***************************************************************************** * 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: St000159 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition. ----------------------------------------------------------------------------- References: [1] Tewari, V. V. Kronecker coefficients for some near-rectangular partitions [[MathSciNet:3320625]] [[arXiv:1403.5327]] ----------------------------------------------------------------------------- Code: def statistic(L): return len(set(L)) ----------------------------------------------------------------------------- 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] => 1 [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] => 1 [3,2,1] => 3 [3,1,1,1] => 2 [2,2,2] => 1 [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] => 2 [3,2,2] => 2 [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] => 1 [4,3,1] => 3 [4,2,2] => 2 [4,2,1,1] => 3 [4,1,1,1,1] => 2 [3,3,2] => 2 [3,3,1,1] => 2 [3,2,2,1] => 3 [3,2,1,1,1] => 3 [3,1,1,1,1,1] => 2 [2,2,2,2] => 1 [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] => 2 [5,2,1,1] => 3 [5,1,1,1,1] => 2 [4,4,1] => 2 [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] => 1 [3,3,2,1] => 3 [3,3,1,1,1] => 2 [3,2,2,2] => 2 [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] => 2 [6,2,1,1] => 3 [6,1,1,1,1] => 2 [5,5] => 1 [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] => 2 [4,4,1,1] => 2 [4,3,3] => 2 [4,3,2,1] => 4 [4,3,1,1,1] => 3 [4,2,2,2] => 2 [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] => 2 [3,3,2,2] => 2 [3,3,2,1,1] => 3 [3,3,1,1,1,1] => 2 [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] => 1 [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 [11] => 1 [10,1] => 2 [9,2] => 2 [9,1,1] => 2 [8,3] => 2 [8,2,1] => 3 [8,1,1,1] => 2 [7,4] => 2 [7,3,1] => 3 [7,2,2] => 2 [7,2,1,1] => 3 [7,1,1,1,1] => 2 [6,5] => 2 [6,4,1] => 3 [6,3,2] => 3 [6,3,1,1] => 3 [6,2,2,1] => 3 [6,2,1,1,1] => 3 [6,1,1,1,1,1] => 2 [5,5,1] => 2 [5,4,2] => 3 [5,4,1,1] => 3 [5,3,3] => 2 [5,3,2,1] => 4 [5,3,1,1,1] => 3 [5,2,2,2] => 2 [5,2,2,1,1] => 3 [5,2,1,1,1,1] => 3 [5,1,1,1,1,1,1] => 2 [4,4,3] => 2 [4,4,2,1] => 3 [4,4,1,1,1] => 2 [4,3,3,1] => 3 [4,3,2,2] => 3 [4,3,2,1,1] => 4 [4,3,1,1,1,1] => 3 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => 3 [4,2,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1] => 2 [3,3,3,2] => 2 [3,3,3,1,1] => 2 [3,3,2,2,1] => 3 [3,3,2,1,1,1] => 3 [3,3,1,1,1,1,1] => 2 [3,2,2,2,2] => 2 [3,2,2,2,1,1] => 3 [3,2,2,1,1,1,1] => 3 [3,2,1,1,1,1,1,1] => 3 [3,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,1] => 2 [2,2,2,2,1,1,1] => 2 [2,2,2,1,1,1,1,1] => 2 [2,2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 2 [10,2] => 2 [10,1,1] => 2 [9,3] => 2 [9,2,1] => 3 [9,1,1,1] => 2 [8,4] => 2 [8,3,1] => 3 [8,2,2] => 2 [8,2,1,1] => 3 [8,1,1,1,1] => 2 [7,5] => 2 [7,4,1] => 3 [7,3,2] => 3 [7,3,1,1] => 3 [7,2,2,1] => 3 [7,2,1,1,1] => 3 [7,1,1,1,1,1] => 2 [6,6] => 1 [6,5,1] => 3 [6,4,2] => 3 [6,4,1,1] => 3 [6,3,3] => 2 [6,3,2,1] => 4 [6,3,1,1,1] => 3 [6,2,2,2] => 2 [6,2,2,1,1] => 3 [6,2,1,1,1,1] => 3 [6,1,1,1,1,1,1] => 2 [5,5,2] => 2 [5,5,1,1] => 2 [5,4,3] => 3 [5,4,2,1] => 4 [5,4,1,1,1] => 3 [5,3,3,1] => 3 [5,3,2,2] => 3 [5,3,2,1,1] => 4 [5,3,1,1,1,1] => 3 [5,2,2,2,1] => 3 [5,2,2,1,1,1] => 3 [5,2,1,1,1,1,1] => 3 [5,1,1,1,1,1,1,1] => 2 [4,4,4] => 1 [4,4,3,1] => 3 [4,4,2,2] => 2 [4,4,2,1,1] => 3 [4,4,1,1,1,1] => 2 [4,3,3,2] => 3 [4,3,3,1,1] => 3 [4,3,2,2,1] => 4 [4,3,2,1,1,1] => 4 [4,3,1,1,1,1,1] => 3 [4,2,2,2,2] => 2 [4,2,2,2,1,1] => 3 [4,2,2,1,1,1,1] => 3 [4,2,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1] => 2 [3,3,3,3] => 1 [3,3,3,2,1] => 3 [3,3,3,1,1,1] => 2 [3,3,2,2,2] => 2 [3,3,2,2,1,1] => 3 [3,3,2,1,1,1,1] => 3 [3,3,1,1,1,1,1,1] => 2 [3,2,2,2,2,1] => 3 [3,2,2,2,1,1,1] => 3 [3,2,2,1,1,1,1,1] => 3 [3,2,1,1,1,1,1,1,1] => 3 [3,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2] => 1 [2,2,2,2,2,1,1] => 2 [2,2,2,2,1,1,1,1] => 2 [2,2,2,1,1,1,1,1,1] => 2 [2,2,1,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Sep 04, 2013 at 14:27 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 20:23 by Martin Rubey