***************************************************************************** * 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: St000714 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Young_tableau#Applications_in_representation_theory]] ----------------------------------------------------------------------------- Code: def statistic(mu): return SemistandardTableaux(shape=mu, max_entry=2).cardinality() ----------------------------------------------------------------------------- Statistic values: [2] => 3 [1,1] => 1 [3] => 4 [2,1] => 2 [1,1,1] => 0 [4] => 5 [3,1] => 3 [2,2] => 1 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 6 [4,1] => 4 [3,2] => 2 [3,1,1] => 0 [2,2,1] => 0 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 7 [5,1] => 5 [4,2] => 3 [4,1,1] => 0 [3,3] => 1 [3,2,1] => 0 [3,1,1,1] => 0 [2,2,2] => 0 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 8 [6,1] => 6 [5,2] => 4 [5,1,1] => 0 [4,3] => 2 [4,2,1] => 0 [4,1,1,1] => 0 [3,3,1] => 0 [3,2,2] => 0 [3,2,1,1] => 0 [3,1,1,1,1] => 0 [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] => 9 [7,1] => 7 [6,2] => 5 [6,1,1] => 0 [5,3] => 3 [5,2,1] => 0 [5,1,1,1] => 0 [4,4] => 1 [4,3,1] => 0 [4,2,2] => 0 [4,2,1,1] => 0 [4,1,1,1,1] => 0 [3,3,2] => 0 [3,3,1,1] => 0 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 0 [2,2,2,2] => 0 [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] => 10 [8,1] => 8 [7,2] => 6 [7,1,1] => 0 [6,3] => 4 [6,2,1] => 0 [6,1,1,1] => 0 [5,4] => 2 [5,3,1] => 0 [5,2,2] => 0 [5,2,1,1] => 0 [5,1,1,1,1] => 0 [4,4,1] => 0 [4,3,2] => 0 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 0 [3,3,3] => 0 [3,3,2,1] => 0 [3,3,1,1,1] => 0 [3,2,2,2] => 0 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 0 [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] => 11 [9,1] => 9 [8,2] => 7 [8,1,1] => 0 [7,3] => 5 [7,2,1] => 0 [7,1,1,1] => 0 [6,4] => 3 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 1 [5,4,1] => 0 [5,3,2] => 0 [5,3,1,1] => 0 [5,2,2,1] => 0 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => 0 [4,4,2] => 0 [4,4,1,1] => 0 [4,3,3] => 0 [4,3,2,1] => 0 [4,3,1,1,1] => 0 [4,2,2,2] => 0 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 0 [4,1,1,1,1,1,1] => 0 [3,3,3,1] => 0 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 0 [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] => 0 [2,2,2,2,2] => 0 [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 [11] => 12 [10,1] => 10 [9,2] => 8 [9,1,1] => 0 [8,3] => 6 [8,2,1] => 0 [8,1,1,1] => 0 [7,4] => 4 [7,3,1] => 0 [7,2,2] => 0 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 2 [6,4,1] => 0 [6,3,2] => 0 [6,3,1,1] => 0 [6,2,2,1] => 0 [6,2,1,1,1] => 0 [6,1,1,1,1,1] => 0 [5,5,1] => 0 [5,4,2] => 0 [5,4,1,1] => 0 [5,3,3] => 0 [5,3,2,1] => 0 [5,3,1,1,1] => 0 [5,2,2,2] => 0 [5,2,2,1,1] => 0 [5,2,1,1,1,1] => 0 [5,1,1,1,1,1,1] => 0 [4,4,3] => 0 [4,4,2,1] => 0 [4,4,1,1,1] => 0 [4,3,3,1] => 0 [4,3,2,2] => 0 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => 0 [4,2,2,2,1] => 0 [4,2,2,1,1,1] => 0 [4,2,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1] => 0 [3,3,3,2] => 0 [3,3,3,1,1] => 0 [3,3,2,2,1] => 0 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => 0 [3,2,2,2,2] => 0 [3,2,2,2,1,1] => 0 [3,2,2,1,1,1,1] => 0 [3,2,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,1] => 0 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 13 [11,1] => 11 [10,2] => 9 [10,1,1] => 0 [9,3] => 7 [9,2,1] => 0 [9,1,1,1] => 0 [8,4] => 5 [8,3,1] => 0 [8,2,2] => 0 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 3 [7,4,1] => 0 [7,3,2] => 0 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 1 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 0 [6,3,3] => 0 [6,3,2,1] => 0 [6,3,1,1,1] => 0 [6,2,2,2] => 0 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => 0 [5,5,2] => 0 [5,5,1,1] => 0 [5,4,3] => 0 [5,4,2,1] => 0 [5,4,1,1,1] => 0 [5,3,3,1] => 0 [5,3,2,2] => 0 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => 0 [5,2,2,2,1] => 0 [5,2,2,1,1,1] => 0 [5,2,1,1,1,1,1] => 0 [5,1,1,1,1,1,1,1] => 0 [4,4,4] => 0 [4,4,3,1] => 0 [4,4,2,2] => 0 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 0 [4,3,3,1,1] => 0 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [4,2,2,2,1,1] => 0 [4,2,2,1,1,1,1] => 0 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => 0 [3,3,3,3] => 0 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 0 [3,3,2,2,2] => 0 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 0 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2] => 0 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Mar 21, 2017 at 07:32 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 21, 2017 at 07:32 by Martin Rubey