***************************************************************************** * 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: St001908 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. For example, there are eight tableaux of shape $[3,2,1]$ with maximal entry $3$, but two of them have the same weight. ----------------------------------------------------------------------------- References: [1] (bad identifier) [[MathOverflow:450294]] ----------------------------------------------------------------------------- Code: def statistic(la): s = SymmetricFunctions(ZZ).s() return len(list(s(la).expand(len(la)))) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 2 [1,1,1] => 1 [4] => 1 [3,1] => 3 [2,2] => 1 [2,1,1] => 3 [1,1,1,1] => 1 [5] => 1 [4,1] => 4 [3,2] => 2 [3,1,1] => 6 [2,2,1] => 3 [2,1,1,1] => 4 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 5 [4,2] => 3 [4,1,1] => 10 [3,3] => 1 [3,2,1] => 7 [3,1,1,1] => 10 [2,2,2] => 1 [2,2,1,1] => 6 [2,1,1,1,1] => 5 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 6 [5,2] => 4 [5,1,1] => 15 [4,3] => 2 [4,2,1] => 12 [4,1,1,1] => 20 [3,3,1] => 6 [3,2,2] => 3 [3,2,1,1] => 16 [3,1,1,1,1] => 15 [2,2,2,1] => 4 [2,2,1,1,1] => 10 [2,1,1,1,1,1] => 6 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 7 [6,2] => 5 [6,1,1] => 21 [5,3] => 3 [5,2,1] => 18 [5,1,1,1] => 35 [4,4] => 1 [4,3,1] => 12 [4,2,2] => 6 [4,2,1,1] => 31 [4,1,1,1,1] => 35 [3,3,2] => 3 [3,3,1,1] => 19 [3,2,2,1] => 13 [3,2,1,1,1] => 30 [3,1,1,1,1,1] => 21 [2,2,2,2] => 1 [2,2,2,1,1] => 10 [2,2,1,1,1,1] => 15 [2,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 8 [7,2] => 6 [7,1,1] => 28 [6,3] => 4 [6,2,1] => 25 [6,1,1,1] => 56 [5,4] => 2 [5,3,1] => 19 [5,2,2] => 10 [5,2,1,1] => 52 [5,1,1,1,1] => 70 [4,4,1] => 10 [4,3,2] => 7 [4,3,1,1] => 40 [4,2,2,1] => 28 [4,2,1,1,1] => 65 [4,1,1,1,1,1] => 56 [3,3,3] => 1 [3,3,2,1] => 16 [3,3,1,1,1] => 45 [3,2,2,2] => 4 [3,2,2,1,1] => 35 [3,2,1,1,1,1] => 50 [3,1,1,1,1,1,1] => 28 [2,2,2,2,1] => 5 [2,2,2,1,1,1] => 20 [2,2,1,1,1,1,1] => 21 [2,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 9 [8,2] => 7 [8,1,1] => 36 [7,3] => 5 [7,2,1] => 33 [7,1,1,1] => 84 [6,4] => 3 [6,3,1] => 27 [6,2,2] => 15 [6,2,1,1] => 80 [6,1,1,1,1] => 126 [5,5] => 1 [5,4,1] => 18 [5,3,2] => 12 [5,3,1,1] => 68 [5,2,2,1] => 50 [5,2,1,1,1] => 121 [5,1,1,1,1,1] => 126 [4,4,2] => 6 [4,4,1,1] => 44 [4,3,3] => 3 [4,3,2,1] => 38 [4,3,1,1,1] => 101 [4,2,2,2] => 10 [4,2,2,1,1] => 81 [4,2,1,1,1,1] => 120 [4,1,1,1,1,1,1] => 84 [3,3,3,1] => 10 [3,3,2,2] => 6 [3,3,2,1,1] => 51 [3,3,1,1,1,1] => 90 [3,2,2,2,1] => 21 [3,2,2,1,1,1] => 75 [3,2,1,1,1,1,1] => 77 [3,1,1,1,1,1,1,1] => 36 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 15 [2,2,2,1,1,1,1] => 35 [2,2,1,1,1,1,1,1] => 28 [2,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 10 [9,2] => 8 [9,1,1] => 45 [8,3] => 6 [8,2,1] => 42 [8,1,1,1] => 120 [7,4] => 4 [7,3,1] => 36 [7,2,2] => 21 [7,2,1,1] => 116 [7,1,1,1,1] => 210 [6,5] => 2 [6,4,1] => 27 [6,3,2] => 18 [6,3,1,1] => 104 [6,2,2,1] => 80 [6,2,1,1,1] => 205 [6,1,1,1,1,1] => 252 [5,5,1] => 15 [5,4,2] => 12 [5,4,1,1] => 80 [5,3,3] => 6 [5,3,2,1] => 68 [5,3,1,1,1] => 185 [5,2,2,2] => 20 [5,2,2,1,1] => 155 [5,2,1,1,1,1] => 246 [5,1,1,1,1,1,1] => 210 [4,4,3] => 3 [4,4,2,1] => 40 [4,4,1,1,1] => 135 [4,3,3,1] => 28 [4,3,2,2] => 16 [4,3,2,1,1] => 125 [4,3,1,1,1,1] => 216 [4,2,2,2,1] => 55 [4,2,2,1,1,1] => 186 [4,2,1,1,1,1,1] => 203 [4,1,1,1,1,1,1,1] => 120 [3,3,3,2] => 4 [3,3,3,1,1] => 45 [3,3,2,2,1] => 35 [3,3,2,1,1,1] => 126 [3,3,1,1,1,1,1] => 161 [3,2,2,2,2] => 5 [3,2,2,2,1,1] => 66 [3,2,2,1,1,1,1] => 140 [3,2,1,1,1,1,1,1] => 112 [3,1,1,1,1,1,1,1,1] => 45 [2,2,2,2,2,1] => 6 [2,2,2,2,1,1,1] => 35 [2,2,2,1,1,1,1,1] => 56 [2,2,1,1,1,1,1,1,1] => 36 [2,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 11 [10,2] => 9 [10,1,1] => 55 [9,3] => 7 [9,2,1] => 52 [9,1,1,1] => 165 [8,4] => 5 [8,3,1] => 46 [8,2,2] => 28 [8,2,1,1] => 161 [8,1,1,1,1] => 330 [7,5] => 3 [7,4,1] => 37 [7,3,2] => 25 [7,3,1,1] => 149 [7,2,2,1] => 119 [7,2,1,1,1] => 325 [7,1,1,1,1,1] => 462 [6,6] => 1 [6,5,1] => 25 [6,4,2] => 19 [6,4,1,1] => 125 [6,3,3] => 10 [6,3,2,1] => 107 [6,3,1,1,1] => 305 [6,2,2,2] => 35 [6,2,2,1,1] => 265 [6,2,1,1,1,1] => 456 [6,1,1,1,1,1,1] => 462 [5,5,2] => 10 [5,5,1,1] => 85 [5,4,3] => 7 [5,4,2,1] => 79 [5,4,1,1,1] => 255 [5,3,3,1] => 55 [5,3,2,2] => 31 [5,3,2,1,1] => 235 [5,3,1,1,1,1] => 426 [5,2,2,2,1] => 115 [5,2,2,1,1,1] => 381 [5,2,1,1,1,1,1] => 455 [5,1,1,1,1,1,1,1] => 330 [4,4,4] => 1 [4,4,3,1] => 31 [4,4,2,2] => 19 [4,4,2,1,1] => 155 [4,4,1,1,1,1] => 336 [4,3,3,2] => 13 [4,3,3,1,1] => 125 [4,3,2,2,1] => 95 [4,3,2,1,1,1] => 321 [4,3,1,1,1,1,1] => 413 [4,2,2,2,2] => 15 [4,2,2,2,1,1] => 181 [4,2,2,1,1,1,1] => 371 [4,2,1,1,1,1,1,1] => 322 [4,1,1,1,1,1,1,1,1] => 165 [3,3,3,3] => 1 [3,3,3,2,1] => 30 [3,3,3,1,1,1] => 141 [3,3,2,2,2] => 10 [3,3,2,2,1,1] => 121 [3,3,2,1,1,1,1] => 266 [3,3,1,1,1,1,1,1] => 266 [3,2,2,2,2,1] => 31 [3,2,2,2,1,1,1] => 161 [3,2,2,1,1,1,1,1] => 238 [3,2,1,1,1,1,1,1,1] => 156 [3,1,1,1,1,1,1,1,1,1] => 55 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 21 [2,2,2,2,1,1,1,1] => 70 [2,2,2,1,1,1,1,1,1] => 84 [2,2,1,1,1,1,1,1,1,1] => 45 [2,1,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [13] => 1 [12,1] => 12 [11,2] => 10 [11,1,1] => 66 [10,3] => 8 [10,2,1] => 63 [10,1,1,1] => 220 [9,4] => 6 [9,3,1] => 57 [9,2,2] => 36 [9,2,1,1] => 216 [9,1,1,1,1] => 495 [8,5] => 4 [8,4,1] => 48 [8,3,2] => 33 [8,3,1,1] => 204 [8,2,2,1] => 168 [8,2,1,1,1] => 490 [8,1,1,1,1,1] => 792 [7,6] => 2 [7,5,1] => 36 [7,4,2] => 27 [7,4,1,1] => 180 [7,3,3] => 15 [7,3,2,1] => 156 [7,3,1,1,1] => 470 [7,2,2,2] => 56 [7,2,2,1,1] => 420 [7,2,1,1,1,1] => 786 [7,1,1,1,1,1,1] => 924 [6,6,1] => 21 [6,5,2] => 18 [6,5,1,1] => 140 [6,4,3] => 12 [6,4,2,1] => 128 [6,4,1,1,1] => 420 [6,3,3,1] => 92 [6,3,2,2] => 52 [6,3,2,1,1] => 390 [6,3,1,1,1,1] => 756 [6,2,2,2,1] => 210 [6,2,2,1,1,1] => 696 [6,2,1,1,1,1,1] => 917 [6,1,1,1,1,1,1,1] => 792 [5,5,3] => 6 [5,5,2,1] => 80 [5,5,1,1,1] => 320 [5,4,4] => 3 [5,4,3,1] => 68 [5,4,2,2] => 40 [5,4,2,1,1] => 310 [5,4,1,1,1,1] => 666 [5,3,3,2] => 28 [5,3,3,1,1] => 250 [5,3,2,2,1] => 190 [5,3,2,1,1,1] => 636 [5,3,1,1,1,1,1] => 875 [5,2,2,2,2] => 35 [5,2,2,2,1,1] => 396 [5,2,2,1,1,1,1] => 812 [5,2,1,1,1,1,1,1] => 784 [5,1,1,1,1,1,1,1,1] => 495 [4,4,4,1] => 20 [4,4,3,2] => 16 [4,4,3,1,1] => 155 [4,4,2,2,1] => 125 [4,4,2,1,1,1] => 456 [4,4,1,1,1,1,1] => 728 [4,3,3,3] => 4 [4,3,3,2,1] => 95 [4,3,3,1,1,1] => 396 [4,3,2,2,2] => 30 [4,3,2,2,1,1] => 336 [4,3,2,1,1,1,1] => 707 [4,3,1,1,1,1,1,1] => 728 [4,2,2,2,2,1] => 96 [4,2,2,2,1,1,1] => 462 [4,2,2,1,1,1,1,1] => 672 [4,2,1,1,1,1,1,1,1] => 486 [4,1,1,1,1,1,1,1,1,1] => 220 [3,3,3,3,1] => 15 [3,3,3,2,2] => 10 [3,3,3,2,1,1] => 126 [3,3,3,1,1,1,1] => 357 [3,3,2,2,2,1] => 66 [3,3,2,2,1,1,1] => 322 [3,3,2,1,1,1,1,1] => 504 [3,3,1,1,1,1,1,1,1] => 414 [3,2,2,2,2,2] => 6 [3,2,2,2,2,1,1] => 112 [3,2,2,2,1,1,1,1] => 336 [3,2,2,1,1,1,1,1,1] => 378 [3,2,1,1,1,1,1,1,1,1] => 210 [3,1,1,1,1,1,1,1,1,1,1] => 66 [2,2,2,2,2,2,1] => 7 [2,2,2,2,2,1,1,1] => 56 [2,2,2,2,1,1,1,1,1] => 126 [2,2,2,1,1,1,1,1,1,1] => 120 [2,2,1,1,1,1,1,1,1,1,1] => 55 [2,1,1,1,1,1,1,1,1,1,1,1] => 12 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 1 [13,1] => 13 [12,2] => 11 [12,1,1] => 78 [11,3] => 9 [11,2,1] => 75 [11,1,1,1] => 286 [10,4] => 7 [10,3,1] => 69 [10,2,2] => 45 [10,2,1,1] => 282 [10,1,1,1,1] => 715 [9,5] => 5 [9,4,1] => 60 [9,3,2] => 42 [9,3,1,1] => 270 [9,2,2,1] => 228 [9,2,1,1,1] => 710 [9,1,1,1,1,1] => 1287 [8,6] => 3 [8,5,1] => 48 [8,4,2] => 36 [8,4,1,1] => 246 [8,3,3] => 21 [8,3,2,1] => 216 [8,3,1,1,1] => 690 [8,2,2,2] => 84 [8,2,2,1,1] => 630 ----------------------------------------------------------------------------- Created: Jul 06, 2023 at 19:05 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 06, 2023 at 19:05 by Martin Rubey