***************************************************************************** * 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: St000704 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard tableaux on a given integer partition with minimal maximal entry. This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$. Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly, $$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$ where the product is over all cells $(i,j) \in L$ 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): return SemistandardTableaux(shape=L,max_entry=len(L)).cardinality() ----------------------------------------------------------------------------- Statistic values: [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] => 8 [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] => 15 [4,1,1,1] => 20 [3,3,1] => 6 [3,2,2] => 3 [3,2,1,1] => 20 [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] => 24 [5,1,1,1] => 35 [4,4] => 1 [4,3,1] => 15 [4,2,2] => 6 [4,2,1,1] => 45 [4,1,1,1,1] => 35 [3,3,2] => 3 [3,3,1,1] => 20 [3,2,2,1] => 15 [3,2,1,1,1] => 40 [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] => 35 [6,1,1,1] => 56 [5,4] => 2 [5,3,1] => 27 [5,2,2] => 10 [5,2,1,1] => 84 [5,1,1,1,1] => 70 [4,4,1] => 10 [4,3,2] => 8 [4,3,1,1] => 60 [4,2,2,1] => 36 [4,2,1,1,1] => 105 [4,1,1,1,1,1] => 56 [3,3,3] => 1 [3,3,2,1] => 20 [3,3,1,1,1] => 50 [3,2,2,2] => 4 [3,2,2,1,1] => 45 [3,2,1,1,1,1] => 70 [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] => 48 [7,1,1,1] => 84 [6,4] => 3 [6,3,1] => 42 [6,2,2] => 15 [6,2,1,1] => 140 [6,1,1,1,1] => 126 [5,5] => 1 [5,4,1] => 24 [5,3,2] => 15 [5,3,1,1] => 126 [5,2,2,1] => 70 [5,2,1,1,1] => 224 [5,1,1,1,1,1] => 126 [4,4,2] => 6 [4,4,1,1] => 50 [4,3,3] => 3 [4,3,2,1] => 64 [4,3,1,1,1] => 175 [4,2,2,2] => 10 [4,2,2,1,1] => 126 [4,2,1,1,1,1] => 210 [4,1,1,1,1,1,1] => 84 [3,3,3,1] => 10 [3,3,2,2] => 6 [3,3,2,1,1] => 75 [3,3,1,1,1,1] => 105 [3,2,2,2,1] => 24 [3,2,2,1,1,1] => 105 [3,2,1,1,1,1,1] => 112 [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] => 63 [8,1,1,1] => 120 [7,4] => 4 [7,3,1] => 60 [7,2,2] => 21 [7,2,1,1] => 216 [7,1,1,1,1] => 210 [6,5] => 2 [6,4,1] => 42 [6,3,2] => 24 [6,3,1,1] => 224 [6,2,2,1] => 120 [6,2,1,1,1] => 420 [6,1,1,1,1,1] => 252 [5,5,1] => 15 [5,4,2] => 15 [5,4,1,1] => 140 [5,3,3] => 6 [5,3,2,1] => 140 [5,3,1,1,1] => 420 [5,2,2,2] => 20 [5,2,2,1,1] => 280 [5,2,1,1,1,1] => 504 [5,1,1,1,1,1,1] => 210 [4,4,3] => 3 [4,4,2,1] => 60 [4,4,1,1,1] => 175 [4,3,3,1] => 36 [4,3,2,2] => 20 [4,3,2,1,1] => 280 [4,3,1,1,1,1] => 420 [4,2,2,2,1] => 70 [4,2,2,1,1,1] => 336 [4,2,1,1,1,1,1] => 378 [4,1,1,1,1,1,1,1] => 120 [3,3,3,2] => 4 [3,3,3,1,1] => 50 [3,3,2,2,1] => 45 [3,3,2,1,1,1] => 210 [3,3,1,1,1,1,1] => 196 [3,2,2,2,2] => 5 [3,2,2,2,1,1] => 84 [3,2,2,1,1,1,1] => 210 [3,2,1,1,1,1,1,1] => 168 [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] => 80 [9,1,1,1] => 165 [8,4] => 5 [8,3,1] => 81 [8,2,2] => 28 [8,2,1,1] => 315 [8,1,1,1,1] => 330 [7,5] => 3 [7,4,1] => 64 [7,3,2] => 35 [7,3,1,1] => 360 [7,2,2,1] => 189 [7,2,1,1,1] => 720 [7,1,1,1,1,1] => 462 [6,6] => 1 [6,5,1] => 35 [6,4,2] => 27 [6,4,1,1] => 280 [6,3,3] => 10 [6,3,2,1] => 256 [6,3,1,1,1] => 840 [6,2,2,2] => 35 [6,2,2,1,1] => 540 [6,2,1,1,1,1] => 1050 [6,1,1,1,1,1,1] => 462 [5,5,2] => 10 [5,5,1,1] => 105 [5,4,3] => 8 [5,4,2,1] => 175 [5,4,1,1,1] => 560 [5,3,3,1] => 84 [5,3,2,2] => 45 [5,3,2,1,1] => 700 [5,3,1,1,1,1] => 1134 [5,2,2,2,1] => 160 [5,2,2,1,1,1] => 840 [5,2,1,1,1,1,1] => 1008 [5,1,1,1,1,1,1,1] => 330 [4,4,4] => 1 [4,4,3,1] => 45 [4,4,2,2] => 20 [4,4,2,1,1] => 315 [4,4,1,1,1,1] => 490 [4,3,3,2] => 15 [4,3,3,1,1] => 210 [4,3,2,2,1] => 175 [4,3,2,1,1,1] => 896 [4,3,1,1,1,1,1] => 882 [4,2,2,2,2] => 15 [4,2,2,2,1,1] => 280 [4,2,2,1,1,1,1] => 756 [4,2,1,1,1,1,1,1] => 630 [4,1,1,1,1,1,1,1,1] => 165 [3,3,3,3] => 1 [3,3,3,2,1] => 40 [3,3,3,1,1,1] => 175 [3,3,2,2,2] => 10 [3,3,2,2,1,1] => 189 [3,3,2,1,1,1,1] => 490 [3,3,1,1,1,1,1,1] => 336 [3,2,2,2,2,1] => 35 [3,2,2,2,1,1,1] => 224 [3,2,2,1,1,1,1,1] => 378 [3,2,1,1,1,1,1,1,1] => 240 [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 ----------------------------------------------------------------------------- Created: Mar 07, 2017 at 09:15 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Mar 07, 2017 at 09:15 by Christian Stump