***************************************************************************** * 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: St000705 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard tableaux on a given integer partition of n with maximal entry n. This is, for an integer partition $\lambda = ( \lambda_1 \geq \cdots \geq \lambda_k \geq 0) \vdash n$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $n$. Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $n$ variables, or, explicitly, $$\prod_{(i,j) \in \lambda} \frac{n+j-i}{\operatorname{hook}(i,j)}$$ where the product is over all cells $(i,j) \in \lambda$ and $\operatorname{hook}(i,j)$ is the hook length of a cell. See [Theorem 6.3, 1] for details. ----------------------------------------------------------------------------- References: [1] Fulton, W., Harris, J. Representation theory [[MathSciNet:1153249]] ----------------------------------------------------------------------------- Code: def statistic(L): if L: return SemistandardTableaux(shape=L, max_entry=sum(L)).cardinality() return 1 ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 3 [1,1] => 1 [3] => 10 [2,1] => 8 [1,1,1] => 1 [4] => 35 [3,1] => 45 [2,2] => 20 [2,1,1] => 15 [1,1,1,1] => 1 [5] => 126 [4,1] => 224 [3,2] => 175 [3,1,1] => 126 [2,2,1] => 75 [2,1,1,1] => 24 [1,1,1,1,1] => 1 [6] => 462 [5,1] => 1050 [4,2] => 1134 [4,1,1] => 840 [3,3] => 490 [3,2,1] => 896 [3,1,1,1] => 280 [2,2,2] => 175 [2,2,1,1] => 189 [2,1,1,1,1] => 35 [1,1,1,1,1,1] => 1 [7] => 1716 [6,1] => 4752 [5,2] => 6468 [5,1,1] => 4950 [4,3] => 4704 [4,2,1] => 7350 [4,1,1,1] => 2400 [3,3,1] => 3528 [3,2,2] => 2646 [3,2,1,1] => 2940 [3,1,1,1,1] => 540 [2,2,2,1] => 784 [2,2,1,1,1] => 392 [2,1,1,1,1,1] => 48 [1,1,1,1,1,1,1] => 1 [8] => 6435 [7,1] => 21021 [6,2] => 34320 [6,1,1] => 27027 [5,3] => 33264 [5,2,1] => 50688 [5,1,1,1] => 17325 [4,4] => 13860 [4,3,1] => 41580 [4,2,2] => 25872 [4,2,1,1] => 29700 [4,1,1,1,1] => 5775 [3,3,2] => 15876 [3,3,1,1] => 15120 [3,2,2,1] => 14700 [3,2,1,1,1] => 7680 [3,1,1,1,1,1] => 945 [2,2,2,2] => 1764 [2,2,2,1,1] => 2352 [2,2,1,1,1,1] => 720 [2,1,1,1,1,1,1] => 63 [1,1,1,1,1,1,1,1] => 1 [9] => 24310 [8,1] => 91520 [7,2] => 173745 [7,1,1] => 140140 [6,3] => 205920 [6,2,1] => 315315 [6,1,1,1] => 112112 [5,4] => 141570 [5,3,1] => 347490 [5,2,2] => 205920 [5,2,1,1] => 243243 [5,1,1,1,1] => 50050 [4,4,1] => 152460 [4,3,2] => 221760 [4,3,1,1] => 213840 [4,2,2,1] => 171072 [4,2,1,1,1] => 93555 [4,1,1,1,1,1] => 12320 [3,3,3] => 41580 [3,3,2,1] => 110880 [3,3,1,1,1] => 49500 [3,2,2,2] => 38808 [3,2,2,1,1] => 53460 [3,2,1,1,1,1] => 17325 [3,1,1,1,1,1,1] => 1540 [2,2,2,2,1] => 8820 [2,2,2,1,1,1] => 5760 [2,2,1,1,1,1,1] => 1215 [2,1,1,1,1,1,1,1] => 80 [1,1,1,1,1,1,1,1,1] => 1 [10] => 92378 [9,1] => 393822 [8,2] => 850850 [8,1,1] => 700128 [7,3] => 1179750 [7,2,1] => 1830400 [7,1,1,1] => 672672 [6,4] => 1061775 [6,3,1] => 2477475 [6,2,2] => 1447875 [6,2,1,1] => 1751750 [6,1,1,1,1] => 378378 [5,5] => 429429 [5,4,1] => 1812096 [5,3,2] => 2123550 [5,3,1,1] => 2081079 [5,2,2,1] => 1576575 [5,2,1,1,1] => 896896 [5,1,1,1,1,1] => 126126 [4,4,2] => 1019304 [4,4,1,1] => 943800 [4,3,3] => 707850 [4,3,2,1] => 1812096 [4,3,1,1,1] => 825825 [4,2,2,2] => 514800 [4,2,2,1,1] => 729729 [4,2,1,1,1,1] => 250250 [4,1,1,1,1,1,1] => 24024 [3,3,3,1] => 381150 [3,3,2,2] => 365904 [3,3,2,1,1] => 490050 [3,3,1,1,1,1] => 136125 [3,2,2,2,1] => 228096 [3,2,2,1,1,1] => 155925 [3,2,1,1,1,1,1] => 35200 [3,1,1,1,1,1,1,1] => 2376 [2,2,2,2,2] => 19404 [2,2,2,2,1,1] => 29700 [2,2,2,1,1,1,1] => 12375 [2,2,1,1,1,1,1,1] => 1925 [2,1,1,1,1,1,1,1,1] => 99 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 352716 [10,1] => 1679600 [9,2] => 4064632 [9,1,1] => 3401190 [8,3] => 6417840 [8,2,1] => 10108098 [8,1,1,1] => 3818880 [7,4] => 6952660 [7,3,1] => 16044600 [7,2,2] => 9359350 [7,2,1,1] => 11552112 [7,1,1,1,1] => 2598960 [6,5] => 4580576 [6,4,1] => 15459444 [6,3,2] => 16988400 [6,3,1,1] => 16912896 [6,2,2,1] => 12584000 [6,2,1,1,1] => 7399392 [6,1,1,1,1,1] => 1100736 [5,5,1] => 6441435 [5,4,2] => 13803075 [5,4,1,1] => 12882870 [5,3,3] => 7786350 [5,3,2,1] => 19819800 [5,3,1,1,1] => 9249240 [5,2,2,2] => 5308875 [5,2,2,1,1] => 7707700 [5,2,1,1,1,1] => 2774772 [5,1,1,1,1,1,1] => 286650 [4,4,3] => 4723719 [4,4,2,1] => 9815520 [4,4,1,1,1] => 4294290 [4,3,3,1] => 7474896 [4,3,2,2] => 6795360 [4,3,2,1,1] => 9249240 [4,3,1,1,1,1] => 2642640 [4,2,2,2,1] => 3468465 [4,2,2,1,1,1] => 2466464 [4,2,1,1,1,1,1] => 594594 [4,1,1,1,1,1,1,1] => 43680 [3,3,3,2] => 1868724 [3,3,3,1,1] => 2076360 [3,3,2,2,1] => 2548260 [3,3,2,1,1,1] => 1698840 [3,3,1,1,1,1,1] => 330330 [3,2,2,2,2] => 566280 [3,2,2,2,1,1] => 891891 [3,2,2,1,1,1,1] => 393250 [3,2,1,1,1,1,1,1] => 66066 [3,1,1,1,1,1,1,1,1] => 3510 [2,2,2,2,2,1] => 104544 [2,2,2,2,1,1,1] => 81675 [2,2,2,1,1,1,1,1] => 24200 [2,2,1,1,1,1,1,1,1] => 2904 [2,1,1,1,1,1,1,1,1,1] => 120 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1352078 [11,1] => 7113106 [10,2] => 19046664 [10,1,1] => 16166150 [9,3] => 33625592 [9,2,1] => 53747200 [9,1,1,1] => 20785050 [8,4] => 42031990 [8,3,1] => 97274034 [8,2,2] => 56904848 [8,2,1,1] => 71424990 [8,1,1,1,1] => 16628040 [7,5] => 35837802 [7,4,1] => 113265152 [7,3,2] => 121671550 [7,3,1,1] => 122872464 [7,2,2,1] => 90972882 [7,2,1,1,1] => 54991872 [7,1,1,1,1,1] => 8576568 [6,6] => 14158144 [6,5,1] => 77427350 [6,4,2] => 131405274 [6,4,1,1] => 123883760 [6,3,3] => 69526600 [6,3,2,1] => 177988096 [6,3,1,1,1] => 84948864 [6,2,2,2] => 46796750 [6,2,2,1,1] => 69312672 [6,2,1,1,1,1] => 25989600 [6,1,1,1,1,1,1] => 2858856 [5,5,2] => 57257200 [5,5,1,1] => 52702650 [5,4,3] => 73289216 [5,4,2,1] => 150300150 [5,4,1,1,1] => 66626560 [5,3,3,1] => 92756664 [5,3,2,2] => 82818450 [5,3,2,1,1] => 114514400 [5,3,1,1,1,1] => 33729696 [5,2,2,2,1] => 40268800 [5,2,2,1,1,1] => 29597568 [5,2,1,1,1,1,1] => 7547904 [5,1,1,1,1,1,1,1] => 600600 [4,4,4] => 13026013 [4,4,3,1] => 57972915 [4,4,2,2] => 42942900 [4,4,2,1,1] => 57972915 [4,4,1,1,1,1] => 15940925 [4,3,3,2] => 41409225 [4,3,3,1,1] => 46378332 [4,3,2,2,1] => 53678625 [4,3,2,1,1,1] => 36644608 [4,3,1,1,1,1,1] => 7378371 [4,2,2,2,2] => 9555975 [4,2,2,2,1,1] => 15415400 [4,2,2,1,1,1,1] => 7135128 [4,2,1,1,1,1,1,1] => 1289925 [4,1,1,1,1,1,1,1,1] => 75075 [3,3,3,3] => 4723719 [3,3,3,2,1] => 15704832 [3,3,3,1,1,1] => 8588580 [3,3,2,2,2] => 7361640 [3,3,2,2,1,1] => 11594583 [3,3,2,1,1,1,1] => 5010005 [3,3,1,1,1,1,1,1] => 728728 [3,2,2,2,2,1] => 3468465 [3,2,2,2,1,1,1] => 2818816 [3,2,2,1,1,1,1,1] => 891891 [3,2,1,1,1,1,1,1,1] => 116480 [3,1,1,1,1,1,1,1,1,1] => 5005 [2,2,2,2,2,2] => 226512 [2,2,2,2,2,1,1] => 382239 [2,2,2,2,1,1,1,1] => 196625 [2,2,2,1,1,1,1,1,1] => 44044 [2,2,1,1,1,1,1,1,1,1] => 4212 [2,1,1,1,1,1,1,1,1,1,1] => 143 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [13] => 5200300 [12,1] => 29953728 [10,3] => 171184832 [9,2,2] => 330142176 [8,5] => 244549760 [8,4,1] => 756575820 [8,3,2] => 807014208 [8,3,1,1] => 825355440 [7,6] => 155900472 [7,5,1] => 672511840 [7,4,2] => 1059206148 [7,3,3] => 546415870 [6,6,1] => 270774504 [6,5,2] => 764539776 [6,5,1,1] => 707907200 [6,4,3] => 776485710 [6,4,2,1] => 1592791200 [6,3,2,2] => 817632816 [6,3,1,1,1,1] => 347518080 [6,2,2,1,1,1] => 297872640 [6,1,1,1,1,1,1,1] => 6785856 [5,5,3] => 368111744 [5,4,4] => 241573332 [5,4,3,1] => 1006555550 [5,4,2,2] => 730029300 [5,4,2,1,1] => 995494500 [5,4,1,1,1,1] => 278738460 [5,3,3,2] => 569422854 [5,3,3,1,1] => 644195552 [5,3,2,2,1] => 730029300 [5,3,2,1,1,1] => 509693184 [5,3,1,1,1,1,1] => 106427412 [5,2,2,2,1,1] => 200236608 [5,2,2,1,1,1,1] => 96532800 [5,2,1,1,1,1,1,1] => 18582564 [4,4,4,1] => 189469280 [4,4,3,2] => 372171800 [4,4,3,1,1] => 411080670 [4,4,2,2,1] => 386486100 [4,3,3,3] => 119094976 [4,3,3,2,1] => 390780390 [3,3,3,3,1] => 50243193 [3,3,3,2,2] => 55825770 [3,3,2,2,2,1] => 51531480 [3,3,2,1,1,1,1,1] => 13117104 [3,2,2,2,2,2] => 8281845 [3,2,2,2,2,1,1] => 14314300 [3,1,1,1,1,1,1,1,1,1,1] => 6930 [2,2,2,2,2,2,1] => 1288287 [2,2,2,2,1,1,1,1,1] => 429429 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 20058300 [13,1] => 125550100 [12,2] => 400423100 [12,1,1] => 347677200 [9,5] => 1539017480 [8,6] => 1259196120 [7,7] => 488259720 [6,2,2,2,2] => 1294585292 [6,1,1,1,1,1,1,1,1] => 14965236 [5,5,4] => 1723110480 [5,5,1,1,1,1] => 1458016560 [5,4,1,1,1,1,1] => 998000640 [5,3,3,3] => 1811799990 [5,3,2,1,1,1,1] => 1900489500 [5,3,1,1,1,1,1,1] => 300179880 [5,2,2,2,2,1] => 919836918 [5,2,2,1,1,1,1,1] => 278738460 [4,4,4,2] => 1486605120 [4,4,4,1,1] => 1548547000 [4,4,3,3] => 1288391104 [4,3,2,2,2,1] => 1334910720 [3,3,3,3,2] => 260520260 [3,3,3,3,1,1] => 319729410 [3,3,3,2,2,1] => 450900450 [3,3,2,2,2,2] => 138030750 [3,3,1,1,1,1,1,1,1,1] => 2866500 [3,2,2,2,2,1,1,1] => 48096048 [2,2,2,2,2,2,2] => 2760615 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [15] => 77558760 [14,1] => 524190240 [5,3,1,1,1,1,1,1,1] => 773587584 [3,3,3,3,3] => 644195552 [3,3,3,2,2,2] => 1390532000 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [16] => 300540195 [2,2,2,2,2,2,2,2] => 34763300 [2,2,2,2,2,2,1,1,1,1] => 45048640 [2,2,2,2,1,1,1,1,1,1,1,1] => 2998800 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [17] => 1166803110 [3,2,2,2,2,2,2,2] => 1796567344 [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 20520 [2,2,2,2,2,2,2,2,2] => 449141836 ----------------------------------------------------------------------------- Created: Mar 07, 2017 at 09:21 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Dec 29, 2023 at 14:37 by Martin Rubey