***************************************************************************** * 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: St000867 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the hook lengths in the first row 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 plus one. This statistic is the sum of the hook lengths of the first row of a partition. Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(L): return L.size() + binomial(L[0], 2) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 3 [1,1] => 2 [3] => 6 [2,1] => 4 [1,1,1] => 3 [4] => 10 [3,1] => 7 [2,2] => 5 [2,1,1] => 5 [1,1,1,1] => 4 [5] => 15 [4,1] => 11 [3,2] => 8 [3,1,1] => 8 [2,2,1] => 6 [2,1,1,1] => 6 [1,1,1,1,1] => 5 [6] => 21 [5,1] => 16 [4,2] => 12 [4,1,1] => 12 [3,3] => 9 [3,2,1] => 9 [3,1,1,1] => 9 [2,2,2] => 7 [2,2,1,1] => 7 [2,1,1,1,1] => 7 [1,1,1,1,1,1] => 6 [7] => 28 [6,1] => 22 [5,2] => 17 [5,1,1] => 17 [4,3] => 13 [4,2,1] => 13 [4,1,1,1] => 13 [3,3,1] => 10 [3,2,2] => 10 [3,2,1,1] => 10 [3,1,1,1,1] => 10 [2,2,2,1] => 8 [2,2,1,1,1] => 8 [2,1,1,1,1,1] => 8 [1,1,1,1,1,1,1] => 7 [8] => 36 [7,1] => 29 [6,2] => 23 [6,1,1] => 23 [5,3] => 18 [5,2,1] => 18 [5,1,1,1] => 18 [4,4] => 14 [4,3,1] => 14 [4,2,2] => 14 [4,2,1,1] => 14 [4,1,1,1,1] => 14 [3,3,2] => 11 [3,3,1,1] => 11 [3,2,2,1] => 11 [3,2,1,1,1] => 11 [3,1,1,1,1,1] => 11 [2,2,2,2] => 9 [2,2,2,1,1] => 9 [2,2,1,1,1,1] => 9 [2,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1] => 8 [9] => 45 [8,1] => 37 [7,2] => 30 [7,1,1] => 30 [6,3] => 24 [6,2,1] => 24 [6,1,1,1] => 24 [5,4] => 19 [5,3,1] => 19 [5,2,2] => 19 [5,2,1,1] => 19 [5,1,1,1,1] => 19 [4,4,1] => 15 [4,3,2] => 15 [4,3,1,1] => 15 [4,2,2,1] => 15 [4,2,1,1,1] => 15 [4,1,1,1,1,1] => 15 [3,3,3] => 12 [3,3,2,1] => 12 [3,3,1,1,1] => 12 [3,2,2,2] => 12 [3,2,2,1,1] => 12 [3,2,1,1,1,1] => 12 [3,1,1,1,1,1,1] => 12 [2,2,2,2,1] => 10 [2,2,2,1,1,1] => 10 [2,2,1,1,1,1,1] => 10 [2,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1] => 9 [10] => 55 [9,1] => 46 [8,2] => 38 [8,1,1] => 38 [7,3] => 31 [7,2,1] => 31 [7,1,1,1] => 31 [6,4] => 25 [6,3,1] => 25 [6,2,2] => 25 [6,2,1,1] => 25 [6,1,1,1,1] => 25 [5,5] => 20 [5,4,1] => 20 [5,3,2] => 20 [5,3,1,1] => 20 [5,2,2,1] => 20 [5,2,1,1,1] => 20 [5,1,1,1,1,1] => 20 [4,4,2] => 16 [4,4,1,1] => 16 [4,3,3] => 16 [4,3,2,1] => 16 [4,3,1,1,1] => 16 [4,2,2,2] => 16 [4,2,2,1,1] => 16 [4,2,1,1,1,1] => 16 [4,1,1,1,1,1,1] => 16 [3,3,3,1] => 13 [3,3,2,2] => 13 [3,3,2,1,1] => 13 [3,3,1,1,1,1] => 13 [3,2,2,2,1] => 13 [3,2,2,1,1,1] => 13 [3,2,1,1,1,1,1] => 13 [3,1,1,1,1,1,1,1] => 13 [2,2,2,2,2] => 11 [2,2,2,2,1,1] => 11 [2,2,2,1,1,1,1] => 11 [2,2,1,1,1,1,1,1] => 11 [2,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1] => 10 [11] => 66 [10,1] => 56 [9,2] => 47 [9,1,1] => 47 [8,3] => 39 [8,2,1] => 39 [8,1,1,1] => 39 [7,4] => 32 [7,3,1] => 32 [7,2,2] => 32 [7,2,1,1] => 32 [7,1,1,1,1] => 32 [6,5] => 26 [6,4,1] => 26 [6,3,2] => 26 [6,3,1,1] => 26 [6,2,2,1] => 26 [6,2,1,1,1] => 26 [6,1,1,1,1,1] => 26 [5,5,1] => 21 [5,4,2] => 21 [5,4,1,1] => 21 [5,3,3] => 21 [5,3,2,1] => 21 [5,3,1,1,1] => 21 [5,2,2,2] => 21 [5,2,2,1,1] => 21 [5,2,1,1,1,1] => 21 [5,1,1,1,1,1,1] => 21 [4,4,3] => 17 [4,4,2,1] => 17 [4,4,1,1,1] => 17 [4,3,3,1] => 17 [4,3,2,2] => 17 [4,3,2,1,1] => 17 [4,3,1,1,1,1] => 17 [4,2,2,2,1] => 17 [4,2,2,1,1,1] => 17 [4,2,1,1,1,1,1] => 17 [4,1,1,1,1,1,1,1] => 17 [3,3,3,2] => 14 [3,3,3,1,1] => 14 [3,3,2,2,1] => 14 [3,3,2,1,1,1] => 14 [3,3,1,1,1,1,1] => 14 [3,2,2,2,2] => 14 [3,2,2,2,1,1] => 14 [3,2,2,1,1,1,1] => 14 [3,2,1,1,1,1,1,1] => 14 [3,1,1,1,1,1,1,1,1] => 14 [2,2,2,2,2,1] => 12 [2,2,2,2,1,1,1] => 12 [2,2,2,1,1,1,1,1] => 12 [2,2,1,1,1,1,1,1,1] => 12 [2,1,1,1,1,1,1,1,1,1] => 12 [1,1,1,1,1,1,1,1,1,1,1] => 11 [12] => 78 [11,1] => 67 [10,2] => 57 [10,1,1] => 57 [9,3] => 48 [9,2,1] => 48 [9,1,1,1] => 48 [8,4] => 40 [8,3,1] => 40 [8,2,2] => 40 [8,2,1,1] => 40 [8,1,1,1,1] => 40 [7,5] => 33 [7,4,1] => 33 [7,3,2] => 33 [7,3,1,1] => 33 [7,2,2,1] => 33 [7,2,1,1,1] => 33 [7,1,1,1,1,1] => 33 [6,6] => 27 [6,5,1] => 27 [6,4,2] => 27 [6,4,1,1] => 27 [6,3,3] => 27 [6,3,2,1] => 27 [6,3,1,1,1] => 27 [6,2,2,2] => 27 [6,2,2,1,1] => 27 [6,2,1,1,1,1] => 27 [6,1,1,1,1,1,1] => 27 [5,5,2] => 22 [5,5,1,1] => 22 [5,4,3] => 22 [5,4,2,1] => 22 [5,4,1,1,1] => 22 [5,3,3,1] => 22 [5,3,2,2] => 22 [5,3,2,1,1] => 22 [5,3,1,1,1,1] => 22 [5,2,2,2,1] => 22 [5,2,2,1,1,1] => 22 [5,2,1,1,1,1,1] => 22 [5,1,1,1,1,1,1,1] => 22 [4,4,4] => 18 [4,4,3,1] => 18 [4,4,2,2] => 18 [4,4,2,1,1] => 18 [4,4,1,1,1,1] => 18 [4,3,3,2] => 18 [4,3,3,1,1] => 18 [4,3,2,2,1] => 18 [4,3,2,1,1,1] => 18 [4,3,1,1,1,1,1] => 18 [4,2,2,2,2] => 18 [4,2,2,2,1,1] => 18 [4,2,2,1,1,1,1] => 18 [4,2,1,1,1,1,1,1] => 18 [4,1,1,1,1,1,1,1,1] => 18 [3,3,3,3] => 15 [3,3,3,2,1] => 15 [3,3,3,1,1,1] => 15 [3,3,2,2,2] => 15 [3,3,2,2,1,1] => 15 [3,3,2,1,1,1,1] => 15 [3,3,1,1,1,1,1,1] => 15 [3,2,2,2,2,1] => 15 [3,2,2,2,1,1,1] => 15 [3,2,2,1,1,1,1,1] => 15 [3,2,1,1,1,1,1,1,1] => 15 [3,1,1,1,1,1,1,1,1,1] => 15 [2,2,2,2,2,2] => 13 [2,2,2,2,2,1,1] => 13 [2,2,2,2,1,1,1,1] => 13 [2,2,2,1,1,1,1,1,1] => 13 [2,2,1,1,1,1,1,1,1,1] => 13 [2,1,1,1,1,1,1,1,1,1,1] => 13 [1,1,1,1,1,1,1,1,1,1,1,1] => 12 [5,4,3,1] => 23 [5,4,2,2] => 23 [5,4,2,1,1] => 23 [5,3,3,2] => 23 [5,3,3,1,1] => 23 [5,3,2,2,1] => 23 [4,4,3,2] => 19 [4,4,3,1,1] => 19 [4,4,2,2,1] => 19 [4,3,3,2,1] => 19 [5,4,3,2] => 24 [5,4,3,1,1] => 24 [5,4,2,2,1] => 24 [5,3,3,2,1] => 24 [4,4,3,2,1] => 20 [5,4,3,2,1] => 25 ----------------------------------------------------------------------------- Created: Jun 27, 2017 at 09:08 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 19, 2020 at 13:57 by Martin Rubey