There are , 5, different equivalence classes For the equivalence class of patterns, {{[3, 2, 1], [1, 2, 3]}} the member , {[3, 2, 1], [1, 2, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {[3, 0], [0, 3]}, {}], [[2, 1], {[1, 0, 0], [0, 2, 0], [0, 1, 2], [0, 0, 3]}, {1}], [[1, 2], {[0, 0, 1], [0, 2, 0], [2, 1, 0], [3, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2]}, {[3, 2, 1], [2, 1, 3]}, {[1, 2, 3], [2, 3, 1]}, {[1, 2, 3], [3, 1, 2]}} the member , {[3, 2, 1], [1, 3, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436] the sequence seems to be polynomial 2 2 - 3/2 n + 1/2 n For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 1]}, {[3, 2, 1], [3, 1, 2]}, {[1, 2, 3], [1, 3, 2]}, {[1, 2, 3], [2, 1, 3]}} the member , {[3, 2, 1], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {[1, 0, 0]}, {1}], [[1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912] This enumerating sequence seems to have the -1 + x rational generating function, -------- -1 + 2 x For the equivalence class of patterns, {{[1, 3, 2], [2, 1, 3]}, {[2, 3, 1], [3, 1, 2]}} the member , {[1, 3, 2], [2, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[1], {}, {}], [[2, 1], {[0, 0, 1]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912] This enumerating sequence seems to have the -1 + x rational generating function, -------- -1 + 2 x For the equivalence class of patterns, {{[1, 3, 2], [2, 3, 1]}, {[1, 3, 2], [3, 1, 2]}, {[2, 1, 3], [2, 3, 1]}, {[2, 1, 3], [3, 1, 2]}} the member , {[1, 3, 2], [2, 3, 1]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[1, 0, 0], [0, 1, 0]}, {1}], [[2, 1], {}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912] This enumerating sequence seems to have the -1 + x rational generating function, -------- -1 + 2 x Out of a total of , 5, cases 5, were successful and , 0, failed Success Rate: , 1. Here are the failures {} {} It took, 5.884, seconds of CPU time .