There all together, 92, different equivalence classes For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [4, 2, 1, 3]}, {[1, 2, 3], [3, 2, 4, 1], [4, 1, 3, 2]}, {[3, 2, 1], [1, 3, 4, 2], [3, 1, 2, 4]}, {[3, 2, 1], [1, 4, 2, 3], [2, 3, 1, 4]}} the member , {[1, 2, 3], [2, 4, 3, 1], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[2, 1, 0], [0, 0, 1]}, {}], [[2, 3, 1], {[1, 2, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]}, {1}], [[1], {}, {}], [[2, 1], {[1, 2, 0], [1, 1, 1], [0, 1, 2]}, {}], [[3, 2, 1], {[1, 2, 0, 0], [1, 1, 0, 1], [0, 0, 1, 0], [0, 1, 0, 2]}, {1}], [[2, 1, 3], {[2, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0], [0, 1, 1, 0]}, {1}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 22, 37, 60, 96, 153, 244, 390, 625, 1004, 1616, 2605, 4204, 6790, 10973, 17740, 28688] For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 4, 2], [2, 3, 4, 1]}, {[1, 3, 2], [3, 2, 1, 4], [4, 2, 1, 3]}, {[3, 1, 2], [2, 3, 1, 4], [2, 3, 4, 1]}, {[2, 1, 3], [1, 4, 3, 2], [4, 1, 3, 2]}, {[1, 3, 2], [3, 2, 1, 4], [3, 2, 4, 1]}, {[2, 3, 1], [1, 4, 2, 3], [4, 1, 2, 3]}, {[2, 3, 1], [3, 1, 2, 4], [4, 1, 2, 3]}, {[2, 1, 3], [1, 4, 3, 2], [2, 4, 3, 1]}} the member , {[3, 1, 2], [1, 3, 4, 2], [2, 3, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1], {}, {}], [[1, 2], {}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {2}], [[2, 1], {[0, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 3, 4], [4, 2, 1, 3]}, {[2, 1, 3], [1, 4, 2, 3], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [4, 1, 3, 2]}, {[2, 1, 3], [1, 3, 4, 2], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [2, 4, 3, 1]}, {[1, 3, 2], [3, 1, 2, 4], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [3, 2, 4, 1]}, {[1, 3, 2], [2, 3, 1, 4], [4, 3, 2, 1]}} the member , {[2, 3, 1], [1, 2, 3, 4], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 0, 1, 0], [1, 1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 1], {[1, 1, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]}, {2}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 23, 38, 57, 80, 107, 138, 173, 212, 255, 302, 353, 408, 467, 530, 597, 668] the sequence seems to be polynomial 2 17 - 11 n + 2 n For the equivalence class of patterns, { {[1, 2, 3], [3, 1, 4, 2], [4, 3, 1, 2]}, {[1, 2, 3], [3, 1, 4, 2], [3, 4, 2, 1]}, {[3, 2, 1], [1, 2, 4, 3], [2, 4, 1, 3]}, {[3, 2, 1], [1, 2, 4, 3], [3, 1, 4, 2]}, {[1, 2, 3], [2, 4, 1, 3], [3, 4, 2, 1]}, {[3, 2, 1], [2, 1, 3, 4], [2, 4, 1, 3]}, {[3, 2, 1], [2, 1, 3, 4], [3, 1, 4, 2]}, {[1, 2, 3], [2, 4, 1, 3], [4, 3, 1, 2]}} the member , {[1, 2, 3], [3, 1, 4, 2], [4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 3, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 1]}, {}], [[1, 2], {[0, 0, 1]}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 1]}, {2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 24, 42, 67, 100, 142, 194] For the equivalence class of patterns, { {[1, 2, 3], [1, 4, 3, 2], [3, 2, 1, 4]}, {[3, 2, 1], [2, 3, 4, 1], [4, 1, 2, 3]}} the member , {[1, 2, 3], [1, 4, 3, 2], [3, 2, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1], {[0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]}, {}], [[2, 1, 3], {[0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {3}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {1}], [[1, 2], {[0, 2, 0], [0, 0, 1]}, {2}], [[1], {[0, 3]}, {}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1]}, {1} ]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 58, 131, 295, 662, 1487, 3342, 7510, 16874, 37915, 85195, 191432, 430143, 966522, 2171756, 4879892] For the equivalence class of patterns, { {[3, 1, 2], [3, 2, 1, 4], [3, 2, 4, 1]}, {[3, 1, 2], [1, 4, 3, 2], [2, 4, 3, 1]}, {[2, 3, 1], [3, 2, 1, 4], [4, 2, 1, 3]}, {[2, 1, 3], [1, 3, 4, 2], [2, 3, 4, 1]}, {[1, 3, 2], [3, 1, 2, 4], [4, 1, 2, 3]}, {[2, 3, 1], [1, 4, 3, 2], [4, 1, 3, 2]}, {[2, 1, 3], [1, 4, 2, 3], [4, 1, 2, 3]}, {[1, 3, 2], [2, 3, 1, 4], [2, 3, 4, 1]}} the member , {[3, 1, 2], [3, 2, 1, 4], [3, 2, 4, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {}, {1}], [[2, 1], {[0, 1, 0], [1, 0, 1]}, {1}], [[1], {}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584] This enumerating sequence seems to have the 2 3 1 - 3 x + 2 x + x rational generating function, ------------------- 2 1 - 4 x + 4 x For the equivalence class of patterns, { {[1, 2, 3], [3, 1, 4, 2], [4, 2, 3, 1]}, {[3, 2, 1], [1, 3, 2, 4], [2, 4, 1, 3]}, {[3, 2, 1], [1, 3, 2, 4], [3, 1, 4, 2]}, {[1, 2, 3], [2, 4, 1, 3], [4, 2, 3, 1]}} the member , {[1, 2, 3], [3, 1, 4, 2], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 1, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]}, {2}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 3, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 1]}, {}], [[1, 2], {[0, 0, 1]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176] the sequence seems to be polynomial 23 2 3 4 2 - 7/4 n + -- n - 1/4 n + 1/24 n 24 For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 3, 4], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [4, 3, 2, 1]}} the member , {[2, 3, 1], [1, 2, 3, 4], [4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 2, 0, 1], [0, 1, 1, 1], [1, 0, 0, 0]}, {}] , [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[4, 2, 1, 3], {[0, 0, 0, 1, 1], [0, 0, 2, 0, 1], [0, 0, 0, 3, 0], [0, 0, 2, 1, 0], [0, 0, 3, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[4, 3, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 3, 1, 2], %1, {1}], [[3, 2, 1, 4], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 0, 3, 0], [0, 2, 0, 1], [1, 0, 0, 0], [0, 0, 1, 1]}, {}], [[1, 3, 2, 4], {[0, 0, 0, 3, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 2, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 4, 3, 2], %1, {2}], [[4, 1, 3, 2], %1, {1}], [[3, 1, 2, 4], { [0, 0, 0, 3, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 2, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[3, 1, 2], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 0, 3, 0], [0, 2, 0, 1], [1, 0, 0, 0], [0, 0, 1, 1]}, {}], [[2, 1], {}, {}]} %1 := {[0, 0, 1, 1, 0], [0, 0, 2, 0, 0], [0, 0, 1, 0, 1], [0, 0, 0, 1, 1], [0, 0, 0, 3, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 23, 23, 15, 5, 1, 0] For the equivalence class of patterns, { {[3, 2, 1], [3, 1, 2, 4], [3, 4, 1, 2]}, {[1, 2, 3], [2, 1, 4, 3], [4, 1, 3, 2]}, {[1, 2, 3], [2, 1, 4, 3], [2, 4, 3, 1]}, {[1, 2, 3], [2, 1, 4, 3], [4, 2, 1, 3]}, {[1, 2, 3], [2, 1, 4, 3], [3, 2, 4, 1]}, {[3, 2, 1], [1, 3, 4, 2], [3, 4, 1, 2]}, {[3, 2, 1], [1, 4, 2, 3], [3, 4, 1, 2]}, {[3, 2, 1], [2, 3, 1, 4], [3, 4, 1, 2]}} the member , {[3, 2, 1], [3, 1, 2, 4], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1], {[0, 2, 1], [1, 0, 0]}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [ [2, 1, 3, 4], {[0, 2, 0, 0, 0], [0, 1, 0, 1, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [1, 0, 0, 0, 0]}, {3}], [[2, 1, 3], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0]}, {}], [[1, 2], {[2, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 0, 0, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 0, 0, 2, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1, 2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 26, 54, 110, 222, 446, 894] For the equivalence class of patterns, { {[1, 3, 2], [3, 1, 2, 4], [4, 3, 1, 2]}, {[2, 1, 3], [1, 3, 4, 2], [3, 4, 2, 1]}, {[2, 3, 1], [1, 2, 4, 3], [4, 1, 3, 2]}, {[2, 3, 1], [2, 1, 3, 4], [4, 2, 1, 3]}, {[3, 1, 2], [2, 1, 3, 4], [3, 2, 4, 1]}, {[2, 1, 3], [1, 4, 2, 3], [4, 3, 1, 2]}, {[1, 3, 2], [2, 3, 1, 4], [3, 4, 2, 1]}, {[3, 1, 2], [1, 2, 4, 3], [2, 4, 3, 1]}} the member , {[1, 3, 2], [3, 1, 2, 4], [4, 3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 1]}, {1}], [[1, 2], {[0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [[2, 1], {[0, 1, 1]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 3, 4], [4, 3, 1, 2]}, {[2, 1, 3], [1, 2, 4, 3], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4], [4, 3, 2, 1]}} the member , {[2, 3, 1], [1, 2, 3, 4], [4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 0, 1, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {2}], [[], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [ [4, 1, 2, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 1]}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}], [[2, 1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 1]}, {}], [[1, 4, 2, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 23, 38, 57, 80, 107, 138] For the equivalence class of patterns, { {[2, 1, 3], [1, 2, 4, 3], [1, 3, 4, 2]}, {[3, 1, 2], [2, 4, 3, 1], [3, 4, 2, 1]}, {[2, 1, 3], [1, 2, 4, 3], [1, 4, 2, 3]}, {[2, 3, 1], [4, 2, 1, 3], [4, 3, 1, 2]}, {[2, 3, 1], [4, 1, 3, 2], [4, 3, 1, 2]}, {[3, 1, 2], [3, 2, 4, 1], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4], [2, 3, 1, 4]}, {[1, 3, 2], [2, 1, 3, 4], [3, 1, 2, 4]}} the member , {[2, 1, 3], [1, 2, 4, 3], [1, 3, 4, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[1, 2], {[0, 1, 1]}, {}], [[1, 3, 2], {[0, 0, 0, 1], [0, 1, 1, 0]}, {2}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[1, 3, 2], [2, 3, 4, 1], [3, 4, 1, 2]}, {[3, 1, 2], [2, 1, 4, 3], [3, 2, 1, 4]}, {[2, 3, 1], [2, 1, 4, 3], [3, 2, 1, 4]}, {[3, 1, 2], [1, 4, 3, 2], [2, 1, 4, 3]}, {[2, 3, 1], [1, 4, 3, 2], [2, 1, 4, 3]}, {[1, 3, 2], [3, 4, 1, 2], [4, 1, 2, 3]}, {[2, 1, 3], [3, 4, 1, 2], [4, 1, 2, 3]}, {[2, 1, 3], [2, 3, 4, 1], [3, 4, 1, 2]}} the member , {[1, 3, 2], [2, 3, 4, 1], [3, 4, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {}, {1}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 4, 3], [2, 1, 3, 4]}, {[1, 3, 2], [3, 4, 2, 1], [4, 3, 1, 2]}, {[2, 1, 3], [3, 4, 2, 1], [4, 3, 1, 2]}, {[3, 1, 2], [1, 2, 4, 3], [2, 1, 3, 4]}} the member , {[2, 3, 1], [1, 2, 4, 3], [2, 1, 3, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 3, 2], {[0, 1, 2, 0], [0, 1, 1, 1], [1, 0, 0, 0], [0, 0, 0, 2]}, {2}], [[3, 2, 1], {[1, 1, 1, 0], [0, 1, 2, 0], [1, 1, 0, 1], [0, 1, 0, 2], [0, 0, 1, 1]}, {1} ], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 2, 0], [0, 1, 1, 1], [1, 0, 0, 0], [0, 0, 0, 2]}, {1}], [[2, 1], {[1, 1, 1], [0, 1, 2]}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[1, 2], {[0, 1, 2], [1, 0, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 4, 3], [2, 1, 4, 3]}, {[1, 3, 2], [3, 4, 1, 2], [4, 3, 1, 2]}, {[3, 1, 2], [2, 1, 3, 4], [2, 1, 4, 3]}, {[1, 3, 2], [3, 4, 1, 2], [3, 4, 2, 1]}, {[2, 3, 1], [2, 1, 3, 4], [2, 1, 4, 3]}, {[2, 1, 3], [3, 4, 1, 2], [3, 4, 2, 1]}, {[2, 1, 3], [3, 4, 1, 2], [4, 3, 1, 2]}, {[3, 1, 2], [1, 2, 4, 3], [2, 1, 4, 3]}} the member , {[2, 3, 1], [1, 2, 4, 3], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[3, 1, 2], {[1, 0, 0, 0]}, {1}], [[], {}, {}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1], {}, {}], [[1, 3, 2], {[1, 0, 0, 0]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584] This enumerating sequence seems to have the 2 3 1 - 3 x + 2 x + x rational generating function, ------------------- 2 1 - 4 x + 4 x For the equivalence class of patterns, { {[1, 3, 2], [2, 3, 4, 1], [3, 1, 2, 4]}, {[2, 3, 1], [1, 4, 3, 2], [4, 2, 1, 3]}, {[3, 1, 2], [1, 4, 3, 2], [3, 2, 4, 1]}, {[3, 1, 2], [2, 4, 3, 1], [3, 2, 1, 4]}, {[2, 1, 3], [1, 4, 2, 3], [2, 3, 4, 1]}, {[2, 3, 1], [3, 2, 1, 4], [4, 1, 3, 2]}, {[2, 1, 3], [1, 3, 4, 2], [4, 1, 2, 3]}, {[1, 3, 2], [2, 3, 1, 4], [4, 1, 2, 3]}} the member , {[1, 3, 2], [2, 3, 4, 1], [3, 1, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 1, 0], [0, 1, 0, 1]}, {}], [ [3, 4, 2, 1], {[0, 1, 1, 0, 0], [0, 0, 1, 0, 1], [0, 1, 0, 0, 1], [0, 0, 0, 1, 0]}, {3}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {3}], [ [2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 4, 1, 2], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1, 2}] , [[2, 1], {[0, 1, 1]}, {1}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [4, 1, 3, 2]}, {[3, 2, 1], [1, 3, 4, 2], [2, 3, 1, 4]}, {[3, 2, 1], [1, 4, 2, 3], [3, 1, 2, 4]}, {[1, 2, 3], [3, 2, 4, 1], [4, 2, 1, 3]}} the member , {[1, 2, 3], [2, 4, 3, 1], [4, 1, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {3}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 2, 0], [0, 1, 2]}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 1, 0, 2], [0, 0, 1, 2], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2} ], [[1, 2], {[1, 2, 0], [0, 0, 1]}, {}], [[2, 1, 3], {[1, 0, 2, 0], [0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 24, 45, 83, 154, 290, 555, 1077, 2112, 4172, 8281, 16487, 32886, 65670, 131223, 262313, 524476] For the equivalence class of patterns, { {[1, 2, 3], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 2, 3], [1, 4, 3, 2], [4, 3, 1, 2]}, {[3, 2, 1], [1, 2, 4, 3], [2, 3, 4, 1]}, {[3, 2, 1], [1, 2, 4, 3], [4, 1, 2, 3]}, {[3, 2, 1], [2, 1, 3, 4], [2, 3, 4, 1]}, {[3, 2, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 2, 3], [3, 2, 1, 4], [3, 4, 2, 1]}, {[1, 2, 3], [1, 4, 3, 2], [3, 4, 2, 1]}} the member , {[1, 2, 3], [3, 2, 1, 4], [4, 3, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 3, 1], {[0, 0, 0, 1]}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 0, 0, 1]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 23, 36, 51, 68, 87, 108] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [3, 2, 1, 4]}, {[1, 2, 3], [1, 4, 3, 2], [3, 2, 4, 1]}, {[1, 2, 3], [3, 2, 1, 4], [4, 1, 3, 2]}, {[1, 2, 3], [1, 4, 3, 2], [4, 2, 1, 3]}, {[3, 2, 1], [1, 3, 4, 2], [4, 1, 2, 3]}, {[3, 2, 1], [1, 4, 2, 3], [2, 3, 4, 1]}, {[3, 2, 1], [2, 3, 1, 4], [4, 1, 2, 3]}, {[3, 2, 1], [2, 3, 4, 1], [3, 1, 2, 4]}} the member , {[1, 2, 3], [2, 4, 3, 1], [3, 2, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[2, 1, 0], [0, 0, 1]}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 2], [0, 1, 2]}, {}], [[3, 2, 1], {[1, 0, 2, 0], [0, 1, 2, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]}, {1}], [[2, 1, 3], {[2, 1, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[3, 1, 2], {[2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 24, 46, 87, 162, 300, 554, 1021, 1880, 3460, 6366, 11711, 21542, 39624, 72882, 134053, 246564] For the equivalence class of patterns, { {[3, 2, 1], [3, 1, 4, 2], [3, 4, 1, 2]}, {[1, 2, 3], [2, 1, 4, 3], [2, 4, 1, 3]}, {[1, 2, 3], [2, 1, 4, 3], [3, 1, 4, 2]}, {[3, 2, 1], [2, 4, 1, 3], [3, 4, 1, 2]}} the member , {[3, 2, 1], [3, 1, 4, 2], [3, 4, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {[2, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[1, 2, 3], [4, 3, 1, 2], [4, 3, 2, 1]}, {[3, 2, 1], [1, 2, 3, 4], [1, 2, 4, 3]}, {[3, 2, 1], [1, 2, 3, 4], [2, 1, 3, 4]}, {[1, 2, 3], [3, 4, 2, 1], [4, 3, 2, 1]}} the member , {[1, 2, 3], [4, 3, 1, 2], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 3, 0], [2, 0, 0]}, {}], [[1, 2], {[0, 3, 0], [0, 0, 1]}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1], [2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0]}, {1} ], [[2, 3, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1], [2, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1], [2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0]}, {1} ], [[2, 1, 3], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1], [2, 0, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 17, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, { {[1, 3, 2], [3, 4, 2, 1], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [2, 1, 3, 4]}, {[3, 1, 2], [1, 3, 2, 4], [2, 1, 3, 4]}, {[2, 1, 3], [4, 2, 3, 1], [4, 3, 1, 2]}, {[1, 3, 2], [4, 2, 3, 1], [4, 3, 1, 2]}, {[2, 3, 1], [1, 2, 4, 3], [1, 3, 2, 4]}, {[2, 1, 3], [3, 4, 2, 1], [4, 2, 3, 1]}, {[3, 1, 2], [1, 2, 4, 3], [1, 3, 2, 4]}} the member , {[1, 3, 2], [3, 4, 2, 1], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1], {}, {}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 52, 99, 184, 340, 632, 1189, 2268, 4382, 8556, 16839, 33328, 66216, 131888, 263113, 525428] For the equivalence class of patterns, { {[1, 3, 2], [3, 2, 4, 1], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [1, 3, 4, 2]}, {[3, 1, 2], [1, 2, 3, 4], [2, 3, 1, 4]}, {[2, 3, 1], [1, 2, 3, 4], [1, 4, 2, 3]}, {[2, 3, 1], [1, 2, 3, 4], [3, 1, 2, 4]}, {[2, 1, 3], [4, 1, 3, 2], [4, 3, 2, 1]}, {[2, 1, 3], [2, 4, 3, 1], [4, 3, 2, 1]}, {[1, 3, 2], [4, 2, 1, 3], [4, 3, 2, 1]}} the member , {[1, 3, 2], [3, 2, 4, 1], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176] the sequence seems to be polynomial 23 2 3 4 2 - 7/4 n + -- n - 1/4 n + 1/24 n 24 For the equivalence class of patterns, { {[2, 3, 1], [4, 2, 1, 3], [4, 3, 2, 1]}, {[2, 3, 1], [4, 1, 3, 2], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [1, 4, 2, 3]}, {[3, 1, 2], [2, 4, 3, 1], [4, 3, 2, 1]}, {[3, 1, 2], [3, 2, 4, 1], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [1, 3, 4, 2]}, {[1, 3, 2], [1, 2, 3, 4], [2, 3, 1, 4]}, {[1, 3, 2], [1, 2, 3, 4], [3, 1, 2, 4]}} the member , {[2, 3, 1], [4, 2, 1, 3], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 1, 0]}, {2}], [[2, 1], {[1, 1, 0]}, {}], [[3, 2, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[1, 3, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, {[3, 1, 2], [1, 3, 4, 2], [2, 3, 1, 4]}, {[2, 3, 1], [1, 4, 2, 3], [3, 1, 2, 4]}, {[2, 1, 3], [2, 4, 3, 1], [4, 1, 3, 2]}} the member , {[1, 3, 2], [3, 2, 4, 1], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 1, 3], [4, 1, 2, 3], [4, 3, 2, 1]}, {[1, 3, 2], [2, 3, 4, 1], [4, 3, 2, 1]}, {[2, 1, 3], [2, 3, 4, 1], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [3, 2, 1, 4]}, {[1, 3, 2], [4, 1, 2, 3], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [1, 4, 3, 2]}, {[3, 1, 2], [1, 2, 3, 4], [1, 4, 3, 2]}, {[3, 1, 2], [1, 2, 3, 4], [3, 2, 1, 4]}} the member , {[1, 3, 2], [2, 3, 4, 1], [4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {}, {}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}], [[2, 3, 1], {[0, 0, 1, 0]}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[2, 1, 3, 4], %1, {1}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], %1, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2, 4], %1, {1}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 23, 38, 57, 80, 107, 138] For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 4, 3], [2, 1, 3, 4]}, {[1, 2, 3], [3, 4, 2, 1], [4, 3, 1, 2]}} the member , {[3, 2, 1], [1, 2, 4, 3], [2, 1, 3, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 2]}, {1}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 2]}, {2}], [[1, 2], {[1, 2, 0]}, {}], [[2, 1], {[0, 0, 3], [1, 0, 0]}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1}], [[1, 2, 3], {[1, 2, 0, 0], [0, 0, 1, 0]}, {2}], [[3, 4, 1, 2], { [0, 0, 0, 2, 0], [0, 0, 0, 0, 2], [0, 0, 0, 1, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [ [2, 3, 1, 4], {[0, 0, 2, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 19, 26, 33, 40, 47, 54] For the equivalence class of patterns, { {[1, 3, 2], [3, 1, 2, 4], [4, 2, 1, 3]}, {[2, 3, 1], [3, 1, 2, 4], [4, 2, 1, 3]}, {[3, 1, 2], [2, 3, 1, 4], [3, 2, 4, 1]}, {[2, 1, 3], [1, 3, 4, 2], [2, 4, 3, 1]}, {[2, 3, 1], [1, 4, 2, 3], [4, 1, 3, 2]}, {[2, 1, 3], [1, 4, 2, 3], [4, 1, 3, 2]}, {[3, 1, 2], [1, 3, 4, 2], [2, 4, 3, 1]}, {[1, 3, 2], [2, 3, 1, 4], [3, 2, 4, 1]}} the member , {[1, 3, 2], [3, 1, 2, 4], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {2}], [[2, 1], {[0, 1, 1]}, {}], [[3, 2, 1], {[0, 0, 1, 0], [0, 1, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584] This enumerating sequence seems to have the 2 3 1 - 3 x + 2 x + x rational generating function, ------------------- 2 1 - 4 x + 4 x For the equivalence class of patterns, { {[1, 2, 3], [3, 4, 1, 2], [3, 4, 2, 1]}, {[3, 2, 1], [1, 2, 4, 3], [2, 1, 4, 3]}, {[3, 2, 1], [2, 1, 3, 4], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2], [4, 3, 1, 2]}} the member , {[3, 2, 1], [1, 2, 4, 3], [2, 1, 4, 3]}, has a scheme of depth , 4 here it is: {[[2, 3, 1], {[1, 0, 0, 0]}, {}], [[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 1], {[1, 0, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [4, 2, 3, 1]}, {[3, 2, 1], [1, 3, 2, 4], [1, 3, 4, 2]}, {[3, 2, 1], [1, 3, 2, 4], [1, 4, 2, 3]}, {[3, 2, 1], [1, 3, 2, 4], [2, 3, 1, 4]}, {[3, 2, 1], [1, 3, 2, 4], [3, 1, 2, 4]}, {[1, 2, 3], [4, 2, 1, 3], [4, 2, 3, 1]}, {[1, 2, 3], [3, 2, 4, 1], [4, 2, 3, 1]}, {[1, 2, 3], [4, 1, 3, 2], [4, 2, 3, 1]}} the member , {[1, 2, 3], [2, 4, 3, 1], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2], {[0, 0, 1]}, {}], [[3, 2, 1], {[0, 1, 0, 2], [0, 0, 1, 2], [0, 1, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[2, 1], {[0, 1, 2]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 2, 4], [2, 1, 4, 3]}, {[1, 3, 2], [3, 4, 1, 2], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [2, 1, 4, 3]}, {[2, 1, 3], [3, 4, 1, 2], [4, 2, 3, 1]}} the member , {[3, 1, 2], [1, 3, 2, 4], [2, 1, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {}, {2}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 3, 1], [1, 4, 2, 3], [1, 4, 3, 2]}, {[2, 1, 3], [4, 1, 2, 3], [4, 1, 3, 2]}, {[3, 1, 2], [1, 3, 4, 2], [1, 4, 3, 2]}, {[3, 1, 2], [2, 3, 1, 4], [3, 2, 1, 4]}, {[1, 3, 2], [2, 3, 4, 1], [3, 2, 4, 1]}, {[2, 1, 3], [2, 3, 4, 1], [2, 4, 3, 1]}, {[2, 3, 1], [3, 1, 2, 4], [3, 2, 1, 4]}, {[1, 3, 2], [4, 1, 2, 3], [4, 2, 1, 3]}} the member , {[3, 1, 2], [1, 3, 4, 2], [1, 4, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 2, 0]}, {1}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {[0, 2, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428] This enumerating sequence seems to have the 2 -1 + x + x rational generating function, ------------- 2 -1 + 2 x + x For the equivalence class of patterns, { {[1, 2, 3], [2, 1, 4, 3], [4, 3, 2, 1]}, {[3, 2, 1], [1, 2, 3, 4], [3, 4, 1, 2]}} the member , {[1, 2, 3], [2, 1, 4, 3], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1, 3], {[3, 0, 0, 0], [0, 3, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}], [[2, 3, 1], {[3, 0, 0, 0], [0, 3, 0, 0], [0, 2, 1, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1} ], [[1, 2], {[0, 3, 0], [0, 0, 1]}, {}], [[3, 1, 2], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {2} ], [[1, 3, 2], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {1} ], [[2, 1], {[3, 0, 0], [0, 3, 0], [0, 0, 2]}, {}], [[3, 2, 1], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 1, 1], [0, 0, 0, 2]}, {1} ]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 17, 9, 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, 4], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2], [4, 2, 3, 1]}} the member , {[3, 2, 1], [1, 3, 2, 4], [2, 1, 4, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 1, 0]}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 1]}, {1}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}], [[2, 1], {[1, 0, 0]}, {}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 4, 1], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[1, 2, 3], [3, 2, 1, 4], [4, 2, 3, 1]}, {[1, 2, 3], [1, 4, 3, 2], [4, 2, 3, 1]}, {[3, 2, 1], [1, 3, 2, 4], [2, 3, 4, 1]}, {[3, 2, 1], [1, 3, 2, 4], [4, 1, 2, 3]}} the member , {[1, 2, 3], [3, 2, 1, 4], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 1, 3], {[2, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[3, 2, 1], {[2, 1, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0], [0, 1, 1, 0]}, {1}], [[2, 3, 1], {[2, 1, 0, 0], [0, 0, 0, 1], [1, 0, 1, 0]}, {1}], [[2, 1], {[2, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 24, 41, 63, 90, 122, 159, 201, 248, 300, 357, 419, 486, 558, 635, 717, 804] the sequence seems to be polynomial 2 27 - 31/2 n + 5/2 n For the equivalence class of patterns, { {[2, 1, 3], [1, 2, 3, 4], [3, 4, 1, 2]}, {[2, 3, 1], [2, 1, 4, 3], [4, 3, 2, 1]}, {[3, 1, 2], [2, 1, 4, 3], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [3, 4, 1, 2]}} the member , {[2, 1, 3], [1, 2, 3, 4], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 4, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {3}], [[2, 3, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 3, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[1, 2, 3], {[0, 0, 0, 1]}, {}], [[1, 3, 2], {[0, 0, 0, 1]}, {2}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 26, 51, 92, 155, 247, 376] For the equivalence class of patterns, { {[1, 2, 3], [3, 4, 2, 1], [4, 1, 3, 2]}, {[1, 2, 3], [3, 4, 2, 1], [4, 2, 1, 3]}, {[3, 2, 1], [1, 2, 4, 3], [2, 3, 1, 4]}, {[3, 2, 1], [1, 2, 4, 3], [3, 1, 2, 4]}, {[1, 2, 3], [2, 4, 3, 1], [4, 3, 1, 2]}, {[3, 2, 1], [1, 3, 4, 2], [2, 1, 3, 4]}, {[3, 2, 1], [1, 4, 2, 3], [2, 1, 3, 4]}, {[1, 2, 3], [3, 2, 4, 1], [4, 3, 1, 2]}} the member , {[1, 2, 3], [3, 4, 2, 1], [4, 1, 3, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 1], {[0, 2, 0]}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[3, 0, 0], [0, 0, 1]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}], [[2, 1, 3], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {}], [[1, 3, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1} ], [[3, 2, 4, 1], {[0, 2, 0, 0, 0], [0, 1, 0, 1, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 2, 0, 0, 0], [3, 0, 0, 0, 0], [2, 1, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {3}], [[3, 1, 4, 2], { [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2], {[3, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 22, 34, 48, 64, 82, 102] For the equivalence class of patterns, { {[1, 2, 3], [2, 1, 4, 3], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 4, 3], [3, 4, 1, 2]}} the member , {[1, 2, 3], [2, 1, 4, 3], [3, 4, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {3}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 0, 2]}, {}], [[3, 2, 1], {[0, 0, 2, 0], [0, 0, 1, 1], [0, 0, 0, 2]}, {1}], [[2, 1, 3], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 1, 3], [4, 3, 2, 1]}, {[3, 2, 1], [1, 2, 3, 4], [2, 4, 1, 3]}, {[3, 2, 1], [1, 2, 3, 4], [3, 1, 4, 2]}, {[1, 2, 3], [3, 1, 4, 2], [4, 3, 2, 1]}} the member , {[1, 2, 3], [2, 4, 1, 3], [4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 3, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0], [0, 2, 0, 0]}, {1} ], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], { [3, 0, 0, 0], [1, 2, 0, 0], [0, 3, 0, 0], [1, 1, 1, 0], [0, 2, 1, 0], [1, 0, 2, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1]}, {}], [[2, 1], {[3, 0, 0], [0, 3, 0]}, {}], [[3, 1, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [1, 1, 0, 0], [0, 2, 0, 0]}, {2} ], [[3, 2, 1], {[0, 1, 2, 0], [0, 0, 3, 0], [1, 0, 0, 0], [0, 2, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 3, 1], {[3, 0, 0, 0], [0, 3, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[1, 1, 0, 0, 0], [0, 2, 0, 0, 0], [3, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 1, 4, 3], { [1, 1, 0, 0, 0], [0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [1, 0, 1, 0, 0], [3, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}], [[3, 2, 4, 1], {[0, 2, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 2], {[1, 2, 0], [0, 3, 0], [0, 0, 1]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 17, 10, 0, 0, 0, 0] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [3, 2, 4, 1]}, {[3, 2, 1], [1, 3, 4, 2], [1, 4, 2, 3]}, {[1, 2, 3], [4, 1, 3, 2], [4, 2, 1, 3]}, {[3, 2, 1], [2, 3, 1, 4], [3, 1, 2, 4]}} the member , {[1, 2, 3], [2, 4, 3, 1], [3, 2, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 1, 2, 0], [0, 0, 1, 2], [0, 1, 0, 1]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2], {[0, 0, 1]}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 1, 2]}, {}], [[2, 3, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 48, 87, 152, 259, 434, 719, 1182, 1933, 3150, 5121, 8312, 13477, 21836, 35363, 57252] For the equivalence class of patterns, { {[1, 3, 2], [3, 4, 1, 2], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [2, 1, 4, 3]}, {[2, 1, 3], [3, 4, 1, 2], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [2, 1, 4, 3]}} the member , {[1, 3, 2], [3, 4, 1, 2], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [2, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0], [2, 0, 0, 0]}, {1}], [[1, 2, 3], {[3, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [2, 0, 0, 0]}, {1}], [[1, 2], {[3, 0, 0], [0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 23, 38, 57, 80, 107, 138, 173, 212, 255, 302, 353, 408, 467, 530, 597, 668] the sequence seems to be polynomial 2 17 - 11 n + 2 n For the equivalence class of patterns, { {[1, 2, 3], [3, 2, 4, 1], [3, 4, 1, 2]}, {[1, 2, 3], [3, 4, 1, 2], [4, 1, 3, 2]}, {[3, 2, 1], [1, 3, 4, 2], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2], [4, 2, 1, 3]}, {[3, 2, 1], [1, 4, 2, 3], [2, 1, 4, 3]}, {[3, 2, 1], [2, 1, 4, 3], [2, 3, 1, 4]}, {[3, 2, 1], [2, 1, 4, 3], [3, 1, 2, 4]}, {[1, 2, 3], [2, 4, 3, 1], [3, 4, 1, 2]}} the member , {[1, 2, 3], [3, 2, 4, 1], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 1], {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[0, 1, 0, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 0, 1]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 2, 4], [2, 3, 1, 4]}, {[2, 1, 3], [2, 4, 3, 1], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [3, 1, 2, 4]}, {[3, 1, 2], [1, 3, 2, 4], [1, 3, 4, 2]}, {[2, 3, 1], [1, 3, 2, 4], [1, 4, 2, 3]}, {[1, 3, 2], [3, 2, 4, 1], [4, 2, 3, 1]}, {[1, 3, 2], [4, 2, 1, 3], [4, 2, 3, 1]}, {[2, 1, 3], [4, 1, 3, 2], [4, 2, 3, 1]}} the member , {[3, 1, 2], [1, 3, 2, 4], [2, 3, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2, 3], {}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[1, 3, 2], [3, 2, 1, 4], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [4, 1, 2, 3]}, {[3, 1, 2], [1, 2, 3, 4], [2, 3, 4, 1]}, {[2, 1, 3], [1, 4, 3, 2], [4, 3, 2, 1]}} the member , {[1, 3, 2], [3, 2, 1, 4], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 2, 1], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 2, 4], [3, 2, 4, 1]}, {[1, 3, 2], [3, 1, 2, 4], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [4, 1, 3, 2]}, {[2, 1, 3], [1, 3, 4, 2], [4, 2, 3, 1]}, {[3, 1, 2], [1, 3, 2, 4], [2, 4, 3, 1]}, {[2, 1, 3], [1, 4, 2, 3], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [4, 2, 1, 3]}, {[1, 3, 2], [2, 3, 1, 4], [4, 2, 3, 1]}} the member , {[3, 1, 2], [1, 3, 2, 4], [3, 2, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {}, {2}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 4, 3], [1, 3, 2, 4]}, {[3, 2, 1], [1, 3, 2, 4], [2, 1, 3, 4]}, {[1, 2, 3], [3, 4, 2, 1], [4, 2, 3, 1]}, {[1, 2, 3], [4, 2, 3, 1], [4, 3, 1, 2]}} the member , {[3, 2, 1], [1, 2, 4, 3], [1, 3, 2, 4]}, has a scheme of depth , 4 here it is: {[[2, 3, 1], {[1, 0, 0, 0]}, {}], [[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 1], {[1, 0, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 24, 42, 67, 100, 142, 194] For the equivalence class of patterns, { {[1, 2, 3], [1, 4, 3, 2], [2, 4, 1, 3]}, {[3, 2, 1], [3, 1, 4, 2], [4, 1, 2, 3]}, {[1, 2, 3], [1, 4, 3, 2], [3, 1, 4, 2]}, {[1, 2, 3], [3, 1, 4, 2], [3, 2, 1, 4]}, {[1, 2, 3], [2, 4, 1, 3], [3, 2, 1, 4]}, {[3, 2, 1], [2, 3, 4, 1], [2, 4, 1, 3]}, {[3, 2, 1], [2, 3, 4, 1], [3, 1, 4, 2]}, {[3, 2, 1], [2, 4, 1, 3], [4, 1, 2, 3]}} the member , {[1, 2, 3], [1, 4, 3, 2], [2, 4, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[0, 3]}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]}, {1}], [[1, 2], {[0, 2, 0], [0, 0, 1]}, {}], [[2, 3, 1], {[0, 3, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 3, 4], [1, 3, 2, 4]}, {[1, 2, 3], [4, 2, 3, 1], [4, 3, 2, 1]}} the member , {[3, 2, 1], [1, 2, 3, 4], [1, 3, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 2], {[0, 3, 0], [0, 1, 2], [0, 0, 3]}, {}], [[2, 3, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [1, 0, 0, 0], [0, 0, 1, 2], [0, 0, 0, 3]}, {}], [[2, 1, 3], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 2], [0, 0, 1, 2], [0, 0, 0, 3]}, {1}] , [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 2], [0, 0, 0, 3], [0, 0, 2, 0]}, {1} ], [[1, 2, 3], {[0, 3, 0, 0], [0, 0, 0, 1], [0, 1, 1, 0], [0, 0, 2, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 1, 0, 2], [0, 0, 0, 1, 2], [0, 0, 0, 0, 3], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 2, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1, 4], { [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 2, 0, 1, 0], [0, 0, 3, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 3, 0, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}], [[2, 1], {[0, 3, 0], [1, 0, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 19, 14, 0, 0, 0, 0] For the equivalence class of patterns, { {[1, 2, 3], [2, 1, 4, 3], [3, 4, 2, 1]}, {[1, 2, 3], [2, 1, 4, 3], [4, 3, 1, 2]}, {[3, 2, 1], [1, 2, 4, 3], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 3, 4], [3, 4, 1, 2]}} the member , {[1, 2, 3], [2, 1, 4, 3], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 2, 1, 0], [0, 2, 0, 1], [0, 0, 2, 0], [0, 0, 1, 1], [0, 0, 0, 2]}, {1} ], [[2, 1, 3], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {2} ], [[1, 2], {[3, 0, 0], [0, 0, 1]}, {}], [[1, 3, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[3, 1, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 2, 1], [0, 0, 2]}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0], [0, 0, 2, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, 112, 122, 132, 142, 152, 162, 172] the sequence seems to be polynomial -38 + 10 n For the equivalence class of patterns, { {[3, 1, 2], [2, 1, 3, 4], [2, 3, 4, 1]}, {[2, 3, 1], [2, 1, 3, 4], [4, 1, 2, 3]}, {[1, 3, 2], [3, 2, 1, 4], [4, 3, 1, 2]}, {[2, 1, 3], [1, 4, 3, 2], [4, 3, 1, 2]}, {[2, 3, 1], [1, 2, 4, 3], [4, 1, 2, 3]}, {[2, 1, 3], [1, 4, 3, 2], [3, 4, 2, 1]}, {[1, 3, 2], [3, 2, 1, 4], [3, 4, 2, 1]}, {[3, 1, 2], [1, 2, 4, 3], [2, 3, 4, 1]}} the member , {[3, 1, 2], [2, 1, 3, 4], [2, 3, 4, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {3}], [ [2, 3, 1, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[2, 3, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, {[2, 1, 3], [4, 1, 3, 2], [4, 3, 1, 2]}, {[2, 3, 1], [1, 2, 4, 3], [1, 4, 2, 3]}, {[1, 3, 2], [3, 2, 4, 1], [3, 4, 2, 1]}, {[1, 3, 2], [4, 2, 1, 3], [4, 3, 1, 2]}, {[3, 1, 2], [2, 1, 3, 4], [2, 3, 1, 4]}, {[3, 1, 2], [1, 2, 4, 3], [1, 3, 4, 2]}, {[2, 1, 3], [2, 4, 3, 1], [3, 4, 2, 1]}} the member , {[2, 3, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1], {[0, 1, 1]}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {3}], [[3, 2, 1], {[0, 1, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[1, 2, 3], [3, 2, 1, 4], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 4, 3], [2, 3, 4, 1]}, {[1, 2, 3], [1, 4, 3, 2], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 4, 3], [4, 1, 2, 3]}} the member , {[1, 2, 3], [3, 2, 1, 4], [3, 4, 1, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {3}], [[3, 4, 2, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {2}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 1], {}, {}], [[2, 3, 1, 4], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 0, 0, 1]}, {}], [[3, 1, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 0, 1]}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 1, 3], [3, 1, 4, 2]}, {[3, 2, 1], [2, 4, 1, 3], [3, 1, 4, 2]}} the member , {[1, 2, 3], [2, 4, 1, 3], [3, 1, 4, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1], {[0, 1, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[3, 2, 1], [3, 1, 2, 4], [4, 1, 2, 3]}, {[1, 2, 3], [1, 4, 3, 2], [2, 4, 3, 1]}, {[1, 2, 3], [3, 2, 1, 4], [3, 2, 4, 1]}, {[1, 2, 3], [1, 4, 3, 2], [4, 1, 3, 2]}, {[1, 2, 3], [3, 2, 1, 4], [4, 2, 1, 3]}, {[3, 2, 1], [1, 3, 4, 2], [2, 3, 4, 1]}, {[3, 2, 1], [1, 4, 2, 3], [4, 1, 2, 3]}, {[3, 2, 1], [2, 3, 1, 4], [2, 3, 4, 1]}} the member , {[3, 2, 1], [3, 1, 2, 4], [4, 1, 2, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 4, 2], { [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 0, 0, 0, 0]}, {3}], [[1], {[3, 0]}, {}], [[2, 1, 3], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0]}, {}], [[2, 1, 3, 4], {[0, 2, 0, 0, 0], [0, 1, 0, 1, 0], [0, 1, 1, 0, 0], [0, 0, 0, 3, 0], [0, 0, 1, 2, 0], [0, 0, 2, 1, 0], [0, 0, 3, 0, 0], [1, 0, 0, 0, 0]}, {3}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 1, 4, 3], {[0, 0, 0, 2, 0], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1, 2}], [[2, 1], {[0, 2, 0], [1, 0, 0]}, {}], [[1, 2], {[3, 0, 0], [2, 1, 0], [1, 2, 0], [0, 3, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014] For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 3, 4], [1, 3, 2, 4]}, {[2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1]}, {[1, 3, 2], [4, 2, 3, 1], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [1, 3, 2, 4]}} the member , {[1, 3, 2], [4, 2, 3, 1], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 4, 2], [3, 4, 2, 1]}, {[2, 3, 1], [3, 1, 2, 4], [4, 3, 1, 2]}, {[2, 3, 1], [1, 4, 2, 3], [4, 3, 1, 2]}, {[2, 1, 3], [1, 2, 4, 3], [2, 4, 3, 1]}, {[2, 1, 3], [1, 2, 4, 3], [4, 1, 3, 2]}, {[1, 3, 2], [2, 1, 3, 4], [3, 2, 4, 1]}, {[1, 3, 2], [2, 1, 3, 4], [4, 2, 1, 3]}, {[3, 1, 2], [2, 3, 1, 4], [3, 4, 2, 1]}} the member , {[3, 1, 2], [1, 3, 4, 2], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0], [2, 0, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [2, 0, 0, 0]}, {2}], [[2, 3, 1], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {[2, 0, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584] This enumerating sequence seems to have the 2 3 1 - 3 x + 2 x + x rational generating function, ------------------- 2 1 - 4 x + 4 x For the equivalence class of patterns, { {[1, 3, 2], [3, 1, 2, 4], [3, 2, 1, 4]}, {[2, 1, 3], [1, 3, 4, 2], [1, 4, 3, 2]}, {[2, 3, 1], [4, 1, 2, 3], [4, 1, 3, 2]}, {[3, 1, 2], [2, 3, 4, 1], [2, 4, 3, 1]}, {[2, 1, 3], [1, 4, 2, 3], [1, 4, 3, 2]}, {[2, 3, 1], [4, 1, 2, 3], [4, 2, 1, 3]}, {[3, 1, 2], [2, 3, 4, 1], [3, 2, 4, 1]}, {[1, 3, 2], [2, 3, 1, 4], [3, 2, 1, 4]}} the member , {[1, 3, 2], [3, 1, 2, 4], [3, 2, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {[0, 1, 1]}, {}], [[3, 2, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[2, 3, 1], [1, 3, 2, 4], [4, 3, 1, 2]}, {[3, 1, 2], [1, 3, 2, 4], [3, 4, 2, 1]}, {[2, 1, 3], [1, 2, 4, 3], [4, 2, 3, 1]}, {[1, 3, 2], [2, 1, 3, 4], [4, 2, 3, 1]}} the member , {[2, 3, 1], [1, 3, 2, 4], [4, 3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 1, 0, 0]}, {2}], [[2, 1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1]}, {1}], [[1, 2], {[1, 0, 0], [0, 1, 1]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 1, 0], [0, 1, 0, 1]}, {2}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 1, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 47, 82, 135, 212, 320, 467, 662, 915, 1237, 1640, 2137, 2742, 3470, 4337, 5360] the sequence seems to be polynomial 115 71 2 3 4 12 - --- n + -- n - 5/12 n + 1/24 n 12 24 For the equivalence class of patterns, { {[2, 3, 1], [1, 4, 3, 2], [4, 1, 2, 3]}, {[1, 3, 2], [2, 3, 4, 1], [3, 2, 1, 4]}, {[3, 1, 2], [1, 4, 3, 2], [2, 3, 4, 1]}, {[2, 3, 1], [3, 2, 1, 4], [4, 1, 2, 3]}, {[1, 3, 2], [3, 2, 1, 4], [4, 1, 2, 3]}, {[2, 1, 3], [1, 4, 3, 2], [4, 1, 2, 3]}, {[3, 1, 2], [2, 3, 4, 1], [3, 2, 1, 4]}, {[2, 1, 3], [1, 4, 3, 2], [2, 3, 4, 1]}} the member , {[2, 3, 1], [1, 4, 3, 2], [4, 1, 2, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 3, 0, 0], [0, 0, 3, 0], [0, 1, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0]}, {1}], [[1, 2], {[0, 3, 0], [1, 0, 0]}, {}], [[2, 1], {[0, 3, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 50, 97, 184, 345, 642, 1189, 2196, 4049, 7458, 13729, 25264, 46481, 85506, 157285, 289308] For the equivalence class of patterns, { {[3, 1, 2], [2, 1, 4, 3], [2, 4, 3, 1]}, {[2, 3, 1], [2, 1, 4, 3], [4, 2, 1, 3]}, {[2, 3, 1], [2, 1, 4, 3], [4, 1, 3, 2]}, {[1, 3, 2], [3, 1, 2, 4], [3, 4, 1, 2]}, {[2, 1, 3], [1, 4, 2, 3], [3, 4, 1, 2]}, {[2, 1, 3], [1, 3, 4, 2], [3, 4, 1, 2]}, {[3, 1, 2], [2, 1, 4, 3], [3, 2, 4, 1]}, {[1, 3, 2], [2, 3, 1, 4], [3, 4, 1, 2]}} the member , {[3, 1, 2], [2, 1, 4, 3], [2, 4, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[2, 1], {[0, 1, 0]}, {}], [[1, 2], {[1, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[1, 3, 2], [3, 4, 2, 1], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4], [2, 1, 3, 4]}, {[3, 1, 2], [1, 2, 3, 4], [2, 1, 3, 4]}, {[2, 3, 1], [1, 2, 3, 4], [1, 2, 4, 3]}, {[1, 3, 2], [4, 3, 1, 2], [4, 3, 2, 1]}, {[2, 1, 3], [3, 4, 2, 1], [4, 3, 2, 1]}, {[2, 1, 3], [4, 3, 1, 2], [4, 3, 2, 1]}, {[3, 1, 2], [1, 2, 3, 4], [1, 2, 4, 3]}} the member , {[1, 3, 2], [3, 4, 2, 1], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1], {}, {}], [[1, 2], {[0, 1, 0]}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[2, 3, 1], [1, 2, 4, 3], [3, 2, 1, 4]}, {[1, 3, 2], [2, 3, 4, 1], [4, 3, 1, 2]}, {[3, 1, 2], [1, 4, 3, 2], [2, 1, 3, 4]}, {[2, 3, 1], [1, 4, 3, 2], [2, 1, 3, 4]}, {[2, 1, 3], [2, 3, 4, 1], [4, 3, 1, 2]}, {[1, 3, 2], [3, 4, 2, 1], [4, 1, 2, 3]}, {[2, 1, 3], [3, 4, 2, 1], [4, 1, 2, 3]}, {[3, 1, 2], [1, 2, 4, 3], [3, 2, 1, 4]}} the member , {[2, 3, 1], [1, 2, 4, 3], [3, 2, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 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, 1, 0, 1]}, {2}], [[3, 2, 1], {[0, 0, 0, 1], [1, 0, 1, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 1]}, {1}], [[2, 1], {[1, 0, 1]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[2, 3, 1], [2, 1, 4, 3], [4, 3, 1, 2]}, {[3, 1, 2], [2, 1, 4, 3], [3, 4, 2, 1]}, {[2, 1, 3], [1, 2, 4, 3], [3, 4, 1, 2]}, {[1, 3, 2], [2, 1, 3, 4], [3, 4, 1, 2]}} the member , {[2, 3, 1], [2, 1, 4, 3], [4, 3, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0]}, {1}], [[2, 1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0]}, {2}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 51, 92, 155, 247, 376, 551, 782, 1080, 1457, 1926, 2501, 3197, 4030, 5017, 6176] the sequence seems to be polynomial 23 2 3 4 2 - 7/4 n + -- n - 1/4 n + 1/24 n 24 For the equivalence class of patterns, { {[2, 3, 1], [4, 1, 3, 2], [4, 2, 1, 3]}, {[2, 1, 3], [1, 3, 4, 2], [1, 4, 2, 3]}, {[3, 1, 2], [2, 4, 3, 1], [3, 2, 4, 1]}, {[1, 3, 2], [2, 3, 1, 4], [3, 1, 2, 4]}} the member , {[2, 3, 1], [4, 1, 3, 2], [4, 2, 1, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[3, 2, 1], {[0, 0, 1, 0], [1, 1, 0, 0]}, {1}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 1], {[1, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[3, 1, 2], [3, 2, 1, 4], [3, 4, 2, 1]}, {[2, 1, 3], [1, 2, 4, 3], [2, 3, 4, 1]}, {[3, 1, 2], [1, 4, 3, 2], [3, 4, 2, 1]}, {[2, 3, 1], [3, 2, 1, 4], [4, 3, 1, 2]}, {[1, 3, 2], [2, 1, 3, 4], [2, 3, 4, 1]}, {[2, 3, 1], [1, 4, 3, 2], [4, 3, 1, 2]}, {[1, 3, 2], [2, 1, 3, 4], [4, 1, 2, 3]}, {[2, 1, 3], [1, 2, 4, 3], [4, 1, 2, 3]}} the member , {[3, 1, 2], [3, 2, 1, 4], [3, 4, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {[2, 0, 0]}, {1}], [[2, 1, 3, 4], {[2, 0, 0, 0, 0], [0, 0, 2, 0, 0], [1, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 1], {[2, 0, 1], [0, 1, 0]}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {3}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 1, 3], {[0, 1, 0, 0], [2, 0, 0, 0]}, {}], [[2, 1, 4, 3], { [2, 0, 0, 0, 0], [1, 0, 1, 0, 1], [0, 0, 2, 0, 1], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1, 2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 59, 126, 265, 551, 1136] For the equivalence class of patterns, { {[1, 2, 3], [2, 1, 4, 3], [4, 2, 3, 1]}, {[3, 2, 1], [1, 3, 2, 4], [3, 4, 1, 2]}} the member , {[1, 2, 3], [2, 1, 4, 3], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[3, 2, 1], {[1, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0], [0, 0, 1, 1], [0, 0, 0, 2]}, {2} ], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 1, 3], {[1, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {3}], [[2, 1], {[1, 2, 0], [0, 0, 2]}, {}], [[2, 3, 1], {[1, 2, 0, 0], [1, 1, 1, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684] the sequence seems to be polynomial 2 12 - 10 n + 2 n For the equivalence class of patterns, { {[3, 1, 2], [1, 2, 4, 3], [3, 2, 4, 1]}, {[1, 3, 2], [3, 1, 2, 4], [3, 4, 2, 1]}, {[3, 1, 2], [2, 1, 3, 4], [2, 4, 3, 1]}, {[2, 3, 1], [1, 2, 4, 3], [4, 2, 1, 3]}, {[2, 1, 3], [1, 4, 2, 3], [3, 4, 2, 1]}, {[2, 3, 1], [2, 1, 3, 4], [4, 1, 3, 2]}, {[2, 1, 3], [1, 3, 4, 2], [4, 3, 1, 2]}, {[1, 3, 2], [2, 3, 1, 4], [4, 3, 1, 2]}} the member , {[3, 1, 2], [1, 2, 4, 3], [3, 2, 4, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {2}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 1], {[0, 1, 0]}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 2, 3], {[0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[1, 2, 3], [3, 2, 4, 1], [3, 4, 2, 1]}, {[1, 2, 3], [4, 2, 1, 3], [4, 3, 1, 2]}, {[3, 2, 1], [1, 2, 4, 3], [1, 3, 4, 2]}, {[3, 2, 1], [1, 2, 4, 3], [1, 4, 2, 3]}, {[1, 2, 3], [2, 4, 3, 1], [3, 4, 2, 1]}, {[3, 2, 1], [2, 1, 3, 4], [2, 3, 1, 4]}, {[3, 2, 1], [2, 1, 3, 4], [3, 1, 2, 4]}, {[1, 2, 3], [4, 1, 3, 2], [4, 3, 1, 2]}} the member , {[1, 2, 3], [3, 2, 4, 1], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 2, 1]}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[3, 0, 0], [0, 0, 1]}, {}], [[1, 3, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[3, 1, 2], {[3, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {1}], [[3, 2, 1], {[0, 2, 1, 0], [0, 0, 2, 1], [0, 1, 0, 1]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 23, 38, 57, 80, 107, 138, 173, 212, 255, 302, 353, 408, 467, 530, 597, 668] the sequence seems to be polynomial 2 17 - 11 n + 2 n For the equivalence class of patterns, { {[3, 1, 2], [1, 2, 4, 3], [4, 3, 2, 1]}, {[3, 1, 2], [2, 1, 3, 4], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 4, 3], [4, 3, 2, 1]}, {[2, 3, 1], [2, 1, 3, 4], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [4, 3, 1, 2]}, {[1, 3, 2], [1, 2, 3, 4], [3, 4, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [4, 3, 1, 2]}, {[2, 1, 3], [1, 2, 3, 4], [3, 4, 2, 1]}} the member , {[3, 1, 2], [1, 2, 4, 3], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {[3, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [2, 0, 0, 0]}, {3}], [[1, 2], {[3, 0, 0], [2, 1, 0], [1, 2, 0], [0, 3, 0]}, {}], [[1, 3, 2], {[0, 0, 1, 0], [2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0]}, {2}], [[2, 1], {[2, 0, 0], [0, 1, 0]}, {1}], [[1, 2, 3], {[3, 0, 0, 0], [2, 1, 0, 0], [1, 2, 0, 0], [0, 3, 0, 0], [0, 0, 1, 0]}, {1} ]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 44, 69, 100, 137, 180, 229, 284, 345, 412, 485, 564, 649, 740, 837, 940] the sequence seems to be polynomial 2 37 - 20 n + 3 n For the equivalence class of patterns, { {[2, 3, 1], [4, 1, 2, 3], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [1, 4, 3, 2]}, {[3, 1, 2], [2, 3, 4, 1], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [3, 2, 1, 4]}} the member , {[2, 3, 1], [4, 1, 2, 3], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 3, 0], [1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0]}, {2}], [[2, 1], {[3, 0, 0], [2, 1, 0], [0, 3, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 54, 120, 265, 580, 1272, 2796, 6143, 13488, 29619, 65053, 142873, 313771, 689095, 1513390, 3323699] For the equivalence class of patterns, { {[1, 3, 2], [2, 3, 4, 1], [4, 2, 1, 3]}, {[2, 3, 1], [1, 4, 2, 3], [3, 2, 1, 4]}, {[3, 1, 2], [1, 3, 4, 2], [3, 2, 1, 4]}, {[2, 3, 1], [1, 4, 3, 2], [3, 1, 2, 4]}, {[1, 3, 2], [3, 2, 4, 1], [4, 1, 2, 3]}, {[2, 1, 3], [2, 3, 4, 1], [4, 1, 3, 2]}, {[2, 1, 3], [2, 4, 3, 1], [4, 1, 2, 3]}, {[3, 1, 2], [1, 4, 3, 2], [2, 3, 1, 4]}} the member , {[1, 3, 2], [2, 3, 4, 1], [4, 2, 1, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1], {[1, 2, 0]}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0], [1, 0, 1, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 2, 1], {[1, 2, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 4, 2, 1], {[1, 2, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}], [[3, 4, 1, 2], {[1, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 3, 1], {[1, 2, 0, 0], [0, 0, 1, 0]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 26, 52, 98, 177, 310, 531] For the equivalence class of patterns, { {[3, 2, 1], [3, 4, 1, 2], [4, 1, 2, 3]}, {[1, 2, 3], [2, 1, 4, 3], [3, 2, 1, 4]}, {[1, 2, 3], [1, 4, 3, 2], [2, 1, 4, 3]}, {[3, 2, 1], [2, 3, 4, 1], [3, 4, 1, 2]}} the member , {[3, 2, 1], [3, 4, 1, 2], [4, 1, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[3, 0]}, {}], [[1, 2], {[1, 2, 0], [0, 3, 0], [2, 0, 0]}, {1}], [[2, 1], {[0, 2, 0], [1, 0, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428] This enumerating sequence seems to have the 2 -1 + x + x rational generating function, ------------- 2 -1 + 2 x + x For the equivalence class of patterns, { {[1, 3, 2], [2, 3, 4, 1], [4, 1, 2, 3]}, {[3, 1, 2], [1, 4, 3, 2], [3, 2, 1, 4]}, {[2, 3, 1], [1, 4, 3, 2], [3, 2, 1, 4]}, {[2, 1, 3], [2, 3, 4, 1], [4, 1, 2, 3]}} the member , {[1, 3, 2], [2, 3, 4, 1], [4, 1, 2, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {[0, 2, 0]}, {}], [[1, 2], {[0, 1, 0]}, {}], [[3, 2, 1], {[0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {2}], [[2, 3, 1], {[0, 0, 1, 0], [0, 2, 0, 0]}, {3}], [[3, 1, 2], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 1, 3], [4, 1, 2, 3], [4, 3, 1, 2]}, {[1, 3, 2], [2, 3, 4, 1], [3, 4, 2, 1]}, {[1, 3, 2], [4, 1, 2, 3], [4, 3, 1, 2]}, {[3, 1, 2], [2, 1, 3, 4], [3, 2, 1, 4]}, {[2, 3, 1], [2, 1, 3, 4], [3, 2, 1, 4]}, {[2, 3, 1], [1, 2, 4, 3], [1, 4, 3, 2]}, {[2, 1, 3], [2, 3, 4, 1], [3, 4, 2, 1]}, {[3, 1, 2], [1, 2, 4, 3], [1, 4, 3, 2]}} the member , {[1, 3, 2], [2, 3, 4, 1], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {}, {1}], [[1, 2], {[0, 1, 0]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 4, 2], [3, 2, 4, 1]}, {[1, 3, 2], [3, 1, 2, 4], [3, 2, 4, 1]}, {[2, 3, 1], [3, 1, 2, 4], [4, 1, 3, 2]}, {[3, 1, 2], [2, 3, 1, 4], [2, 4, 3, 1]}, {[2, 1, 3], [1, 4, 2, 3], [2, 4, 3, 1]}, {[2, 3, 1], [1, 4, 2, 3], [4, 2, 1, 3]}, {[2, 1, 3], [1, 3, 4, 2], [4, 1, 3, 2]}, {[1, 3, 2], [2, 3, 1, 4], [4, 2, 1, 3]}} the member , {[3, 1, 2], [1, 3, 4, 2], [3, 2, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1, 2], {}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[1, 2, 3], [4, 1, 3, 2], [4, 3, 2, 1]}, {[3, 2, 1], [1, 2, 3, 4], [1, 3, 4, 2]}, {[3, 2, 1], [1, 2, 3, 4], [1, 4, 2, 3]}, {[3, 2, 1], [1, 2, 3, 4], [2, 3, 1, 4]}, {[3, 2, 1], [1, 2, 3, 4], [3, 1, 2, 4]}, {[1, 2, 3], [2, 4, 3, 1], [4, 3, 2, 1]}, {[1, 2, 3], [3, 2, 4, 1], [4, 3, 2, 1]}, {[1, 2, 3], [4, 2, 1, 3], [4, 3, 2, 1]}} the member , {[1, 2, 3], [4, 1, 3, 2], [4, 3, 2, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[2, 1, 3, 4], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 3, 0], [0, 0, 1]}, {}], [ [1, 3, 2], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {1} ], [[3, 2, 4, 1], {[0, 2, 0, 0, 0], [0, 1, 0, 1, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [0, 0, 0, 0, 1], [1, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[3, 0, 0, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {}] , [[3, 2, 1], {[1, 0, 0, 0], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1}], [[3, 1, 2], {[3, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[2, 1, 4, 3], {[0, 2, 0, 0, 0], [0, 1, 1, 0, 0], [0, 0, 2, 0, 0], [3, 0, 0, 0, 0], [2, 1, 0, 0, 0], [2, 0, 1, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0]}, {1}], [[3, 1, 4, 2], {[3, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1], {[3, 0, 0], [0, 2, 0]}, {}], [[2, 3, 1], {[3, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {1} ]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 18, 9, 0, 0, 0, 0] For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 2, 4], [2, 3, 4, 1]}, {[1, 3, 2], [3, 2, 1, 4], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [4, 1, 2, 3]}, {[2, 1, 3], [1, 4, 3, 2], [4, 2, 3, 1]}} the member , {[3, 1, 2], [1, 3, 2, 4], [2, 3, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[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], [0, 0, 0, 1]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 1]}, {1}], [[1, 2], {[1, 1, 1]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 47, 82, 135, 212, 320, 467, 662, 915, 1237, 1640, 2137, 2742, 3470, 4337, 5360] the sequence seems to be polynomial 115 71 2 3 4 12 - --- n + -- n - 5/12 n + 1/24 n 12 24 For the equivalence class of patterns, { {[2, 1, 3], [1, 2, 4, 3], [1, 4, 3, 2]}, {[3, 1, 2], [2, 3, 4, 1], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4], [3, 2, 1, 4]}, {[2, 3, 1], [4, 1, 2, 3], [4, 3, 1, 2]}} the member , {[2, 1, 3], [1, 2, 4, 3], [1, 4, 3, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 3, 2], {[0, 1, 0, 0], [0, 0, 0, 1]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, 128801, 299426, 696081, 1618192, 3761840, 8745217] This enumerating sequence seems to have the 2 1 - 2 x + x rational generating function, - -------------------- 2 3 -1 + 3 x - 2 x + x For the equivalence class of patterns, { {[3, 1, 2], [2, 1, 4, 3], [2, 3, 1, 4]}, {[1, 3, 2], [3, 4, 1, 2], [4, 2, 1, 3]}, {[2, 3, 1], [2, 1, 4, 3], [3, 1, 2, 4]}, {[1, 3, 2], [3, 2, 4, 1], [3, 4, 1, 2]}, {[2, 1, 3], [3, 4, 1, 2], [4, 1, 3, 2]}, {[2, 3, 1], [1, 4, 2, 3], [2, 1, 4, 3]}, {[3, 1, 2], [1, 3, 4, 2], [2, 1, 4, 3]}, {[2, 1, 3], [2, 4, 3, 1], [3, 4, 1, 2]}} the member , {[3, 1, 2], [2, 1, 4, 3], [2, 3, 1, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 1], {[0, 1, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2, 3], {}, {1}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[2, 4, 3, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014] For the equivalence class of patterns, { {[1, 2, 3], [3, 1, 4, 2], [3, 4, 1, 2]}, {[1, 2, 3], [2, 4, 1, 3], [3, 4, 1, 2]}, {[3, 2, 1], [2, 1, 4, 3], [2, 4, 1, 3]}, {[3, 2, 1], [2, 1, 4, 3], [3, 1, 4, 2]}} the member , {[1, 2, 3], [3, 1, 4, 2], [3, 4, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 0, 1]}, {}], [[2, 1], {[0, 1, 1]}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[2, 1, 3], [1, 2, 3, 4], [2, 3, 4, 1]}, {[2, 3, 1], [1, 4, 3, 2], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [4, 1, 2, 3]}, {[3, 1, 2], [1, 4, 3, 2], [4, 3, 2, 1]}, {[2, 3, 1], [3, 2, 1, 4], [4, 3, 2, 1]}, {[3, 1, 2], [3, 2, 1, 4], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [2, 3, 4, 1]}, {[1, 3, 2], [1, 2, 3, 4], [4, 1, 2, 3]}} the member , {[2, 1, 3], [1, 2, 3, 4], [2, 3, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[1, 3, 2], {[0, 0, 0, 1], [1, 0, 1, 0]}, {2}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 0, 0, 1]}, {3}], [[1, 2], {[1, 0, 1]}, {}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, 3328, 7168, 15360, 32768, 69632, 147456, 311296, 655360, 1376256, 2883584] This enumerating sequence seems to have the 2 3 1 - 3 x + 2 x + x rational generating function, ------------------- 2 1 - 4 x + 4 x For the equivalence class of patterns, { {[3, 1, 2], [1, 3, 2, 4], [4, 3, 2, 1]}, {[2, 3, 1], [1, 3, 2, 4], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [4, 2, 3, 1]}, {[1, 3, 2], [1, 2, 3, 4], [4, 2, 3, 1]}} the member , {[3, 1, 2], [1, 3, 2, 4], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 2, 3], {[3, 0, 0, 0], [2, 1, 0, 0], [1, 2, 0, 0], [0, 3, 0, 0], [2, 0, 1, 0], [1, 1, 1, 0], [0, 2, 1, 0], [1, 0, 2, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {2}], [[1], {[3, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0], [2, 0, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [0, 0, 0, 1], [2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0]}, {1} ], [[1, 2], {[3, 0, 0], [2, 1, 0], [1, 2, 0], [0, 3, 0]}, {}], [[2, 1], {[2, 0, 0], [0, 1, 0]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 47, 82, 135, 212, 320, 467, 662, 915, 1237, 1640, 2137, 2742, 3470, 4337, 5360] the sequence seems to be polynomial 115 71 2 3 4 12 - --- n + -- n - 5/12 n + 1/24 n 12 24 For the equivalence class of patterns, { {[2, 3, 1], [1, 4, 2, 3], [2, 1, 3, 4]}, {[3, 1, 2], [1, 3, 4, 2], [2, 1, 3, 4]}, {[1, 3, 2], [3, 4, 2, 1], [4, 2, 1, 3]}, {[2, 3, 1], [1, 2, 4, 3], [3, 1, 2, 4]}, {[2, 1, 3], [2, 4, 3, 1], [4, 3, 1, 2]}, {[1, 3, 2], [3, 2, 4, 1], [4, 3, 1, 2]}, {[2, 1, 3], [3, 4, 2, 1], [4, 1, 3, 2]}, {[3, 1, 2], [1, 2, 4, 3], [2, 3, 1, 4]}} the member , {[2, 3, 1], [1, 4, 2, 3], [2, 1, 3, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[4, 1, 3, 2], {[0, 0, 0, 1, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [ [3, 1, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {[1, 0, 0]}, {}], [[2, 3, 1], {[0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 1, 1, 1], [1, 0, 0, 0]}, {}], [[2, 4, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 1], {[1, 1, 1]}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0]}, {}], [[3, 2, 1], {[1, 1, 1, 0], [1, 1, 0, 1], [0, 0, 1, 1]}, {1}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [ [1, 3, 2, 4], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {2}], [[4, 1, 2, 3], {[0, 0, 1, 1, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {2}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 26, 51, 92, 155, 247, 376] For the equivalence class of patterns, { {[1, 3, 2], [2, 3, 4, 1], [4, 2, 3, 1]}, {[2, 1, 3], [4, 1, 2, 3], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4], [3, 2, 1, 4]}, {[1, 3, 2], [4, 1, 2, 3], [4, 2, 3, 1]}, {[3, 1, 2], [1, 3, 2, 4], [1, 4, 3, 2]}, {[2, 3, 1], [1, 3, 2, 4], [1, 4, 3, 2]}, {[3, 1, 2], [1, 3, 2, 4], [3, 2, 1, 4]}, {[2, 1, 3], [2, 3, 4, 1], [4, 2, 3, 1]}} the member , {[1, 3, 2], [2, 3, 4, 1], [4, 2, 3, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1], {}, {2}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 1], {}, {}], [[1, 2], {[0, 1, 0]}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 3, 1], [3, 1, 4, 2]}, {[1, 2, 3], [3, 1, 4, 2], [4, 2, 1, 3]}, {[1, 2, 3], [2, 4, 1, 3], [3, 2, 4, 1]}, {[3, 2, 1], [1, 3, 4, 2], [2, 4, 1, 3]}, {[3, 2, 1], [1, 4, 2, 3], [3, 1, 4, 2]}, {[1, 2, 3], [2, 4, 1, 3], [4, 1, 3, 2]}, {[3, 2, 1], [2, 3, 1, 4], [3, 1, 4, 2]}, {[3, 2, 1], [2, 4, 1, 3], [3, 1, 2, 4]}} the member , {[1, 2, 3], [2, 4, 3, 1], [3, 1, 4, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 2], {[0, 0, 1]}, {}], [[2, 1], {[0, 1, 1]}, {1}], [[2, 3, 1], {[0, 0, 0, 1], [0, 1, 1, 0]}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}] , [[2, 4, 1, 3], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 3, 1, 4], {[0, 0, 0, 0, 0]}, {3}], [[3, 4, 2, 1], {[0, 1, 0, 1, 0], [0, 0, 1, 1, 0], [0, 1, 1, 0, 0], [0, 0, 0, 0, 1]}, {3}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014] For the equivalence class of patterns, { {[1, 2, 3], [1, 4, 3, 2], [4, 3, 2, 1]}, {[3, 2, 1], [1, 2, 3, 4], [2, 3, 4, 1]}, {[3, 2, 1], [1, 2, 3, 4], [4, 1, 2, 3]}, {[1, 2, 3], [3, 2, 1, 4], [4, 3, 2, 1]}} the member , {[1, 2, 3], [1, 4, 3, 2], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 1, 3], {[3, 0, 0, 0], [0, 0, 0, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 0, 2, 0]}, {3} ], [[3, 2, 1], {[0, 1, 2, 0], [0, 0, 3, 0], [0, 1, 1, 1], [0, 0, 2, 1], [1, 0, 0, 0], [0, 1, 0, 2], [0, 0, 1, 2], [0, 0, 0, 3], [0, 2, 0, 0]}, {1}] , [[1], {[0, 3]}, {}], [[3, 1, 2], {[3, 0, 0, 0], [2, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], [0, 2, 0, 0]}, {2} ], [[2, 1], {[3, 0, 0], [0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]}, {}], [[1, 2], {[0, 2, 0], [0, 0, 1]}, {1}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 20, 12, 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, 1, 3], [1, 2, 3, 4], [2, 4, 3, 1]}, {[3, 1, 2], [2, 3, 1, 4], [4, 3, 2, 1]}, {[2, 1, 3], [1, 2, 3, 4], [4, 1, 3, 2]}, {[3, 1, 2], [1, 3, 4, 2], [4, 3, 2, 1]}, {[2, 3, 1], [1, 4, 2, 3], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [3, 2, 4, 1]}, {[2, 3, 1], [3, 1, 2, 4], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [4, 2, 1, 3]}} the member , {[2, 1, 3], [1, 2, 3, 4], [2, 4, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1, 3, 2], {[0, 1, 2, 0], [1, 0, 0, 0], [0, 0, 0, 1]}, {2}], [[1], {}, {}], [[1, 2], {[1, 1, 0], [1, 0, 2], [0, 1, 2]}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2, 3], {[0, 0, 0, 1], [1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 1, 0]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 26, 55, 113, 227, 449, 877, 1696, 3254, 6203, 11762, 22205, 41766, 78316, 146467, 273313, 509041] For the equivalence class of patterns, { {[1, 2, 3], [2, 4, 1, 3], [2, 4, 3, 1]}, {[3, 2, 1], [3, 1, 2, 4], [3, 1, 4, 2]}, {[1, 2, 3], [3, 1, 4, 2], [4, 1, 3, 2]}, {[1, 2, 3], [3, 1, 4, 2], [3, 2, 4, 1]}, {[3, 2, 1], [1, 3, 4, 2], [3, 1, 4, 2]}, {[3, 2, 1], [1, 4, 2, 3], [2, 4, 1, 3]}, {[3, 2, 1], [2, 3, 1, 4], [2, 4, 1, 3]}, {[1, 2, 3], [2, 4, 1, 3], [4, 2, 1, 3]}} the member , {[1, 2, 3], [2, 4, 1, 3], [2, 4, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 2]}, {1}], [[1, 2], {[1, 1, 0], [0, 0, 1]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556] This enumerating sequence seems to have the 2 1 - 3 x + 3 x rational generating function, - ---------------------- 2 3 -1 + 4 x - 5 x + 2 x For the equivalence class of patterns, { {[3, 2, 1], [1, 2, 3, 4], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2], [4, 3, 2, 1]}} the member , {[3, 2, 1], [1, 2, 3, 4], [2, 1, 4, 3]}, has a scheme of depth , 4 here it is: {[[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]}, {2}], [[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[2, 1, 3], {[0, 3, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2]}, {1}], [[1, 2, 3], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 0, 0, 1], [0, 0, 2, 0]}, {1}], [[2, 3, 1, 4], {[0, 3, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 1, 1, 0], [0, 0, 0, 2, 0], [0, 0, 2, 0, 0], [0, 0, 0, 0, 2], [0, 0, 0, 1, 1], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 2, 0], [0, 0, 0, 0, 2], [0, 0, 0, 1, 1], [0, 0, 1, 0, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[2, 1], {[0, 3, 0], [0, 0, 3], [1, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]}, {1}], [[1, 2], {[0, 3, 0], [0, 0, 3]}, {}], [[2, 3, 1], {[0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0], [0, 1, 1, 1], [0, 0, 2, 1], [1, 0, 0, 0], [0, 0, 0, 2]}, {}]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 16, 9, 0, 0, 0, 0] For the equivalence class of patterns, { {[3, 1, 2], [2, 1, 4, 3], [2, 3, 4, 1]}, {[2, 1, 3], [1, 4, 3, 2], [3, 4, 1, 2]}, {[1, 3, 2], [3, 2, 1, 4], [3, 4, 1, 2]}, {[2, 3, 1], [2, 1, 4, 3], [4, 1, 2, 3]}} the member , {[3, 1, 2], [2, 1, 4, 3], [2, 3, 4, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0]}, {1}], [[3, 1, 4, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 1, 2], {[0, 0, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 2], {}, {}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[2, 4, 3, 1], %1, {2}], [[1, 4, 3, 2], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {2}], [[2, 1, 4, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], %1, {3}], [[2, 1], {[0, 1, 0]}, {}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[3, 2, 4, 1], %1, {1}]} %1 := {[0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 1, 0, 0, 0]} Using the scheme, the first, , 11, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250] For the equivalence class of patterns, { {[3, 1, 2], [1, 2, 4, 3], [3, 4, 2, 1]}, {[2, 3, 1], [1, 2, 4, 3], [4, 3, 1, 2]}, {[3, 1, 2], [2, 1, 3, 4], [3, 4, 2, 1]}, {[2, 1, 3], [1, 2, 4, 3], [4, 3, 1, 2]}, {[2, 3, 1], [2, 1, 3, 4], [4, 3, 1, 2]}, {[2, 1, 3], [1, 2, 4, 3], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4], [3, 4, 2, 1]}, {[1, 3, 2], [2, 1, 3, 4], [4, 3, 1, 2]}} the member , {[3, 1, 2], [1, 2, 4, 3], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 1, 0], [2, 0, 0, 0]}, {2}], [[1, 2], {[2, 0, 0]}, {}], [[1, 2, 3], {[0, 0, 1, 0], [2, 0, 0, 0], [1, 1, 0, 0], [0, 2, 0, 0]}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 25, 46, 77, 120, 177, 250, 341, 452, 585, 742, 925, 1136, 1377, 1650, 1957, 2300] the sequence seems to be polynomial 2 3 -3 + 14/3 n - 2 n + 1/3 n For the equivalence class of patterns, { {[2, 1, 3], [1, 2, 3, 4], [1, 2, 4, 3]}, {[3, 1, 2], [3, 4, 2, 1], [4, 3, 2, 1]}, {[1, 3, 2], [1, 2, 3, 4], [2, 1, 3, 4]}, {[2, 3, 1], [4, 3, 1, 2], [4, 3, 2, 1]}} the member , {[2, 1, 3], [1, 2, 3, 4], [1, 2, 4, 3]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[0, 0, 1]}, {1}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2], {[0, 0, 2]}, {}], [[1, 2, 3], {[0, 0, 1, 0], [0, 0, 0, 1]}, {2}], [[1, 3, 2], {[0, 0, 0, 1], [0, 0, 2, 0]}, {2}]} Using the scheme, the first, , 21, terms are [1, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428] This enumerating sequence seems to have the 2 -1 + x + x rational generating function, ------------- 2 -1 + 2 x + x Out of a total of , 92, cases 92, were successful and , 0, failed Success Rate: , 1. Here are the failures {} {}