There are , 18, different equivalence classes For the equivalence class of patterns, {{[3, 2, 1], [1, 2, 3, 4]}, {[1, 2, 3], [4, 3, 2, 1]}} the member , {[3, 2, 1], [1, 2, 3, 4]}, has a scheme of depth , 4 here it is: {[[1, 2, 3], {[0, 0, 2, 0], [0, 0, 0, 1], [0, 3, 0, 0], [0, 2, 1, 0]}, {1}], [ [2, 3, 1], {[0, 0, 0, 3], [1, 0, 0, 0], [0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {}], [[], {}, {}], [[2, 1, 3], {[0, 0, 0, 3], [1, 0, 0, 0], [0, 3, 0, 0], [0, 2, 1, 0], [0, 1, 2, 0], [0, 0, 3, 0]}, {1}] , [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 2, 0], [0, 3, 0, 0, 0], [0, 2, 1, 0, 0], [0, 1, 2, 0, 0], [0, 0, 3, 0, 0], [0, 2, 0, 1, 0], [0, 1, 1, 1, 0], [0, 0, 2, 1, 0]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 2], [0, 0, 0, 1, 2], [0, 0, 0, 0, 3], [0, 0, 2, 0, 0], [0, 0, 1, 1, 0], [0, 0, 0, 2, 0]}, {1}], [[2, 1], {[1, 0, 0], [0, 3, 0]}, {}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 2], [0, 0, 0, 0, 3], [0, 0, 0, 2, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 3], [0, 0, 2, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 2]}, {2} ], [[1, 2], {[0, 3, 0], [0, 0, 3]}, {}], [[3, 1, 2], {[0, 0, 0, 3], [0, 0, 2, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 2]}, {1} ], [[1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 25, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] the sequence is finite For the equivalence class of patterns, {{[3, 2, 1], [1, 2, 4, 3]}, {[3, 2, 1], [2, 1, 3, 4]}, {[1, 2, 3], [3, 4, 2, 1]}, {[1, 2, 3], [4, 3, 1, 2]}} the member , {[3, 2, 1], [1, 2, 4, 3]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {2}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 0, 1, 0]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 30, 61, 112, 190, 303, 460, 671, 947, 1300, 1743, 2290, 2956, 3757, 4710, 5833, 7145, 8666, 10417, 12420, 14698, 17275, 20176, 23427, 27055, 31088, 35555] the sequence seems to be polynomial 31 13 2 3 4 -2 + -- n - -- n - 1/12 n + 1/24 n 12 24 For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 2, 4]}, {[1, 2, 3], [4, 2, 3, 1]}} the member , {[3, 2, 1], [1, 3, 2, 4]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[1, 2, 3], {[0, 1, 1, 0]}, {2}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]}, {1}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 0, 1]}, {1}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 32, 72, 148, 281, 499, 838, 1343, 2069, 3082, 4460, 6294, 8689, 11765, 15658, 20521, 26525, 33860, 42736, 53384, 66057, 81031, 98606, 119107, 142885, 170318, 201812] the sequence seems to be polynomial 31 13 2 3 4 5 2 - -- n + -- n + 1/24 n - 1/24 n + 1/120 n 20 24 For the equivalence class of patterns, {{[3, 2, 1], [1, 3, 4, 2]}, {[3, 2, 1], [1, 4, 2, 3]}, {[3, 2, 1], [2, 3, 1, 4]}, {[3, 2, 1], [3, 1, 2, 4]}, {[1, 2, 3], [2, 4, 3, 1]}, {[1, 2, 3], [3, 2, 4, 1]}, {[1, 2, 3], [4, 1, 3, 2]}, {[1, 2, 3], [4, 2, 1, 3]}} the member , {[3, 2, 1], [1, 3, 4, 2]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {3}], [[3, 2, 1], {[0, 0, 0, 0]}, {1}], [[2, 1, 3], {[1, 0, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 0, 0]}, {1}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {2}], [[3, 4, 1, 2], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[2, 4, 1, 3], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0]}, {3}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 0, 1, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 32, 74, 163, 347, 722, 1480, 3005, 6065, 12196, 24470, 49031, 98167, 196454, 393044, 786241, 1572653, 3145496, 6291202, 12582635, 25165523, 50331322, 100662944, 201326213, 402652777, 805305932, 1610612270] This enumerating sequence seems to have the 2 3 4 1 - 4 x + 6 x - 3 x + x rational generating function, ---------------------------- 2 3 4 1 - 5 x + 9 x - 7 x + 2 x For the equivalence class of patterns, {{[3, 2, 1], [2, 3, 4, 1]}, {[3, 2, 1], [4, 1, 2, 3]}, {[1, 2, 3], [1, 4, 3, 2]}, {[1, 2, 3], [3, 2, 1, 4]}} the member , {[3, 2, 1], [2, 3, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[2, 3, 1], {[1, 0, 0, 0]}, {3}], [[1], {}, {}], [[1, 2], {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[1, 3, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[2, 1], {[1, 0, 0]}, {2}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[3, 2, 1], [2, 4, 1, 3]}, {[3, 2, 1], [3, 1, 4, 2]}, {[1, 2, 3], [2, 4, 1, 3]}, {[1, 2, 3], [3, 1, 4, 2]}} the member , {[3, 2, 1], [2, 4, 1, 3]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {[1, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[3, 2, 1], [3, 4, 1, 2]}, {[1, 2, 3], [2, 1, 4, 3]}} the member , {[3, 2, 1], [3, 4, 1, 2]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[2, 1], {[1, 0, 0]}, {2}], [[1, 2], {[2, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [1, 2, 3, 4]}, {[2, 1, 3], [1, 2, 3, 4]}, {[2, 3, 1], [4, 3, 2, 1]}, {[3, 1, 2], [4, 3, 2, 1]}} the member , {[1, 3, 2], [1, 2, 3, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1, 2], {[0, 1, 0], [0, 0, 2]}, {1}], [[2, 1], {[0, 3, 0], [0, 2, 1], [0, 1, 2], [0, 0, 3]}, {1}], [[1], {[0, 3]}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [2, 1, 3, 4]}, {[2, 1, 3], [1, 2, 4, 3]}, {[2, 3, 1], [4, 3, 1, 2]}, {[3, 1, 2], [3, 4, 2, 1]}} the member , {[1, 3, 2], [2, 1, 3, 4]}, has a scheme of depth , 2 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {[0, 0, 2]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [2, 3, 1, 4]}, {[1, 3, 2], [3, 1, 2, 4]}, {[2, 1, 3], [1, 3, 4, 2]}, {[2, 1, 3], [1, 4, 2, 3]}, {[2, 3, 1], [4, 1, 3, 2]}, {[2, 3, 1], [4, 2, 1, 3]}, {[3, 1, 2], [2, 4, 3, 1]}, {[3, 1, 2], [3, 2, 4, 1]}} the member , {[1, 3, 2], [2, 3, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[2, 3, 1], {[0, 0, 1, 0], [0, 0, 0, 1]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1], {}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [2, 3, 4, 1]}, {[1, 3, 2], [4, 1, 2, 3]}, {[2, 1, 3], [2, 3, 4, 1]}, {[2, 1, 3], [4, 1, 2, 3]}, {[2, 3, 1], [1, 4, 3, 2]}, {[2, 3, 1], [3, 2, 1, 4]}, {[3, 1, 2], [1, 4, 3, 2]}, {[3, 1, 2], [3, 2, 1, 4]}} the member , {[1, 3, 2], [2, 3, 4, 1]}, has a scheme of depth , 4 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[3, 4, 2, 1], {[0, 0, 0, 1, 0]}, {3}], [[2, 1], {}, {1}], [[3, 4, 1, 2], {[0, 1, 0, 0, 0], [0, 0, 0, 1, 0]}, {1, 2}], [[2, 3, 1], {[0, 0, 1, 0]}, {}], [[1, 2, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 0, 0, 0]}, {3}], [[2, 3, 1, 4], {[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [3, 2, 1, 4]}, {[2, 1, 3], [1, 4, 3, 2]}, {[2, 3, 1], [4, 1, 2, 3]}, {[3, 1, 2], [2, 3, 4, 1]}} the member , {[1, 3, 2], [3, 2, 1, 4]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[2, 1], {}, {}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 33, 82, 202, 497, 1224, 3017, 7439, 18343, 45228, 111514, 274945, 677894, 1671393, 4120937, 10160465, 25051354, 61765902, 152288233, 375477484, 925766477, 2282543187, 5627772815, 13875674756, 34211464510, 84350802705, 207972912538] This enumerating sequence seems to have the 2 3 -1 + 3 x - 3 x + x rational generating function, ---------------------- 2 3 -1 + 4 x - 5 x + 3 x For the equivalence class of patterns, {{[1, 3, 2], [3, 2, 4, 1]}, {[1, 3, 2], [4, 2, 1, 3]}, {[2, 1, 3], [2, 4, 3, 1]}, {[2, 1, 3], [4, 1, 3, 2]}, {[2, 3, 1], [1, 4, 2, 3]}, {[2, 3, 1], [3, 1, 2, 4]}, {[3, 1, 2], [1, 3, 4, 2]}, {[3, 1, 2], [2, 3, 1, 4]}} the member , {[1, 3, 2], [3, 2, 4, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 1], {}, {}], [[2, 1, 3], {[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[3, 2, 1], {}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [3, 4, 1, 2]}, {[2, 1, 3], [3, 4, 1, 2]}, {[2, 3, 1], [2, 1, 4, 3]}, {[3, 1, 2], [2, 1, 4, 3]}} the member , {[1, 3, 2], [3, 4, 1, 2]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1], {}, {1}], [[2, 3, 1], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041] This enumerating sequence seems to have the 2 x - 1 rational generating function, - ------------ 2 1 - 3 x + x For the equivalence class of patterns, {{[1, 3, 2], [3, 4, 2, 1]}, {[1, 3, 2], [4, 3, 1, 2]}, {[2, 1, 3], [3, 4, 2, 1]}, {[2, 1, 3], [4, 3, 1, 2]}, {[2, 3, 1], [1, 2, 4, 3]}, {[2, 3, 1], [2, 1, 3, 4]}, {[3, 1, 2], [1, 2, 4, 3]}, {[3, 1, 2], [2, 1, 3, 4]}} the member , {[1, 3, 2], [3, 4, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}], [[1, 3, 2], {[0, 0, 0, 0]}, {1}], [[1, 2], {[0, 1, 0]}, {}], [[2, 1], {}, {1}], [[2, 3, 1], {[1, 0, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689, 245761, 524289, 1114113, 2359297, 4980737, 10485761, 22020097, 46137345, 96468993, 201326593, 419430401, 872415233, 1811939329, 3758096385, 7784628225] This enumerating sequence seems to have the 2 3 -1 + 4 x - 5 x + x rational generating function, ---------------------- 2 3 -1 + 5 x - 8 x + 4 x For the equivalence class of patterns, {{[1, 3, 2], [4, 2, 3, 1]}, {[2, 1, 3], [4, 2, 3, 1]}, {[2, 3, 1], [1, 3, 2, 4]}, {[3, 1, 2], [1, 3, 2, 4]}} the member , {[1, 3, 2], [4, 2, 3, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[3, 1, 2], {[1, 0, 0, 0], [0, 1, 0, 0]}, {1}], [[1], {}, {}], [[3, 2, 1], {}, {2}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1], {}, {}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 33, 81, 193, 449, 1025, 2305, 5121, 11265, 24577, 53249, 114689, 245761, 524289, 1114113, 2359297, 4980737, 10485761, 22020097, 46137345, 96468993, 201326593, 419430401, 872415233, 1811939329, 3758096385, 7784628225] This enumerating sequence seems to have the 2 3 -1 + 4 x - 5 x + x rational generating function, ---------------------- 2 3 -1 + 5 x - 8 x + 4 x For the equivalence class of patterns, {{[1, 3, 2], [4, 3, 2, 1]}, {[2, 1, 3], [4, 3, 2, 1]}, {[2, 3, 1], [1, 2, 3, 4]}, {[3, 1, 2], [1, 2, 3, 4]}} the member , {[1, 3, 2], [4, 3, 2, 1]}, has a scheme of depth , 3 here it is: {[[], {}, {}], [[1], {}, {}], [[1, 2], {[0, 1, 0]}, {1}], [[2, 1, 3], {[0, 1, 0, 0], [0, 0, 1, 0]}, {3}], [[3, 1, 2], {[0, 1, 0, 0]}, {2}], [[2, 1], {}, {}], [[3, 2, 1], {[1, 0, 0, 0]}, {1}]} Using the scheme, the first, , 31, terms are [1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, 3291, 4321, 5577, 7090, 8893, 11021, 13511, 16402, 19735, 23553, 27901, 32826, 38377, 44605, 51563, 59306] the sequence seems to be polynomial 29 2 3 4 3 - 23/6 n + -- n - 2/3 n + 1/12 n 12 Out of a total of , 18, cases 17, were successful and , 1, failed Success Rate: , 0.944 Here are the failures {{{[3, 2, 1], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2]}}} {{{[3, 2, 1], [2, 1, 4, 3]}, {[1, 2, 3], [3, 4, 1, 2]}}} It took, 456.156, seconds of CPU time .