***************************************************************************** * 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: St000183 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The side length of the Durfee square of an integer partition. Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$. This is also known as the Frobenius rank. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Durfee_square]] [2] Andrews, G. E., Eriksson, K. Integer partitions [[MathSciNet:2122332]] ----------------------------------------------------------------------------- Code: def statistic(p): s = 0 while len(p) > s and p[s] >= s+1: s += 1 return s ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 1 [1,1,1] => 1 [4] => 1 [3,1] => 1 [2,2] => 2 [2,1,1] => 1 [1,1,1,1] => 1 [5] => 1 [4,1] => 1 [3,2] => 2 [3,1,1] => 1 [2,2,1] => 2 [2,1,1,1] => 1 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 1 [4,2] => 2 [4,1,1] => 1 [3,3] => 2 [3,2,1] => 2 [3,1,1,1] => 1 [2,2,2] => 2 [2,2,1,1] => 2 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 1 [5,2] => 2 [5,1,1] => 1 [4,3] => 2 [4,2,1] => 2 [4,1,1,1] => 1 [3,3,1] => 2 [3,2,2] => 2 [3,2,1,1] => 2 [3,1,1,1,1] => 1 [2,2,2,1] => 2 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 1 [6,2] => 2 [6,1,1] => 1 [5,3] => 2 [5,2,1] => 2 [5,1,1,1] => 1 [4,4] => 2 [4,3,1] => 2 [4,2,2] => 2 [4,2,1,1] => 2 [4,1,1,1,1] => 1 [3,3,2] => 2 [3,3,1,1] => 2 [3,2,2,1] => 2 [3,2,1,1,1] => 2 [3,1,1,1,1,1] => 1 [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] => 1 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 1 [7,2] => 2 [7,1,1] => 1 [6,3] => 2 [6,2,1] => 2 [6,1,1,1] => 1 [5,4] => 2 [5,3,1] => 2 [5,2,2] => 2 [5,2,1,1] => 2 [5,1,1,1,1] => 1 [4,4,1] => 2 [4,3,2] => 2 [4,3,1,1] => 2 [4,2,2,1] => 2 [4,2,1,1,1] => 2 [4,1,1,1,1,1] => 1 [3,3,3] => 3 [3,3,2,1] => 2 [3,3,1,1,1] => 2 [3,2,2,2] => 2 [3,2,2,1,1] => 2 [3,2,1,1,1,1] => 2 [3,1,1,1,1,1,1] => 1 [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] => 1 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 1 [8,2] => 2 [8,1,1] => 1 [7,3] => 2 [7,2,1] => 2 [7,1,1,1] => 1 [6,4] => 2 [6,3,1] => 2 [6,2,2] => 2 [6,2,1,1] => 2 [6,1,1,1,1] => 1 [5,5] => 2 [5,4,1] => 2 [5,3,2] => 2 [5,3,1,1] => 2 [5,2,2,1] => 2 [5,2,1,1,1] => 2 [5,1,1,1,1,1] => 1 [4,4,2] => 2 [4,4,1,1] => 2 [4,3,3] => 3 [4,3,2,1] => 2 [4,3,1,1,1] => 2 [4,2,2,2] => 2 [4,2,2,1,1] => 2 [4,2,1,1,1,1] => 2 [4,1,1,1,1,1,1] => 1 [3,3,3,1] => 3 [3,3,2,2] => 2 [3,3,2,1,1] => 2 [3,3,1,1,1,1] => 2 [3,2,2,2,1] => 2 [3,2,2,1,1,1] => 2 [3,2,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1] => 1 [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] => 1 [1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: May 05, 2014 at 06:22 by Lahiru Kariyawasam ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 16:33 by Martin Rubey