***************************************************************************** * 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: St000185 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The weighted size of a partition. Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is $$\sum_{i=0}^m i \cdot \lambda_i.$$ This is also the sum of the leg lengths of the cells in $\lambda$, or $$ \sum_i \binom{\lambda^{\prime}_i}{2} $$ where $\lambda^{\prime}$ is the conjugate partition of $\lambda$. This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2]. This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103]. ----------------------------------------------------------------------------- References: [1] Hohlweg, C. Minimal and maximal elements in Kazhdan-Lusztig double sided cells of $S_n$ and Robinson-Schensted correspondance [[arXiv:math/0304059]] [2] Stanley, R. P. Enumerative combinatorics. Vol. 2 [[MathSciNet:1676282]] ----------------------------------------------------------------------------- Code: def statistic(L): return L.weighted_size() ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 1 [1,1,1] => 3 [4] => 0 [3,1] => 1 [2,2] => 2 [2,1,1] => 3 [1,1,1,1] => 6 [5] => 0 [4,1] => 1 [3,2] => 2 [3,1,1] => 3 [2,2,1] => 4 [2,1,1,1] => 6 [1,1,1,1,1] => 10 [6] => 0 [5,1] => 1 [4,2] => 2 [4,1,1] => 3 [3,3] => 3 [3,2,1] => 4 [3,1,1,1] => 6 [2,2,2] => 6 [2,2,1,1] => 7 [2,1,1,1,1] => 10 [1,1,1,1,1,1] => 15 [7] => 0 [6,1] => 1 [5,2] => 2 [5,1,1] => 3 [4,3] => 3 [4,2,1] => 4 [4,1,1,1] => 6 [3,3,1] => 5 [3,2,2] => 6 [3,2,1,1] => 7 [3,1,1,1,1] => 10 [2,2,2,1] => 9 [2,2,1,1,1] => 11 [2,1,1,1,1,1] => 15 [1,1,1,1,1,1,1] => 21 [8] => 0 [7,1] => 1 [6,2] => 2 [6,1,1] => 3 [5,3] => 3 [5,2,1] => 4 [5,1,1,1] => 6 [4,4] => 4 [4,3,1] => 5 [4,2,2] => 6 [4,2,1,1] => 7 [4,1,1,1,1] => 10 [3,3,2] => 7 [3,3,1,1] => 8 [3,2,2,1] => 9 [3,2,1,1,1] => 11 [3,1,1,1,1,1] => 15 [2,2,2,2] => 12 [2,2,2,1,1] => 13 [2,2,1,1,1,1] => 16 [2,1,1,1,1,1,1] => 21 [1,1,1,1,1,1,1,1] => 28 [9] => 0 [8,1] => 1 [7,2] => 2 [7,1,1] => 3 [6,3] => 3 [6,2,1] => 4 [6,1,1,1] => 6 [5,4] => 4 [5,3,1] => 5 [5,2,2] => 6 [5,2,1,1] => 7 [5,1,1,1,1] => 10 [4,4,1] => 6 [4,3,2] => 7 [4,3,1,1] => 8 [4,2,2,1] => 9 [4,2,1,1,1] => 11 [4,1,1,1,1,1] => 15 [3,3,3] => 9 [3,3,2,1] => 10 [3,3,1,1,1] => 12 [3,2,2,2] => 12 [3,2,2,1,1] => 13 [3,2,1,1,1,1] => 16 [3,1,1,1,1,1,1] => 21 [2,2,2,2,1] => 16 [2,2,2,1,1,1] => 18 [2,2,1,1,1,1,1] => 22 [2,1,1,1,1,1,1,1] => 28 [1,1,1,1,1,1,1,1,1] => 36 [10] => 0 [9,1] => 1 [8,2] => 2 [8,1,1] => 3 [7,3] => 3 [7,2,1] => 4 [7,1,1,1] => 6 [6,4] => 4 [6,3,1] => 5 [6,2,2] => 6 [6,2,1,1] => 7 [6,1,1,1,1] => 10 [5,5] => 5 [5,4,1] => 6 [5,3,2] => 7 [5,3,1,1] => 8 [5,2,2,1] => 9 [5,2,1,1,1] => 11 [5,1,1,1,1,1] => 15 [4,4,2] => 8 [4,4,1,1] => 9 [4,3,3] => 9 [4,3,2,1] => 10 [4,3,1,1,1] => 12 [4,2,2,2] => 12 [4,2,2,1,1] => 13 [4,2,1,1,1,1] => 16 [4,1,1,1,1,1,1] => 21 [3,3,3,1] => 12 [3,3,2,2] => 13 [3,3,2,1,1] => 14 [3,3,1,1,1,1] => 17 [3,2,2,2,1] => 16 [3,2,2,1,1,1] => 18 [3,2,1,1,1,1,1] => 22 [3,1,1,1,1,1,1,1] => 28 [2,2,2,2,2] => 20 [2,2,2,2,1,1] => 21 [2,2,2,1,1,1,1] => 24 [2,2,1,1,1,1,1,1] => 29 [2,1,1,1,1,1,1,1,1] => 36 [1,1,1,1,1,1,1,1,1,1] => 45 [8,3] => 3 [7,4] => 4 [6,5] => 5 [6,4,1] => 6 [5,5,1] => 7 [5,4,2] => 8 [5,4,1,1] => 9 [5,3,3] => 9 [5,3,2,1] => 10 [5,3,1,1,1] => 12 [5,2,2,2] => 12 [5,2,2,1,1] => 13 [4,4,3] => 10 [4,4,2,1] => 11 [4,4,1,1,1] => 13 [4,3,3,1] => 12 [4,3,2,2] => 13 [4,3,2,1,1] => 14 [4,2,2,2,1] => 16 [3,3,3,2] => 15 [3,3,3,1,1] => 16 [3,3,2,2,1] => 17 [3,2,2,2,2] => 20 [2,2,2,2,2,1] => 25 [7,5] => 5 [7,4,1] => 6 [6,6] => 6 [6,4,2] => 8 [5,5,2] => 9 [5,4,3] => 10 [5,4,2,1] => 11 [5,4,1,1,1] => 13 [5,3,3,1] => 12 [5,3,2,2] => 13 [5,3,2,1,1] => 14 [5,2,2,2,1] => 16 [4,4,4] => 12 [4,4,3,1] => 13 [4,4,2,2] => 14 [4,4,2,1,1] => 15 [4,3,3,2] => 15 [4,3,3,1,1] => 16 [4,3,2,2,1] => 17 [3,3,3,3] => 18 [3,3,3,2,1] => 19 [3,3,2,2,2] => 21 [3,3,2,2,1,1] => 22 [2,2,2,2,2,2] => 30 [8,5] => 5 [7,5,1] => 7 [7,4,2] => 8 [5,5,3] => 11 [5,4,4] => 12 [5,4,3,1] => 13 [5,4,2,2] => 14 [5,4,2,1,1] => 15 [5,3,3,2] => 15 [5,3,3,1,1] => 16 [5,3,2,2,1] => 17 [4,4,4,1] => 15 [4,4,3,2] => 16 [4,4,3,1,1] => 17 [4,4,2,2,1] => 18 [4,3,3,3] => 18 [4,3,3,2,1] => 19 [3,3,3,3,1] => 22 [3,3,3,2,2] => 23 [9,5] => 5 [8,5,1] => 7 [7,5,2] => 9 [7,4,3] => 10 [5,5,4] => 13 [5,4,3,2] => 16 [5,4,3,1,1] => 17 [5,4,2,2,1] => 18 [5,3,3,2,1] => 19 [5,3,2,2,2] => 21 [4,4,4,2] => 18 [4,4,3,3] => 19 [4,4,3,2,1] => 20 [3,3,3,3,2] => 26 [9,5,1] => 7 [8,5,2] => 9 [7,5,3] => 11 [5,5,5] => 15 [5,4,3,2,1] => 20 [5,3,2,2,2,1] => 26 [4,4,4,3] => 21 [3,3,3,3,3] => 30 [8,5,3] => 11 [7,5,3,1] => 14 [4,4,4,4] => 24 [8,6,3] => 12 [9,6,3] => 12 [8,6,4] => 14 [9,6,4] => 14 [8,5,4,2] => 19 [8,5,5,1] => 18 [7,5,4,3,1] => 26 [8,6,4,2] => 20 [10,6,4] => 14 [10,7,3] => 13 [9,7,4] => 15 [9,5,5,1] => 18 [6,5,4,3,2,1] => 35 [11,7,3] => 13 [9,6,4,3] => 23 [9,6,5,3] => 25 [8,6,5,3,1] => 29 [11,7,5,1] => 20 [9,7,5,3] => 26 [9,7,5,3,1] => 30 [10,7,5,3] => 26 [9,7,5,4,1] => 33 [7,6,5,4,3,2,1] => 56 [10,7,6,4,1] => 35 [9,7,6,4,2] => 39 [10,8,5,4,1] => 34 [10,8,6,4,1] => 36 [9,7,5,5,3,1] => 49 [11,8,6,4,1] => 36 [10,8,6,4,2] => 40 [11,8,6,5,1] => 39 [12,9,7,5,1] => 42 [13,9,7,5,1] => 42 [11,9,7,5,3,1] => 55 [11,8,7,5,4,1] => 58 [8,7,6,5,4,3,2,1] => 84 [11,9,7,5,5,3] => 73 [11,9,7,7,5,3,3] => 97 [11,9,7,6,5,3,1] => 82 [13,11,9,7,5,3,1] => 91 [13,11,9,7,7,5,3,1] => 128 [17,13,11,9,7,5,1] => 121 [15,13,11,9,7,5,3,1] => 140 [29,23,19,17,13,11,7,1] => 268 ----------------------------------------------------------------------------- Created: May 07, 2014 at 03:07 by Lahiru Kariyawasam ----------------------------------------------------------------------------- Last Updated: Oct 11, 2023 at 15:39 by Martin Rubey