***************************************************************************** * 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: St000567 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the products of all pairs of parts. This is the evaluation of the second elementary symmetric polynomial which is equal to $$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$ for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1]. ----------------------------------------------------------------------------- References: [1] Kopitzke, G. The Gini Index of an Integer Partition [[arXiv:2005.04284]] ----------------------------------------------------------------------------- Code: def statistic(L): return sum(L[a]*L[b] for a,b in Subsets(range(len(L)), 2)) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 2 [1,1,1] => 3 [4] => 0 [3,1] => 3 [2,2] => 4 [2,1,1] => 5 [1,1,1,1] => 6 [5] => 0 [4,1] => 4 [3,2] => 6 [3,1,1] => 7 [2,2,1] => 8 [2,1,1,1] => 9 [1,1,1,1,1] => 10 [6] => 0 [5,1] => 5 [4,2] => 8 [4,1,1] => 9 [3,3] => 9 [3,2,1] => 11 [3,1,1,1] => 12 [2,2,2] => 12 [2,2,1,1] => 13 [2,1,1,1,1] => 14 [1,1,1,1,1,1] => 15 [7] => 0 [6,1] => 6 [5,2] => 10 [5,1,1] => 11 [4,3] => 12 [4,2,1] => 14 [4,1,1,1] => 15 [3,3,1] => 15 [3,2,2] => 16 [3,2,1,1] => 17 [3,1,1,1,1] => 18 [2,2,2,1] => 18 [2,2,1,1,1] => 19 [2,1,1,1,1,1] => 20 [1,1,1,1,1,1,1] => 21 [8] => 0 [7,1] => 7 [6,2] => 12 [6,1,1] => 13 [5,3] => 15 [5,2,1] => 17 [5,1,1,1] => 18 [4,4] => 16 [4,3,1] => 19 [4,2,2] => 20 [4,2,1,1] => 21 [4,1,1,1,1] => 22 [3,3,2] => 21 [3,3,1,1] => 22 [3,2,2,1] => 23 [3,2,1,1,1] => 24 [3,1,1,1,1,1] => 25 [2,2,2,2] => 24 [2,2,2,1,1] => 25 [2,2,1,1,1,1] => 26 [2,1,1,1,1,1,1] => 27 [1,1,1,1,1,1,1,1] => 28 [9] => 0 [8,1] => 8 [7,2] => 14 [7,1,1] => 15 [6,3] => 18 [6,2,1] => 20 [6,1,1,1] => 21 [5,4] => 20 [5,3,1] => 23 [5,2,2] => 24 [5,2,1,1] => 25 [5,1,1,1,1] => 26 [4,4,1] => 24 [4,3,2] => 26 [4,3,1,1] => 27 [4,2,2,1] => 28 [4,2,1,1,1] => 29 [4,1,1,1,1,1] => 30 [3,3,3] => 27 [3,3,2,1] => 29 [3,3,1,1,1] => 30 [3,2,2,2] => 30 [3,2,2,1,1] => 31 [3,2,1,1,1,1] => 32 [3,1,1,1,1,1,1] => 33 [2,2,2,2,1] => 32 [2,2,2,1,1,1] => 33 [2,2,1,1,1,1,1] => 34 [2,1,1,1,1,1,1,1] => 35 [1,1,1,1,1,1,1,1,1] => 36 [10] => 0 [9,1] => 9 [8,2] => 16 [8,1,1] => 17 [7,3] => 21 [7,2,1] => 23 [7,1,1,1] => 24 [6,4] => 24 [6,3,1] => 27 [6,2,2] => 28 [6,2,1,1] => 29 [6,1,1,1,1] => 30 [5,5] => 25 [5,4,1] => 29 [5,3,2] => 31 [5,3,1,1] => 32 [5,2,2,1] => 33 [5,2,1,1,1] => 34 [5,1,1,1,1,1] => 35 [4,4,2] => 32 [4,4,1,1] => 33 [4,3,3] => 33 [4,3,2,1] => 35 [4,3,1,1,1] => 36 [4,2,2,2] => 36 [4,2,2,1,1] => 37 [4,2,1,1,1,1] => 38 [4,1,1,1,1,1,1] => 39 [3,3,3,1] => 36 [3,3,2,2] => 37 [3,3,2,1,1] => 38 [3,3,1,1,1,1] => 39 [3,2,2,2,1] => 39 [3,2,2,1,1,1] => 40 [3,2,1,1,1,1,1] => 41 [3,1,1,1,1,1,1,1] => 42 [2,2,2,2,2] => 40 [2,2,2,2,1,1] => 41 [2,2,2,1,1,1,1] => 42 [2,2,1,1,1,1,1,1] => 43 [2,1,1,1,1,1,1,1,1] => 44 [1,1,1,1,1,1,1,1,1,1] => 45 [11] => 0 [10,1] => 10 [9,2] => 18 [9,1,1] => 19 [8,3] => 24 [8,2,1] => 26 [8,1,1,1] => 27 [7,4] => 28 [7,3,1] => 31 [7,2,2] => 32 [7,2,1,1] => 33 [7,1,1,1,1] => 34 [6,5] => 30 [6,4,1] => 34 [6,3,2] => 36 [6,3,1,1] => 37 [6,2,2,1] => 38 [6,2,1,1,1] => 39 [6,1,1,1,1,1] => 40 [5,5,1] => 35 [5,4,2] => 38 [5,4,1,1] => 39 [5,3,3] => 39 [5,3,2,1] => 41 [5,3,1,1,1] => 42 [5,2,2,2] => 42 [5,2,2,1,1] => 43 [5,2,1,1,1,1] => 44 [5,1,1,1,1,1,1] => 45 [4,4,3] => 40 [4,4,2,1] => 42 [4,4,1,1,1] => 43 [4,3,3,1] => 43 [4,3,2,2] => 44 [4,3,2,1,1] => 45 [4,3,1,1,1,1] => 46 [4,2,2,2,1] => 46 [4,2,2,1,1,1] => 47 [4,2,1,1,1,1,1] => 48 [4,1,1,1,1,1,1,1] => 49 [3,3,3,2] => 45 [3,3,3,1,1] => 46 [3,3,2,2,1] => 47 [3,3,2,1,1,1] => 48 [3,3,1,1,1,1,1] => 49 [3,2,2,2,2] => 48 [3,2,2,2,1,1] => 49 [3,2,2,1,1,1,1] => 50 [3,2,1,1,1,1,1,1] => 51 [3,1,1,1,1,1,1,1,1] => 52 [2,2,2,2,2,1] => 50 [2,2,2,2,1,1,1] => 51 [2,2,2,1,1,1,1,1] => 52 [2,2,1,1,1,1,1,1,1] => 53 [2,1,1,1,1,1,1,1,1,1] => 54 [1,1,1,1,1,1,1,1,1,1,1] => 55 [12] => 0 [11,1] => 11 [10,2] => 20 [10,1,1] => 21 [9,3] => 27 [9,2,1] => 29 [9,1,1,1] => 30 [8,4] => 32 [8,3,1] => 35 [8,2,2] => 36 [8,2,1,1] => 37 [8,1,1,1,1] => 38 [7,5] => 35 [7,4,1] => 39 [7,3,2] => 41 [7,3,1,1] => 42 [7,2,2,1] => 43 [7,2,1,1,1] => 44 [7,1,1,1,1,1] => 45 [6,6] => 36 [6,5,1] => 41 [6,4,2] => 44 [6,4,1,1] => 45 [6,3,3] => 45 [6,3,2,1] => 47 [6,3,1,1,1] => 48 [6,2,2,2] => 48 [6,2,2,1,1] => 49 [6,2,1,1,1,1] => 50 [6,1,1,1,1,1,1] => 51 [5,5,2] => 45 [5,5,1,1] => 46 [5,4,3] => 47 [5,4,2,1] => 49 [5,4,1,1,1] => 50 [5,3,3,1] => 50 [5,3,2,2] => 51 [5,3,2,1,1] => 52 [5,3,1,1,1,1] => 53 [5,2,2,2,1] => 53 [5,2,2,1,1,1] => 54 [5,2,1,1,1,1,1] => 55 [5,1,1,1,1,1,1,1] => 56 [4,4,4] => 48 [4,4,3,1] => 51 [4,4,2,2] => 52 [4,4,2,1,1] => 53 [4,4,1,1,1,1] => 54 [4,3,3,2] => 53 [4,3,3,1,1] => 54 [4,3,2,2,1] => 55 [4,3,2,1,1,1] => 56 [4,3,1,1,1,1,1] => 57 [4,2,2,2,2] => 56 [4,2,2,2,1,1] => 57 [4,2,2,1,1,1,1] => 58 [4,2,1,1,1,1,1,1] => 59 [4,1,1,1,1,1,1,1,1] => 60 [3,3,3,3] => 54 [3,3,3,2,1] => 56 [3,3,3,1,1,1] => 57 [3,3,2,2,2] => 57 [3,3,2,2,1,1] => 58 [3,3,2,1,1,1,1] => 59 [3,3,1,1,1,1,1,1] => 60 [3,2,2,2,2,1] => 59 [3,2,2,2,1,1,1] => 60 [3,2,2,1,1,1,1,1] => 61 [3,2,1,1,1,1,1,1,1] => 62 [3,1,1,1,1,1,1,1,1,1] => 63 [2,2,2,2,2,2] => 60 [2,2,2,2,2,1,1] => 61 [2,2,2,2,1,1,1,1] => 62 [2,2,2,1,1,1,1,1,1] => 63 [2,2,1,1,1,1,1,1,1,1] => 64 [2,1,1,1,1,1,1,1,1,1,1] => 65 [1,1,1,1,1,1,1,1,1,1,1,1] => 66 ----------------------------------------------------------------------------- Created: Aug 07, 2016 at 13:18 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 12, 2020 at 14:23 by Christian Stump