***************************************************************************** * 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: St000870 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(L): return prod( L.hook_length(*c) for c in L.cells() if c[0] == c[1] ) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 2 [1,1] => 2 [3] => 3 [2,1] => 3 [1,1,1] => 3 [4] => 4 [3,1] => 4 [2,2] => 3 [2,1,1] => 4 [1,1,1,1] => 4 [5] => 5 [4,1] => 5 [3,2] => 4 [3,1,1] => 5 [2,2,1] => 4 [2,1,1,1] => 5 [1,1,1,1,1] => 5 [6] => 6 [5,1] => 6 [4,2] => 5 [4,1,1] => 6 [3,3] => 8 [3,2,1] => 5 [3,1,1,1] => 6 [2,2,2] => 8 [2,2,1,1] => 5 [2,1,1,1,1] => 6 [1,1,1,1,1,1] => 6 [7] => 7 [6,1] => 7 [5,2] => 6 [5,1,1] => 7 [4,3] => 10 [4,2,1] => 6 [4,1,1,1] => 7 [3,3,1] => 10 [3,2,2] => 10 [3,2,1,1] => 6 [3,1,1,1,1] => 7 [2,2,2,1] => 10 [2,2,1,1,1] => 6 [2,1,1,1,1,1] => 7 [1,1,1,1,1,1,1] => 7 [8] => 8 [7,1] => 8 [6,2] => 7 [6,1,1] => 8 [5,3] => 12 [5,2,1] => 7 [5,1,1,1] => 8 [4,4] => 15 [4,3,1] => 12 [4,2,2] => 12 [4,2,1,1] => 7 [4,1,1,1,1] => 8 [3,3,2] => 15 [3,3,1,1] => 12 [3,2,2,1] => 12 [3,2,1,1,1] => 7 [3,1,1,1,1,1] => 8 [2,2,2,2] => 15 [2,2,2,1,1] => 12 [2,2,1,1,1,1] => 7 [2,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1] => 8 [9] => 9 [8,1] => 9 [7,2] => 8 [7,1,1] => 9 [6,3] => 14 [6,2,1] => 8 [6,1,1,1] => 9 [5,4] => 18 [5,3,1] => 14 [5,2,2] => 14 [5,2,1,1] => 8 [5,1,1,1,1] => 9 [4,4,1] => 18 [4,3,2] => 18 [4,3,1,1] => 14 [4,2,2,1] => 14 [4,2,1,1,1] => 8 [4,1,1,1,1,1] => 9 [3,3,3] => 15 [3,3,2,1] => 18 [3,3,1,1,1] => 14 [3,2,2,2] => 18 [3,2,2,1,1] => 14 [3,2,1,1,1,1] => 8 [3,1,1,1,1,1,1] => 9 [2,2,2,2,1] => 18 [2,2,2,1,1,1] => 14 [2,2,1,1,1,1,1] => 8 [2,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1] => 9 [10] => 10 [9,1] => 10 [8,2] => 9 [8,1,1] => 10 [7,3] => 16 [7,2,1] => 9 [7,1,1,1] => 10 [6,4] => 21 [6,3,1] => 16 [6,2,2] => 16 [6,2,1,1] => 9 [6,1,1,1,1] => 10 [5,5] => 24 [5,4,1] => 21 [5,3,2] => 21 [5,3,1,1] => 16 [5,2,2,1] => 16 [5,2,1,1,1] => 9 [5,1,1,1,1,1] => 10 [4,4,2] => 24 [4,4,1,1] => 21 [4,3,3] => 18 [4,3,2,1] => 21 [4,3,1,1,1] => 16 [4,2,2,2] => 21 [4,2,2,1,1] => 16 [4,2,1,1,1,1] => 9 [4,1,1,1,1,1,1] => 10 [3,3,3,1] => 18 [3,3,2,2] => 24 [3,3,2,1,1] => 21 [3,3,1,1,1,1] => 16 [3,2,2,2,1] => 21 [3,2,2,1,1,1] => 16 [3,2,1,1,1,1,1] => 9 [3,1,1,1,1,1,1,1] => 10 [2,2,2,2,2] => 24 [2,2,2,2,1,1] => 21 [2,2,2,1,1,1,1] => 16 [2,2,1,1,1,1,1,1] => 9 [2,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1] => 10 [11] => 11 [10,1] => 11 [9,2] => 10 [9,1,1] => 11 [8,3] => 18 [8,2,1] => 10 [8,1,1,1] => 11 [7,4] => 24 [7,3,1] => 18 [7,2,2] => 18 [7,2,1,1] => 10 [7,1,1,1,1] => 11 [6,5] => 28 [6,4,1] => 24 [6,3,2] => 24 [6,3,1,1] => 18 [6,2,2,1] => 18 [6,2,1,1,1] => 10 [6,1,1,1,1,1] => 11 [5,5,1] => 28 [5,4,2] => 28 [5,4,1,1] => 24 [5,3,3] => 21 [5,3,2,1] => 24 [5,3,1,1,1] => 18 [5,2,2,2] => 24 [5,2,2,1,1] => 18 [5,2,1,1,1,1] => 10 [5,1,1,1,1,1,1] => 11 [4,4,3] => 24 [4,4,2,1] => 28 [4,4,1,1,1] => 24 [4,3,3,1] => 21 [4,3,2,2] => 28 [4,3,2,1,1] => 24 [4,3,1,1,1,1] => 18 [4,2,2,2,1] => 24 [4,2,2,1,1,1] => 18 [4,2,1,1,1,1,1] => 10 [4,1,1,1,1,1,1,1] => 11 [3,3,3,2] => 24 [3,3,3,1,1] => 21 [3,3,2,2,1] => 28 [3,3,2,1,1,1] => 24 [3,3,1,1,1,1,1] => 18 [3,2,2,2,2] => 28 [3,2,2,2,1,1] => 24 [3,2,2,1,1,1,1] => 18 [3,2,1,1,1,1,1,1] => 10 [3,1,1,1,1,1,1,1,1] => 11 [2,2,2,2,2,1] => 28 [2,2,2,2,1,1,1] => 24 [2,2,2,1,1,1,1,1] => 18 [2,2,1,1,1,1,1,1,1] => 10 [2,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1] => 11 [12] => 12 [11,1] => 12 [10,2] => 11 [10,1,1] => 12 [9,3] => 20 [9,2,1] => 11 [9,1,1,1] => 12 [8,4] => 27 [8,3,1] => 20 [8,2,2] => 20 [8,2,1,1] => 11 [8,1,1,1,1] => 12 [7,5] => 32 [7,4,1] => 27 [7,3,2] => 27 [7,3,1,1] => 20 [7,2,2,1] => 20 [7,2,1,1,1] => 11 [7,1,1,1,1,1] => 12 [6,6] => 35 [6,5,1] => 32 [6,4,2] => 32 [6,4,1,1] => 27 [6,3,3] => 24 [6,3,2,1] => 27 [6,3,1,1,1] => 20 [6,2,2,2] => 27 [6,2,2,1,1] => 20 [6,2,1,1,1,1] => 11 [6,1,1,1,1,1,1] => 12 [5,5,2] => 35 [5,5,1,1] => 32 [5,4,3] => 28 [5,4,2,1] => 32 [5,4,1,1,1] => 27 [5,3,3,1] => 24 [5,3,2,2] => 32 [5,3,2,1,1] => 27 [5,3,1,1,1,1] => 20 [5,2,2,2,1] => 27 [5,2,2,1,1,1] => 20 [5,2,1,1,1,1,1] => 11 [5,1,1,1,1,1,1,1] => 12 [4,4,4] => 48 [4,4,3,1] => 28 [4,4,2,2] => 35 [4,4,2,1,1] => 32 [4,4,1,1,1,1] => 27 [4,3,3,2] => 28 [4,3,3,1,1] => 24 [4,3,2,2,1] => 32 [4,3,2,1,1,1] => 27 [4,3,1,1,1,1,1] => 20 [4,2,2,2,2] => 32 [4,2,2,2,1,1] => 27 [4,2,2,1,1,1,1] => 20 [4,2,1,1,1,1,1,1] => 11 [4,1,1,1,1,1,1,1,1] => 12 [3,3,3,3] => 48 [3,3,3,2,1] => 28 [3,3,3,1,1,1] => 24 [3,3,2,2,2] => 35 [3,3,2,2,1,1] => 32 [3,3,2,1,1,1,1] => 27 [3,3,1,1,1,1,1,1] => 20 [3,2,2,2,2,1] => 32 [3,2,2,2,1,1,1] => 27 [3,2,2,1,1,1,1,1] => 20 [3,2,1,1,1,1,1,1,1] => 11 [3,1,1,1,1,1,1,1,1,1] => 12 [2,2,2,2,2,2] => 35 [2,2,2,2,2,1,1] => 32 [2,2,2,2,1,1,1,1] => 27 [2,2,2,1,1,1,1,1,1] => 20 [2,2,1,1,1,1,1,1,1,1] => 11 [2,1,1,1,1,1,1,1,1,1,1] => 12 [1,1,1,1,1,1,1,1,1,1,1,1] => 12 ----------------------------------------------------------------------------- Created: Jun 27, 2017 at 09:05 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jul 06, 2021 at 07:55 by Martin Rubey