***************************************************************************** * 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: St000869 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the hook lengths of 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 sum of all hook lengths of a partition. ----------------------------------------------------------------------------- References: [1] Amdeberhan, T. Average on sum of hooks in a rectangle [[MathOverflow:273095]] ----------------------------------------------------------------------------- Code: def statistic(L): return sum( L.hook_length(*c) for c in L.cells() ) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 3 [1,1] => 3 [3] => 6 [2,1] => 5 [1,1,1] => 6 [4] => 10 [3,1] => 8 [2,2] => 8 [2,1,1] => 8 [1,1,1,1] => 10 [5] => 15 [4,1] => 12 [3,2] => 11 [3,1,1] => 11 [2,2,1] => 11 [2,1,1,1] => 12 [1,1,1,1,1] => 15 [6] => 21 [5,1] => 17 [4,2] => 15 [4,1,1] => 15 [3,3] => 15 [3,2,1] => 14 [3,1,1,1] => 15 [2,2,2] => 15 [2,2,1,1] => 15 [2,1,1,1,1] => 17 [1,1,1,1,1,1] => 21 [7] => 28 [6,1] => 23 [5,2] => 20 [5,1,1] => 20 [4,3] => 19 [4,2,1] => 18 [4,1,1,1] => 19 [3,3,1] => 18 [3,2,2] => 18 [3,2,1,1] => 18 [3,1,1,1,1] => 20 [2,2,2,1] => 19 [2,2,1,1,1] => 20 [2,1,1,1,1,1] => 23 [1,1,1,1,1,1,1] => 28 [8] => 36 [7,1] => 30 [6,2] => 26 [6,1,1] => 26 [5,3] => 24 [5,2,1] => 23 [5,1,1,1] => 24 [4,4] => 24 [4,3,1] => 22 [4,2,2] => 22 [4,2,1,1] => 22 [4,1,1,1,1] => 24 [3,3,2] => 22 [3,3,1,1] => 22 [3,2,2,1] => 22 [3,2,1,1,1] => 23 [3,1,1,1,1,1] => 26 [2,2,2,2] => 24 [2,2,2,1,1] => 24 [2,2,1,1,1,1] => 26 [2,1,1,1,1,1,1] => 30 [1,1,1,1,1,1,1,1] => 36 [9] => 45 [8,1] => 38 [7,2] => 33 [7,1,1] => 33 [6,3] => 30 [6,2,1] => 29 [6,1,1,1] => 30 [5,4] => 29 [5,3,1] => 27 [5,2,2] => 27 [5,2,1,1] => 27 [5,1,1,1,1] => 29 [4,4,1] => 27 [4,3,2] => 26 [4,3,1,1] => 26 [4,2,2,1] => 26 [4,2,1,1,1] => 27 [4,1,1,1,1,1] => 30 [3,3,3] => 27 [3,3,2,1] => 26 [3,3,1,1,1] => 27 [3,2,2,2] => 27 [3,2,2,1,1] => 27 [3,2,1,1,1,1] => 29 [3,1,1,1,1,1,1] => 33 [2,2,2,2,1] => 29 [2,2,2,1,1,1] => 30 [2,2,1,1,1,1,1] => 33 [2,1,1,1,1,1,1,1] => 38 [1,1,1,1,1,1,1,1,1] => 45 [10] => 55 [9,1] => 47 [8,2] => 41 [8,1,1] => 41 [7,3] => 37 [7,2,1] => 36 [7,1,1,1] => 37 [6,4] => 35 [6,3,1] => 33 [6,2,2] => 33 [6,2,1,1] => 33 [6,1,1,1,1] => 35 [5,5] => 35 [5,4,1] => 32 [5,3,2] => 31 [5,3,1,1] => 31 [5,2,2,1] => 31 [5,2,1,1,1] => 32 [5,1,1,1,1,1] => 35 [4,4,2] => 31 [4,4,1,1] => 31 [4,3,3] => 31 [4,3,2,1] => 30 [4,3,1,1,1] => 31 [4,2,2,2] => 31 [4,2,2,1,1] => 31 [4,2,1,1,1,1] => 33 [4,1,1,1,1,1,1] => 37 [3,3,3,1] => 31 [3,3,2,2] => 31 [3,3,2,1,1] => 31 [3,3,1,1,1,1] => 33 [3,2,2,2,1] => 32 [3,2,2,1,1,1] => 33 [3,2,1,1,1,1,1] => 36 [3,1,1,1,1,1,1,1] => 41 [2,2,2,2,2] => 35 [2,2,2,2,1,1] => 35 [2,2,2,1,1,1,1] => 37 [2,2,1,1,1,1,1,1] => 41 [2,1,1,1,1,1,1,1,1] => 47 [1,1,1,1,1,1,1,1,1,1] => 55 [11] => 66 [10,1] => 57 [9,2] => 50 [9,1,1] => 50 [8,3] => 45 [8,2,1] => 44 [8,1,1,1] => 45 [7,4] => 42 [7,3,1] => 40 [7,2,2] => 40 [7,2,1,1] => 40 [7,1,1,1,1] => 42 [6,5] => 41 [6,4,1] => 38 [6,3,2] => 37 [6,3,1,1] => 37 [6,2,2,1] => 37 [6,2,1,1,1] => 38 [6,1,1,1,1,1] => 41 [5,5,1] => 38 [5,4,2] => 36 [5,4,1,1] => 36 [5,3,3] => 36 [5,3,2,1] => 35 [5,3,1,1,1] => 36 [5,2,2,2] => 36 [5,2,2,1,1] => 36 [5,2,1,1,1,1] => 38 [5,1,1,1,1,1,1] => 42 [4,4,3] => 36 [4,4,2,1] => 35 [4,4,1,1,1] => 36 [4,3,3,1] => 35 [4,3,2,2] => 35 [4,3,2,1,1] => 35 [4,3,1,1,1,1] => 37 [4,2,2,2,1] => 36 [4,2,2,1,1,1] => 37 [4,2,1,1,1,1,1] => 40 [4,1,1,1,1,1,1,1] => 45 [3,3,3,2] => 36 [3,3,3,1,1] => 36 [3,3,2,2,1] => 36 [3,3,2,1,1,1] => 37 [3,3,1,1,1,1,1] => 40 [3,2,2,2,2] => 38 [3,2,2,2,1,1] => 38 [3,2,2,1,1,1,1] => 40 [3,2,1,1,1,1,1,1] => 44 [3,1,1,1,1,1,1,1,1] => 50 [2,2,2,2,2,1] => 41 [2,2,2,2,1,1,1] => 42 [2,2,2,1,1,1,1,1] => 45 [2,2,1,1,1,1,1,1,1] => 50 [2,1,1,1,1,1,1,1,1,1] => 57 [1,1,1,1,1,1,1,1,1,1,1] => 66 [12] => 78 [11,1] => 68 [10,2] => 60 [10,1,1] => 60 [9,3] => 54 [9,2,1] => 53 [9,1,1,1] => 54 [8,4] => 50 [8,3,1] => 48 [8,2,2] => 48 [8,2,1,1] => 48 [8,1,1,1,1] => 50 [7,5] => 48 [7,4,1] => 45 [7,3,2] => 44 [7,3,1,1] => 44 [7,2,2,1] => 44 [7,2,1,1,1] => 45 [7,1,1,1,1,1] => 48 [6,6] => 48 [6,5,1] => 44 [6,4,2] => 42 [6,4,1,1] => 42 [6,3,3] => 42 [6,3,2,1] => 41 [6,3,1,1,1] => 42 [6,2,2,2] => 42 [6,2,2,1,1] => 42 [6,2,1,1,1,1] => 44 [6,1,1,1,1,1,1] => 48 [5,5,2] => 42 [5,5,1,1] => 42 [5,4,3] => 41 [5,4,2,1] => 40 [5,4,1,1,1] => 41 [5,3,3,1] => 40 [5,3,2,2] => 40 [5,3,2,1,1] => 40 [5,3,1,1,1,1] => 42 [5,2,2,2,1] => 41 [5,2,2,1,1,1] => 42 [5,2,1,1,1,1,1] => 45 [5,1,1,1,1,1,1,1] => 50 [4,4,4] => 42 [4,4,3,1] => 40 [4,4,2,2] => 40 [4,4,2,1,1] => 40 [4,4,1,1,1,1] => 42 [4,3,3,2] => 40 [4,3,3,1,1] => 40 [4,3,2,2,1] => 40 [4,3,2,1,1,1] => 41 [4,3,1,1,1,1,1] => 44 [4,2,2,2,2] => 42 [4,2,2,2,1,1] => 42 [4,2,2,1,1,1,1] => 44 [4,2,1,1,1,1,1,1] => 48 [4,1,1,1,1,1,1,1,1] => 54 [3,3,3,3] => 42 [3,3,3,2,1] => 41 [3,3,3,1,1,1] => 42 [3,3,2,2,2] => 42 [3,3,2,2,1,1] => 42 [3,3,2,1,1,1,1] => 44 [3,3,1,1,1,1,1,1] => 48 [3,2,2,2,2,1] => 44 [3,2,2,2,1,1,1] => 45 [3,2,2,1,1,1,1,1] => 48 [3,2,1,1,1,1,1,1,1] => 53 [3,1,1,1,1,1,1,1,1,1] => 60 [2,2,2,2,2,2] => 48 [2,2,2,2,2,1,1] => 48 [2,2,2,2,1,1,1,1] => 50 [2,2,2,1,1,1,1,1,1] => 54 [2,2,1,1,1,1,1,1,1,1] => 60 [2,1,1,1,1,1,1,1,1,1,1] => 68 [1,1,1,1,1,1,1,1,1,1,1,1] => 78 [5,4,3,1] => 45 [5,4,2,2] => 45 [5,4,2,1,1] => 45 [5,3,3,2] => 45 [5,3,3,1,1] => 45 [5,3,2,2,1] => 45 [4,4,3,2] => 45 [4,4,3,1,1] => 45 [4,4,2,2,1] => 45 [4,3,3,2,1] => 45 [5,4,3,2] => 50 [5,4,3,1,1] => 50 [5,4,2,2,1] => 50 [5,3,3,2,1] => 50 [4,4,3,2,1] => 50 [5,4,3,2,1] => 55 ----------------------------------------------------------------------------- Created: Jun 27, 2017 at 09:02 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Apr 26, 2018 at 07:39 by Martin Rubey