***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000277 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The number of ribbon shaped standard tableaux. A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape. This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed. For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals $$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$ where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1] ----------------------------------------------------------------------------- References: [1] Stanley, R. P. Enumerative combinatorics. Volume 1 [[MathSciNet:2868112]] ----------------------------------------------------------------------------- Code: def composition_to_ribbon(c): inner = [] outer = [] indent = 0 for p in reversed(c): if indent > 0: inner.append(indent) outer.append(p+indent) indent += p-1 return SkewPartition([outer[::-1], inner[::-1]]) def statistic(c): return StandardSkewTableaux(composition_to_ribbon(c)).cardinality() ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,1] => 1 [2] => 1 [1,1,1] => 1 [1,2] => 2 [2,1] => 2 [3] => 1 [1,1,1,1] => 1 [1,1,2] => 3 [1,2,1] => 5 [1,3] => 3 [2,1,1] => 3 [2,2] => 5 [3,1] => 3 [4] => 1 [1,1,1,1,1] => 1 [1,1,1,2] => 4 [1,1,2,1] => 9 [1,1,3] => 6 [1,2,1,1] => 9 [1,2,2] => 16 [1,3,1] => 11 [1,4] => 4 [2,1,1,1] => 4 [2,1,2] => 11 [2,2,1] => 16 [2,3] => 9 [3,1,1] => 6 [3,2] => 9 [4,1] => 4 [5] => 1 [1,1,1,1,1,1] => 1 [1,1,1,1,2] => 5 [1,1,1,2,1] => 14 [1,1,1,3] => 10 [1,1,2,1,1] => 19 [1,1,2,2] => 35 [1,1,3,1] => 26 [1,1,4] => 10 [1,2,1,1,1] => 14 [1,2,1,2] => 40 [1,2,2,1] => 61 [1,2,3] => 35 [1,3,1,1] => 26 [1,3,2] => 40 [1,4,1] => 19 [1,5] => 5 [2,1,1,1,1] => 5 [2,1,1,2] => 19 [2,1,2,1] => 40 [2,1,3] => 26 [2,2,1,1] => 35 [2,2,2] => 61 [2,3,1] => 40 [2,4] => 14 [3,1,1,1] => 10 [3,1,2] => 26 [3,2,1] => 35 [3,3] => 19 [4,1,1] => 10 [4,2] => 14 [5,1] => 5 [6] => 1 [1,1,1,1,1,1,1] => 1 [1,1,1,1,1,2] => 6 [1,1,1,1,2,1] => 20 [1,1,1,1,3] => 15 [1,1,1,2,1,1] => 34 [1,1,1,2,2] => 64 [1,1,1,3,1] => 50 [1,1,1,4] => 20 [1,1,2,1,1,1] => 34 [1,1,2,1,2] => 99 [1,1,2,2,1] => 155 [1,1,2,3] => 90 [1,1,3,1,1] => 71 [1,1,3,2] => 111 [1,1,4,1] => 55 [1,1,5] => 15 [1,2,1,1,1,1] => 20 [1,2,1,1,2] => 78 [1,2,1,2,1] => 169 [1,2,1,3] => 111 [1,2,2,1,1] => 155 [1,2,2,2] => 272 [1,2,3,1] => 181 [1,2,4] => 64 [1,3,1,1,1] => 50 [1,3,1,2] => 132 [1,3,2,1] => 181 [1,3,3] => 99 [1,4,1,1] => 55 [1,4,2] => 78 [1,5,1] => 29 [1,6] => 6 [2,1,1,1,1,1] => 6 [2,1,1,1,2] => 29 [2,1,1,2,1] => 78 [2,1,1,3] => 55 [2,1,2,1,1] => 99 [2,1,2,2] => 181 [2,1,3,1] => 132 [2,1,4] => 50 [2,2,1,1,1] => 64 [2,2,1,2] => 181 [2,2,2,1] => 272 [2,2,3] => 155 [2,3,1,1] => 111 [2,3,2] => 169 [2,4,1] => 78 [2,5] => 20 [3,1,1,1,1] => 15 [3,1,1,2] => 55 [3,1,2,1] => 111 [3,1,3] => 71 [3,2,1,1] => 90 [3,2,2] => 155 [3,3,1] => 99 [3,4] => 34 [4,1,1,1] => 20 [4,1,2] => 50 [4,2,1] => 64 [4,3] => 34 [5,1,1] => 15 [5,2] => 20 [6,1] => 6 [7] => 1 [1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,2] => 7 [1,1,1,1,1,2,1] => 27 [1,1,1,1,1,3] => 21 [1,1,1,1,2,1,1] => 55 [1,1,1,1,2,2] => 105 [1,1,1,1,3,1] => 85 [1,1,1,1,4] => 35 [1,1,1,2,1,1,1] => 69 [1,1,1,2,1,2] => 203 [1,1,1,2,2,1] => 323 [1,1,1,2,3] => 189 [1,1,1,3,1,1] => 155 [1,1,1,3,2] => 245 [1,1,1,4,1] => 125 [1,1,1,5] => 35 [1,1,2,1,1,1,1] => 55 [1,1,2,1,1,2] => 217 [1,1,2,1,2,1] => 477 [1,1,2,1,3] => 315 [1,1,2,2,1,1] => 449 [1,1,2,2,2] => 791 [1,1,2,3,1] => 531 [1,1,2,4] => 189 [1,1,3,1,1,1] => 155 [1,1,3,1,2] => 413 [1,1,3,2,1] => 573 [1,1,3,3] => 315 [1,1,4,1,1] => 181 [1,1,4,2] => 259 [1,1,5,1] => 99 [1,1,6] => 21 [1,2,1,1,1,1,1] => 27 [1,2,1,1,1,2] => 133 [1,2,1,1,2,1] => 365 [1,2,1,1,3] => 259 [1,2,1,2,1,1] => 477 [1,2,1,2,2] => 875 [1,2,1,3,1] => 643 [1,2,1,4] => 245 [1,2,2,1,1,1] => 323 [1,2,2,1,2] => 917 [1,2,2,2,1] => 1385 [1,2,2,3] => 791 [1,2,3,1,1] => 573 [1,2,3,2] => 875 [1,2,4,1] => 407 [1,2,5] => 105 [1,3,1,1,1,1] => 85 [1,3,1,1,2] => 315 [1,3,1,2,1] => 643 [1,3,1,3] => 413 [1,3,2,1,1] => 531 [1,3,2,2] => 917 [1,3,3,1] => 589 [1,3,4] => 203 [1,4,1,1,1] => 125 [1,4,1,2] => 315 [1,4,2,1] => 407 [1,4,3] => 217 [1,5,1,1] => 99 [1,5,2] => 133 [1,6,1] => 41 [1,7] => 7 [2,1,1,1,1,1,1] => 7 [2,1,1,1,1,2] => 41 [2,1,1,1,2,1] => 133 [2,1,1,1,3] => 99 [2,1,1,2,1,1] => 217 [2,1,1,2,2] => 407 [2,1,1,3,1] => 315 [2,1,1,4] => 125 [2,1,2,1,1,1] => 203 [2,1,2,1,2] => 589 [2,1,2,2,1] => 917 [2,1,2,3] => 531 [2,1,3,1,1] => 413 [2,1,3,2] => 643 [2,1,4,1] => 315 [2,1,5] => 85 [2,2,1,1,1,1] => 105 [2,2,1,1,2] => 407 [2,2,1,2,1] => 875 [2,2,1,3] => 573 [2,2,2,1,1] => 791 [2,2,2,2] => 1385 [2,2,3,1] => 917 [2,2,4] => 323 [2,3,1,1,1] => 245 [2,3,1,2] => 643 [2,3,2,1] => 875 [2,3,3] => 477 [2,4,1,1] => 259 [2,4,2] => 365 [2,5,1] => 133 [2,6] => 27 [3,1,1,1,1,1] => 21 [3,1,1,1,2] => 99 [3,1,1,2,1] => 259 [3,1,1,3] => 181 [3,1,2,1,1] => 315 [3,1,2,2] => 573 [3,1,3,1] => 413 [3,1,4] => 155 [3,2,1,1,1] => 189 [3,2,1,2] => 531 [3,2,2,1] => 791 [3,2,3] => 449 [3,3,1,1] => 315 [3,3,2] => 477 [3,4,1] => 217 [3,5] => 55 [4,1,1,1,1] => 35 [4,1,1,2] => 125 [4,1,2,1] => 245 [4,1,3] => 155 [4,2,1,1] => 189 [4,2,2] => 323 [4,3,1] => 203 [4,4] => 69 [5,1,1,1] => 35 [5,1,2] => 85 [5,2,1] => 105 [5,3] => 55 [6,1,1] => 21 [6,2] => 27 [7,1] => 7 [8] => 1 [1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,2] => 8 [1,1,1,1,1,1,2,1] => 35 [1,1,1,1,1,1,3] => 28 [1,1,1,1,1,2,1,1] => 83 [1,1,1,1,1,2,2] => 160 [1,1,1,1,1,3,1] => 133 [1,1,1,1,1,4] => 56 [1,1,1,1,2,1,1,1] => 125 [1,1,1,1,2,1,2] => 370 [1,1,1,1,2,2,1] => 595 [1,1,1,1,2,3] => 350 [1,1,1,1,3,1,1] => 295 [1,1,1,1,3,2] => 470 [1,1,1,1,4,1] => 245 [1,1,1,1,5] => 70 [1,1,1,2,1,1,1,1] => 125 [1,1,1,2,1,1,2] => 496 [1,1,1,2,1,2,1] => 1099 [1,1,1,2,1,3] => 728 [1,1,1,2,2,1,1] => 1051 [1,1,1,2,2,2] => 1856 [1,1,1,2,3,1] => 1253 [1,1,1,2,4] => 448 [1,1,1,3,1,1,1] => 379 [1,1,1,3,1,2] => 1016 [1,1,1,3,2,1] => 1421 [1,1,1,3,3] => 784 [1,1,1,4,1,1] => 461 [1,1,1,4,2] => 664 [1,1,1,5,1] => 259 [1,1,1,6] => 56 [1,1,2,1,1,1,1,1] => 83 [1,1,2,1,1,1,2] => 412 [1,1,2,1,1,2,1] => 1141 [1,1,2,1,1,3] => 812 [1,1,2,1,2,1,1] => 1513 [1,1,2,1,2,2] => 2780 [1,1,2,1,3,1] => 2051 [1,1,2,1,4] => 784 [1,1,2,2,1,1,1] => 1051 [1,1,2,2,1,2] => 2990 [1,1,2,2,2,1] => 4529 [1,1,2,2,3] => 2590 [1,1,2,3,1,1] => 1889 [1,1,2,3,2] => 2890 [1,1,2,4,1] => 1351 [1,1,2,5] => 350 [1,1,3,1,1,1,1] => 295 [1,1,3,1,1,2] => 1100 [1,1,3,1,2,1] => 2261 [1,1,3,1,3] => 1456 [1,1,3,2,1,1] => 1889 [1,1,3,2,2] => 3268 [1,1,3,3,1] => 2107 [1,1,3,4] => 728 [1,1,4,1,1,1] => 461 [1,1,4,1,2] => 1168 [1,1,4,2,1] => 1519 [1,1,4,3] => 812 [1,1,5,1,1] => 379 [1,1,5,2] => 512 [1,1,6,1] => 161 [1,1,7] => 28 [1,2,1,1,1,1,1,1] => 35 [1,2,1,1,1,1,2] => 208 [1,2,1,1,1,2,1] => 685 [1,2,1,1,1,3] => 512 [1,2,1,1,2,1,1] => 1141 [1,2,1,1,2,2] => 2144 [1,2,1,1,3,1] => 1667 [1,2,1,1,4] => 664 [1,2,1,2,1,1,1] => 1099 [1,2,1,2,1,2] => 3194 [1,2,1,2,2,1] => 4985 [1,2,1,2,3] => 2890 [1,2,1,3,1,1] => 2261 [1,2,1,3,2] => 3526 [1,2,1,4,1] => 1735 [1,2,1,5] => 470 [1,2,2,1,1,1,1] => 595 [1,2,2,1,1,2] => 2312 [1,2,2,1,2,1] => 4985 [1,2,2,1,3] => 3268 [1,2,2,2,1,1] => 4529 [1,2,2,2,2] => 7936 [1,2,2,3,1] => 5263 [1,2,2,4] => 1856 [1,2,3,1,1,1] => 1421 [1,2,3,1,2] => 3736 [1,2,3,2,1] => 5095 [1,2,3,3] => 2780 [1,2,4,1,1] => 1519 [1,2,4,2] => 2144 [1,2,5,1] => 785 [1,2,6] => 160 [1,3,1,1,1,1,1] => 133 [1,3,1,1,1,2] => 632 [1,3,1,1,2,1] => 1667 [1,3,1,1,3] => 1168 [1,3,1,2,1,1] => 2051 [1,3,1,2,2] => 3736 [1,3,1,3,1] => 2701 [1,3,1,4] => 1016 [1,3,2,1,1,1] => 1253 [1,3,2,1,2] => 3526 [1,3,2,2,1] => 5263 [1,3,2,3] => 2990 [1,3,3,1,1] => 2107 [1,3,3,2] => 3194 [1,3,4,1] => 1457 [1,3,5] => 370 [1,4,1,1,1,1] => 245 [1,4,1,1,2] => 880 [1,4,1,2,1] => 1735 [1,4,1,3] => 1100 [1,4,2,1,1] => 1351 [1,4,2,2] => 2312 [1,4,3,1] => 1457 [1,4,4] => 496 [1,5,1,1,1] => 259 [1,5,1,2] => 632 [1,5,2,1] => 785 [1,5,3] => 412 [1,6,1,1] => 161 [1,6,2] => 208 [1,7,1] => 55 [1,8] => 8 [2,1,1,1,1,1,1,1] => 8 [2,1,1,1,1,1,2] => 55 [2,1,1,1,1,2,1] => 208 [2,1,1,1,1,3] => 161 [2,1,1,1,2,1,1] => 412 [2,1,1,1,2,2] => 785 [2,1,1,1,3,1] => 632 [2,1,1,1,4] => 259 [2,1,1,2,1,1,1] => 496 [2,1,1,2,1,2] => 1457 [2,1,1,2,2,1] => 2312 [2,1,1,2,3] => 1351 [2,1,1,3,1,1] => 1100 [2,1,1,3,2] => 1735 [2,1,1,4,1] => 880 [2,1,1,5] => 245 [2,1,2,1,1,1,1] => 370 [2,1,2,1,1,2] => 1457 [2,1,2,1,2,1] => 3194 [2,1,2,1,3] => 2107 [2,1,2,2,1,1] => 2990 [2,1,2,2,2] => 5263 [2,1,2,3,1] => 3526 [2,1,2,4] => 1253 [2,1,3,1,1,1] => 1016 [2,1,3,1,2] => 2701 [2,1,3,2,1] => 3736 [2,1,3,3] => 2051 [2,1,4,1,1] => 1168 [2,1,4,2] => 1667 [2,1,5,1] => 632 [2,1,6] => 133 [2,2,1,1,1,1,1] => 160 [2,2,1,1,1,2] => 785 [2,2,1,1,2,1] => 2144 [2,2,1,1,3] => 1519 [2,2,1,2,1,1] => 2780 [2,2,1,2,2] => 5095 [2,2,1,3,1] => 3736 [2,2,1,4] => 1421 [2,2,2,1,1,1] => 1856 [2,2,2,1,2] => 5263 [2,2,2,2,1] => 7936 [2,2,2,3] => 4529 [2,2,3,1,1] => 3268 [2,2,3,2] => 4985 [2,2,4,1] => 2312 [2,2,5] => 595 [2,3,1,1,1,1] => 470 [2,3,1,1,2] => 1735 [2,3,1,2,1] => 3526 [2,3,1,3] => 2261 [2,3,2,1,1] => 2890 [2,3,2,2] => 4985 [2,3,3,1] => 3194 [2,3,4] => 1099 [2,4,1,1,1] => 664 [2,4,1,2] => 1667 [2,4,2,1] => 2144 [2,4,3] => 1141 [2,5,1,1] => 512 [2,5,2] => 685 [2,6,1] => 208 [2,7] => 35 [3,1,1,1,1,1,1] => 28 [3,1,1,1,1,2] => 161 [3,1,1,1,2,1] => 512 [3,1,1,1,3] => 379 [3,1,1,2,1,1] => 812 [3,1,1,2,2] => 1519 [3,1,1,3,1] => 1168 [3,1,1,4] => 461 [3,1,2,1,1,1] => 728 [3,1,2,1,2] => 2107 [3,1,2,2,1] => 3268 [3,1,2,3] => 1889 [3,1,3,1,1] => 1456 [3,1,3,2] => 2261 [3,1,4,1] => 1100 [3,1,5] => 295 [3,2,1,1,1,1] => 350 [3,2,1,1,2] => 1351 [3,2,1,2,1] => 2890 [3,2,1,3] => 1889 [3,2,2,1,1] => 2590 [3,2,2,2] => 4529 [3,2,3,1] => 2990 [3,2,4] => 1051 [3,3,1,1,1] => 784 [3,3,1,2] => 2051 [3,3,2,1] => 2780 [3,3,3] => 1513 [3,4,1,1] => 812 [3,4,2] => 1141 [3,5,1] => 412 [3,6] => 83 [4,1,1,1,1,1] => 56 [4,1,1,1,2] => 259 [4,1,1,2,1] => 664 [4,1,1,3] => 461 [4,1,2,1,1] => 784 [4,1,2,2] => 1421 [4,1,3,1] => 1016 [4,1,4] => 379 [4,2,1,1,1] => 448 [4,2,1,2] => 1253 [4,2,2,1] => 1856 [4,2,3] => 1051 [4,3,1,1] => 728 [4,3,2] => 1099 [4,4,1] => 496 [4,5] => 125 [5,1,1,1,1] => 70 [5,1,1,2] => 245 [5,1,2,1] => 470 [5,1,3] => 295 [5,2,1,1] => 350 [5,2,2] => 595 [5,3,1] => 370 [5,4] => 125 [6,1,1,1] => 56 [6,1,2] => 133 [6,2,1] => 160 [6,3] => 83 [7,1,1] => 28 [7,2] => 35 [8,1] => 8 [9] => 1 [1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,2,2] => 231 [1,1,1,1,2,1,1,2] => 999 [1,1,1,1,2,2,1,1] => 2149 [1,1,1,1,3,3] => 1674 [1,1,1,1,5,1] => 574 [1,1,2,1,1,1,1,2] => 711 [1,1,2,1,1,2,1,1] => 3961 [1,1,2,1,2,3] => 10206 [1,1,2,2,1,1,1,1] => 2149 [1,1,2,2,2,2] => 28839 [1,1,3,1,1,3] => 4536 [1,1,3,2,1,2] => 13941 [1,1,3,3,1,1] => 8371 [1,1,4,4] => 2064 [1,1,7,1] => 244 [1,2,1,1,4,1] => 5191 [1,2,6,1] => 1369 [1,3,1,1,3,1] => 8371 [1,3,5,1] => 3079 [1,4,1,1,2,1] => 5191 [1,4,4,1] => 3961 [1,5,1,1,1,1] => 574 [1,5,3,1] => 3079 [1,6,1,2] => 1134 [1,6,2,1] => 1369 [1,7,1,1] => 244 [1,8,1] => 71 [1,9] => 9 [2,1,1,1,1,1,1,2] => 71 [2,1,1,1,1,2,1,1] => 711 [2,1,1,1,2,3] => 2906 [2,1,1,2,1,1,1,1] => 999 [2,1,1,2,2,2] => 14759 [2,1,2,1,1,3] => 6056 [2,1,2,2,1,2] => 22121 [2,1,2,3,1,1] => 13941 [2,1,3,4] => 5264 [2,1,6,1] => 1134 [2,2,1,1,1,1,1,1] => 231 [2,2,1,1,2,2] => 13991 [2,2,2,1,1,2] => 14759 [2,2,2,2,1,1] => 28839 [2,2,3,3] => 17594 [3,1,1,1,1,3] => 701 [3,1,1,2,1,2] => 6056 [3,1,1,3,1,1] => 4536 [3,1,2,4] => 4949 [3,2,1,1,1,2] => 2906 [3,2,1,2,1,1] => 10206 [3,2,2,3] => 16451 [3,3,1,1,1,1] => 1674 [3,3,2,2] => 17594 [4,1,1,4] => 1301 [4,2,1,3] => 4949 [4,3,1,2] => 5264 [4,4,1,1] => 2064 [5,5] => 251 [8,1,1] => 36 [9,1] => 9 [1,10] => 10 [1,8,1,1] => 351 [1,7,2,1] => 2221 [1,6,3,1] => 5851 [1,5,4,1] => 9151 [1,4,5,1] => 9151 [1,3,6,1] => 5851 [1,2,7,1] => 2221 [1,1,8,1] => 351 [10,1] => 10 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,2,2] => 429 [1,1,1,1,1,1,2,2,1,1] => 6909 [1,1,1,1,1,1,2,1,1,2] => 3157 [1,1,1,1,1,1,3,3] => 5698 [1,1,1,1,2,2,1,1,1,1] => 15049 [1,1,1,1,2,2,2,2] => 203181 [1,1,1,1,2,1,1,2,1,1] => 26709 [1,1,1,1,2,1,1,1,1,2] => 4741 [1,1,1,1,2,1,2,3] => 69850 [1,1,1,1,3,3,1,1] => 65154 [1,1,1,1,3,2,1,2] => 107866 [1,1,1,1,3,1,1,3] => 34375 [1,1,1,1,4,4] => 16995 [1,1,2,2,1,1,1,1,1,1] => 6909 [1,1,2,2,1,1,2,2] => 421421 [1,1,2,2,2,2,1,1] => 880529 [1,1,2,2,2,1,1,2] => 450021 [1,1,2,2,3,3] => 538890 [1,1,2,1,1,2,1,1,1,1] => 26709 [1,1,2,1,1,2,2,2] => 396077 [1,1,2,1,1,1,1,2,1,1] => 18041 [1,1,2,1,1,1,1,1,1,2] => 1749 [1,1,2,1,1,1,2,3] => 75570 [1,1,2,1,2,3,1,1] => 392822 [1,1,2,1,2,2,1,2] => 622314 [1,1,2,1,2,1,1,3] => 169455 [1,1,2,1,3,4] => 150227 [1,1,3,3,1,1,1,1] => 65154 [1,1,3,3,2,2] => 686906 [1,1,3,2,1,2,1,1] => 392822 [1,1,3,2,1,1,1,2] => 111474 [1,1,3,2,2,3] => 635745 [1,1,3,1,1,3,1,1] => 166541 [1,1,3,1,1,2,1,2] => 221529 [1,1,3,1,1,1,1,3] => 25080 [1,1,3,1,2,4] => 185108 [1,1,4,4,1,1] => 90509 [1,1,4,3,1,2] => 230241 [1,1,4,2,1,3] => 215160 [1,1,4,1,1,4] => 55220 [1,1,5,5] => 11825 [2,2,1,1,1,1,1,1,1,1] => 429 [2,2,1,1,1,1,2,2] => 66989 [2,2,1,1,2,2,1,1] => 421421 [2,2,1,1,2,1,1,2] => 199341 [2,2,1,1,3,3] => 312102 [2,2,2,2,1,1,1,1] => 203181 [2,2,2,2,2,2] => 2702765 [2,2,2,1,1,2,1,1] => 396077 [2,2,2,1,1,1,1,2] => 72621 [2,2,2,1,2,3] => 996390 [2,2,3,3,1,1] => 686906 [2,2,3,2,1,2] => 1152546 [2,2,3,1,1,3] => 385395 [2,2,4,4] => 157475 [2,1,1,2,1,1,1,1,1,1] => 3157 [2,1,1,2,1,1,2,2] => 199341 [2,1,1,2,2,2,1,1] => 450021 [2,1,1,2,2,1,1,2] => 228205 [2,1,1,2,3,3] => 280774 [2,1,1,1,1,2,1,1,1,1] => 4741 [2,1,1,1,1,2,2,2] => 72621 [2,1,1,1,1,1,1,2,1,1] => 1749 [2,1,1,1,1,1,1,1,1,2] => 109 [2,1,1,1,1,1,2,3] => 9910 [2,1,1,1,2,3,1,1] => 111474 [2,1,1,1,2,2,1,2] => 174514 [2,1,1,1,2,1,1,3] => 45715 [2,1,1,1,3,4] => 46947 [2,1,2,3,1,1,1,1] => 107866 [2,1,2,3,2,2] => 1152546 [2,1,2,2,1,2,1,1] => 622314 [2,1,2,2,1,1,1,2] => 174514 [2,1,2,2,2,3] => 1022845 [2,1,2,1,1,3,1,1] => 221529 [2,1,2,1,1,2,1,2] => 291169 [2,1,2,1,1,1,1,3] => 30700 [2,1,2,1,2,4] => 262932 [2,1,3,4,1,1] => 230241 [2,1,3,3,1,2] => 578665 [2,1,3,2,1,3] => 527284 [2,1,3,1,1,4] => 123540 [2,1,4,5] => 39225 [3,3,1,1,1,1,1,1] => 5698 [3,3,1,1,2,2] => 312102 [3,3,2,2,1,1] => 538890 [3,3,2,1,1,2] => 280774 [3,3,3,3] => 315523 [3,2,1,2,1,1,1,1] => 69850 [3,2,1,2,2,2] => 996390 [3,2,1,1,1,2,1,1] => 75570 [3,2,1,1,1,1,1,2] => 9910 [3,2,1,1,2,3] => 251371 [3,2,2,3,1,1] => 635745 [3,2,2,2,1,2] => 1022845 [3,2,2,1,1,3] => 294106 [3,2,3,4] => 215346 [3,1,1,3,1,1,1,1] => 34375 [3,1,1,3,2,2] => 385395 [3,1,1,2,1,2,1,1] => 169455 [3,1,1,2,1,1,1,2] => 45715 [3,1,1,2,2,3] => 294106 [3,1,1,1,1,3,1,1] => 25080 [3,1,1,1,1,2,1,2] => 30700 [3,1,1,1,1,1,1,3] => 1891 [3,1,1,1,2,4] => 43251 [3,1,2,4,1,1] => 215160 [3,1,2,3,1,2] => 527284 [3,1,2,2,1,3] => 458083 [3,1,2,1,1,4] => 90771 [3,1,3,5] => 54966 [4,4,1,1,1,1] => 16995 [4,4,2,2] => 157475 [4,3,1,2,1,1] => 150227 [4,3,1,1,1,2] => 46947 [4,3,2,3] => 215346 [4,2,1,3,1,1] => 185108 [4,2,1,2,1,2] => 262932 [4,2,1,1,1,3] => 43251 [4,2,2,4] => 147443 [4,1,1,4,1,1] => 55220 [4,1,1,3,1,2] => 123540 [4,1,1,2,1,3] => 90771 [4,1,1,1,1,4] => 8051 [4,1,2,5] => 36926 [5,5,1,1] => 11825 [5,4,1,2] => 39225 [5,3,1,3] => 54966 [5,2,1,4] => 36926 [5,1,1,5] => 8051 [6,6] => 923 ----------------------------------------------------------------------------- Created: Sep 11, 2015 at 21:12 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 20, 2021 at 17:25 by Martin Rubey