***************************************************************************** * 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: St000715 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard Young tableaux of given shape and entries at most 3. This is also the dimension of the corresponding irreducible representation of $GL_3$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Young_tableau#Applications_in_representation_theory]] ----------------------------------------------------------------------------- Code: def statistic(mu): return SemistandardTableaux(shape=mu, max_entry=3).cardinality() ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 3 [2] => 6 [1,1] => 3 [3] => 10 [2,1] => 8 [1,1,1] => 1 [4] => 15 [3,1] => 15 [2,2] => 6 [2,1,1] => 3 [1,1,1,1] => 0 [5] => 21 [4,1] => 24 [3,2] => 15 [3,1,1] => 6 [2,2,1] => 3 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 28 [5,1] => 35 [4,2] => 27 [4,1,1] => 10 [3,3] => 10 [3,2,1] => 8 [3,1,1,1] => 0 [2,2,2] => 1 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 36 [6,1] => 48 [5,2] => 42 [5,1,1] => 15 [4,3] => 24 [4,2,1] => 15 [4,1,1,1] => 0 [3,3,1] => 6 [3,2,2] => 3 [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] => 45 [7,1] => 63 [6,2] => 60 [6,1,1] => 21 [5,3] => 42 [5,2,1] => 24 [5,1,1,1] => 0 [4,4] => 15 [4,3,1] => 15 [4,2,2] => 6 [4,2,1,1] => 0 [4,1,1,1,1] => 0 [3,3,2] => 3 [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] => 55 [8,1] => 80 [7,2] => 81 [7,1,1] => 28 [6,3] => 64 [6,2,1] => 35 [6,1,1,1] => 0 [5,4] => 35 [5,3,1] => 27 [5,2,2] => 10 [5,2,1,1] => 0 [5,1,1,1,1] => 0 [4,4,1] => 10 [4,3,2] => 8 [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] => 1 [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] => 66 [9,1] => 99 [8,2] => 105 [8,1,1] => 36 [7,3] => 90 [7,2,1] => 48 [7,1,1,1] => 0 [6,4] => 60 [6,3,1] => 42 [6,2,2] => 15 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 21 [5,4,1] => 24 [5,3,2] => 15 [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] => 6 [4,4,1,1] => 0 [4,3,3] => 3 [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] => 78 [10,1] => 120 [9,2] => 132 [9,1,1] => 45 [8,3] => 120 [8,2,1] => 63 [8,1,1,1] => 0 [7,4] => 90 [7,3,1] => 60 [7,2,2] => 21 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 48 [6,4,1] => 42 [6,3,2] => 24 [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] => 15 [5,4,2] => 15 [5,4,1,1] => 0 [5,3,3] => 6 [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] => 3 [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] => 91 [11,1] => 143 [10,2] => 162 [10,1,1] => 55 [9,3] => 154 [9,2,1] => 80 [9,1,1,1] => 0 [8,4] => 125 [8,3,1] => 81 [8,2,2] => 28 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 81 [7,4,1] => 64 [7,3,2] => 35 [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] => 28 [6,5,1] => 35 [6,4,2] => 27 [6,4,1,1] => 0 [6,3,3] => 10 [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] => 10 [5,5,1,1] => 0 [5,4,3] => 8 [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] => 1 [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 [5,4,3,1] => 0 [5,4,2,2] => 0 [5,4,2,1,1] => 0 [5,3,3,2] => 0 [5,3,3,1,1] => 0 [5,3,2,2,1] => 0 [4,4,3,2] => 0 [4,4,3,1,1] => 0 [4,4,2,2,1] => 0 [4,3,3,2,1] => 0 [5,4,3,2] => 0 [5,4,3,1,1] => 0 [5,4,2,2,1] => 0 [5,3,3,2,1] => 0 [4,4,3,2,1] => 0 [5,4,3,2,1] => 0 ----------------------------------------------------------------------------- Created: Mar 21, 2017 at 07:34 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Apr 26, 2018 at 07:24 by Martin Rubey