***************************************************************************** * 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: St000149 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1]. ----------------------------------------------------------------------------- References: [1] Tim Statistics on partitions equidistributed with number of even parts [[MathOverflow:134577]] ----------------------------------------------------------------------------- Code: def good_cells(L): return [ c for c in L.cells() if L.leg_length(*c) == 0 and L.arm_length(*c) % 2 == 1 ] def statistic(x): return len(good_cells(x)) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [2] => 1 [1,1] => 0 [3] => 1 [2,1] => 0 [1,1,1] => 0 [4] => 2 [3,1] => 1 [2,2] => 1 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 2 [4,1] => 1 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 0 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 3 [5,1] => 2 [4,2] => 2 [4,1,1] => 1 [3,3] => 1 [3,2,1] => 0 [3,1,1,1] => 1 [2,2,2] => 1 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 3 [6,1] => 2 [5,2] => 2 [5,1,1] => 2 [4,3] => 1 [4,2,1] => 1 [4,1,1,1] => 1 [3,3,1] => 1 [3,2,2] => 1 [3,2,1,1] => 0 [3,1,1,1,1] => 1 [2,2,2,1] => 0 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 0 [8] => 4 [7,1] => 3 [6,2] => 3 [6,1,1] => 2 [5,3] => 2 [5,2,1] => 1 [5,1,1,1] => 2 [4,4] => 2 [4,3,1] => 1 [4,2,2] => 2 [4,2,1,1] => 1 [4,1,1,1,1] => 1 [3,3,2] => 1 [3,3,1,1] => 1 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 1 [2,2,2,2] => 1 [2,2,2,1,1] => 0 [2,2,1,1,1,1] => 0 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => 0 [9] => 4 [8,1] => 3 [7,2] => 3 [7,1,1] => 3 [6,3] => 2 [6,2,1] => 2 [6,1,1,1] => 2 [5,4] => 2 [5,3,1] => 2 [5,2,2] => 2 [5,2,1,1] => 1 [5,1,1,1,1] => 2 [4,4,1] => 1 [4,3,2] => 1 [4,3,1,1] => 1 [4,2,2,1] => 1 [4,2,1,1,1] => 1 [4,1,1,1,1,1] => 1 [3,3,3] => 1 [3,3,2,1] => 0 [3,3,1,1,1] => 1 [3,2,2,2] => 1 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 1 [2,2,2,2,1] => 0 [2,2,2,1,1,1] => 0 [2,2,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1] => 0 [10] => 5 [9,1] => 4 [8,2] => 4 [8,1,1] => 3 [7,3] => 3 [7,2,1] => 2 [7,1,1,1] => 3 [6,4] => 3 [6,3,1] => 2 [6,2,2] => 3 [6,2,1,1] => 2 [6,1,1,1,1] => 2 [5,5] => 2 [5,4,1] => 1 [5,3,2] => 2 [5,3,1,1] => 2 [5,2,2,1] => 1 [5,2,1,1,1] => 1 [5,1,1,1,1,1] => 2 [4,4,2] => 2 [4,4,1,1] => 1 [4,3,3] => 1 [4,3,2,1] => 0 [4,3,1,1,1] => 1 [4,2,2,2] => 2 [4,2,2,1,1] => 1 [4,2,1,1,1,1] => 1 [4,1,1,1,1,1,1] => 1 [3,3,3,1] => 1 [3,3,2,2] => 1 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 1 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 1 [2,2,2,2,2] => 1 [2,2,2,2,1,1] => 0 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Jul 08, 2013 at 13:27 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 16:06 by Martin Rubey