***************************************************************************** * 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: St000783 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The side length of the largest staircase partition fitting into a partition. For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$. In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions. ----------------------------------------------------------------------------- References: [1] Chow, T. Coloring a Ferrers diagram [[MathOverflow:203962]] ----------------------------------------------------------------------------- Code: def statistic(la): return min(p + i for i, p in enumerate(la + [0])) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 2 [1,1,1] => 1 [4] => 1 [3,1] => 2 [2,2] => 2 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 1 [4,1] => 2 [3,2] => 2 [3,1,1] => 2 [2,2,1] => 2 [2,1,1,1] => 2 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 2 [4,2] => 2 [4,1,1] => 2 [3,3] => 2 [3,2,1] => 3 [3,1,1,1] => 2 [2,2,2] => 2 [2,2,1,1] => 2 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 2 [5,2] => 2 [5,1,1] => 2 [4,3] => 2 [4,2,1] => 3 [4,1,1,1] => 2 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 3 [3,1,1,1,1] => 2 [2,2,2,1] => 2 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 2 [6,2] => 2 [6,1,1] => 2 [5,3] => 2 [5,2,1] => 3 [5,1,1,1] => 2 [4,4] => 2 [4,3,1] => 3 [4,2,2] => 3 [4,2,1,1] => 3 [4,1,1,1,1] => 2 [3,3,2] => 3 [3,3,1,1] => 3 [3,2,2,1] => 3 [3,2,1,1,1] => 3 [3,1,1,1,1,1] => 2 [2,2,2,2] => 2 [2,2,2,1,1] => 2 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 2 [7,2] => 2 [7,1,1] => 2 [6,3] => 2 [6,2,1] => 3 [6,1,1,1] => 2 [5,4] => 2 [5,3,1] => 3 [5,2,2] => 3 [5,2,1,1] => 3 [5,1,1,1,1] => 2 [4,4,1] => 3 [4,3,2] => 3 [4,3,1,1] => 3 [4,2,2,1] => 3 [4,2,1,1,1] => 3 [4,1,1,1,1,1] => 2 [3,3,3] => 3 [3,3,2,1] => 3 [3,3,1,1,1] => 3 [3,2,2,2] => 3 [3,2,2,1,1] => 3 [3,2,1,1,1,1] => 3 [3,1,1,1,1,1,1] => 2 [2,2,2,2,1] => 2 [2,2,2,1,1,1] => 2 [2,2,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 2 [8,2] => 2 [8,1,1] => 2 [7,3] => 2 [7,2,1] => 3 [7,1,1,1] => 2 [6,4] => 2 [6,3,1] => 3 [6,2,2] => 3 [6,2,1,1] => 3 [6,1,1,1,1] => 2 [5,5] => 2 [5,4,1] => 3 [5,3,2] => 3 [5,3,1,1] => 3 [5,2,2,1] => 3 [5,2,1,1,1] => 3 [5,1,1,1,1,1] => 2 [4,4,2] => 3 [4,4,1,1] => 3 [4,3,3] => 3 [4,3,2,1] => 4 [4,3,1,1,1] => 3 [4,2,2,2] => 3 [4,2,2,1,1] => 3 [4,2,1,1,1,1] => 3 [4,1,1,1,1,1,1] => 2 [3,3,3,1] => 3 [3,3,2,2] => 3 [3,3,2,1,1] => 3 [3,3,1,1,1,1] => 3 [3,2,2,2,1] => 3 [3,2,2,1,1,1] => 3 [3,2,1,1,1,1,1] => 3 [3,1,1,1,1,1,1,1] => 2 [2,2,2,2,2] => 2 [2,2,2,2,1,1] => 2 [2,2,2,1,1,1,1] => 2 [2,2,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1] => 1 [5,4,2] => 3 [5,4,1,1] => 3 [5,3,3] => 3 [5,3,2,1] => 4 [5,3,1,1,1] => 3 [5,2,2,2] => 3 [5,2,2,1,1] => 3 [4,4,3] => 3 [4,4,2,1] => 4 [4,4,1,1,1] => 3 [4,3,3,1] => 4 [4,3,2,2] => 4 [4,3,2,1,1] => 4 [4,2,2,2,1] => 3 [3,3,3,2] => 3 [3,3,3,1,1] => 3 [3,3,2,2,1] => 3 [6,4,2] => 3 [5,4,3] => 3 [5,4,2,1] => 4 [5,4,1,1,1] => 3 [5,3,3,1] => 4 [5,3,2,2] => 4 [5,3,2,1,1] => 4 [5,2,2,2,1] => 3 [4,4,3,1] => 4 [4,4,2,2] => 4 [4,4,2,1,1] => 4 [4,3,3,2] => 4 [4,3,3,1,1] => 4 [4,3,2,2,1] => 4 [3,3,3,2,1] => 3 [3,3,2,2,1,1] => 3 [5,4,3,1] => 4 [5,4,2,2] => 4 [5,4,2,1,1] => 4 [5,3,3,2] => 4 [5,3,3,1,1] => 4 [5,3,2,2,1] => 4 [4,4,3,2] => 4 [4,4,3,1,1] => 4 [4,4,2,2,1] => 4 [4,3,3,2,1] => 4 [5,4,3,2] => 4 [5,4,3,1,1] => 4 [5,4,2,2,1] => 4 [5,3,3,2,1] => 4 [4,4,3,2,1] => 4 [5,4,3,2,1] => 5 ----------------------------------------------------------------------------- Created: Apr 19, 2017 at 10:21 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 28, 2019 at 18:15 by Martin Rubey