, 1], [3, 1, 2], [1, 3, 4, 2]}, {[3, 2, 1], [3, 1, 2], [2, 3, 1, 4]}, {[1, 2, 3], [1, 3, 2], [3, 2, 4, 1]}, {[1, 2, 3], [1, 3, 2], [4, 2, 1, 3]}, {[1, 2, 3], [2, 1, 3], [2, 4, 3, 1]}, {[1, 2, 3], [2, 1, 3], [4, 1, 3, 2]}} the member , {[3, 2, 1], [2, 3, 1], [1, 4, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {[1, 0, 0], [0, 2, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308] This enumerating sequence seems to have the 3 1 - x + x rational generating function, ------------ 3 1 - 2 x + x For the equivalence class of patterns, {{[2, 1, 3], [2, 3, 1], [4, 3, 2, 1]}, {[2, 1, 3], [3, 1, 2], [4, 3, 2, 1]}, {[1, 3, 2], [2, 3, 1], [1, 2, 3, 4]}, {[2, 1, 3], [3, 1, 2], [1, 2, 3, 4]}, {[1, 3, 2], [2, 3, 1], [4, 3, 2, 1]}, {[1, 3, 2], [3, 1, 2], [1, 2, 3, 4]}, {[1, 3, 2], [3, 1, 2], [4, 3, 2, 1]}, {[2, 1, 3], [2, 3, 1], [1, 2, 3, 4]}} the member , {[2, 1, 3], [2, 3, 1], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 0, 0]}, {1}], [[1], {[1, 1]}, {}], [[2, 1], {[0, 0, 1], [1, 1, 0]}, {}], [[3, 2, 1], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [1, 2, 3, 4]}, {[3, 2, 1], [3, 1, 2], [1, 2, 3, 4]}, {[1, 2, 3], [1, 3, 2], [4, 3, 2, 1]}, {[1, 2, 3], [2, 1, 3], [4, 3, 2, 1]}} the member , {[3, 2, 1], [2, 3, 1], [1, 2, 3, 4]}, has a scheme of depth , 3 here it is: {[[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 3], [0, 0, 2, 0], [0, 0, 1, 1]}, {1} ], [[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0], [0, 3, 0], [0, 2, 1]}, {}], [[1, 2], {[1, 0, 0], [0, 3, 0], [0, 2, 1], [0, 0, 3]}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 3], [0, 0, 2, 0], [0, 0, 1, 1]}, {2} ], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 3, 0], [0, 0, 2, 1], [0, 0, 0, 3]}, {1} ]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 3], [2, 1, 4, 3]}, {[3, 2, 1], [1, 2, 3], [3, 4, 1, 2]}} the member , {[3, 2, 1], [1, 2, 3], [2, 1, 4, 3]}, has a scheme of depth , 2 here it is: {[[1, 2], {[0, 0, 1], [0, 2, 0], [2, 1, 0], [3, 0, 0]}, {1}], [[], {}, {}], [[1], {[3, 0], [0, 3]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 0, 2]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, { {[2, 3, 1], [3, 1, 2], [4, 3, 2, 1]}, {[1, 3, 2], [2, 1, 3], [1, 2, 3, 4]}} the member , {[2, 3, 1], [3, 1, 2], [4, 3, 2, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 3, 0]}, {1}], [[1], {[3, 0]}, {}], [[2, 1], {[0, 1, 0], [2, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064, 15902591, 29249425, 53798080] This enumerating sequence seems to have the 1 rational generating function, - ---------------- 2 3 -1 + x + x + x For the equivalence class of patterns, {{[2, 3, 1], [3, 1, 2], [1, 2, 4, 3]}, {[2, 3, 1], [3, 1, 2], [2, 1, 3, 4]}, {[1, 3, 2], [2, 1, 3], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3], [4, 3, 1, 2]}} the member , {[2, 3, 1], [3, 1, 2], [1, 2, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [2, 1, 4, 3]}, {[3, 2, 1], [3, 1, 2], [2, 1, 4, 3]}, {[1, 2, 3], [1, 3, 2], [3, 4, 1, 2]}, {[1, 2, 3], [2, 1, 3], [3, 4, 1, 2]}} the member , {[3, 2, 1], [2, 3, 1], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 1], {[1, 0, 0]}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[2, 1, 3], [3, 1, 2], [1, 3, 4, 2]}, {[2, 1, 3], [2, 3, 1], [1, 4, 2, 3]}, {[2, 1, 3], [3, 1, 2], [2, 4, 3, 1]}, {[2, 1, 3], [2, 3, 1], [4, 1, 3, 2]}, {[1, 3, 2], [2, 3, 1], [3, 1, 2, 4]}, {[1, 3, 2], [2, 3, 1], [4, 2, 1, 3]}, {[1, 3, 2], [3, 1, 2], [2, 3, 1, 4]}, {[1, 3, 2], [3, 1, 2], [3, 2, 4, 1]}} the member , {[2, 1, 3], [3, 1, 2], [1, 3, 4, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 1]}, {1}], [[2, 1], {[0, 1, 0], [0, 0, 1]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1], [1, 3, 2, 4]}, {[3, 2, 1], [3, 1, 2], [1, 3, 2, 4]}, {[1, 2, 3], [1, 3, 2], [4, 2, 3, 1]}, {[1, 2, 3], [2, 1, 3], [4, 2, 3, 1]}} the member , {[3, 2, 1], [2, 3, 1], [1, 3, 2, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {1}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, { {[2, 3, 1], [3, 1, 2], [1, 2, 3, 4]}, {[1, 3, 2], [2, 1, 3], [4, 3, 2, 1]}} the member , {[2, 3, 1], [3, 1, 2], [1, 2, 3, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[2, 3, 1], [3, 1, 2], [1, 4, 3, 2]}, {[2, 3, 1], [3, 1, 2], [3, 2, 1, 4]}, {[1, 3, 2], [2, 1, 3], [2, 3, 4, 1]}, {[1, 3, 2], [2, 1, 3], [4, 1, 2, 3]}} the member , {[2, 3, 1], [3, 1, 2], [1, 4, 3, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2], {[1, 0, 0], [0, 2, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308] This enumerating sequence seems to have the 3 1 - x + x rational generating function, ------------ 3 1 - 2 x + x Out of a total of , 23, cases 23, were successful and , 0, failed Success Rate: , 1. Here are the failures {} {} It took, 31.041, seconds of CPU time .