***************************************************************************** * 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: St000712 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of semistandard Young tableau of given shape, with entries at most 4. This is also the dimension of the corresponding irreducible representation of \$GL_4\$. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Young_tableau#Applications_in_representation_theory]] ----------------------------------------------------------------------------- Code: def statistic(mu): return SemistandardTableaux(shape=mu, max_entry=4).cardinality() ----------------------------------------------------------------------------- Statistic values: [2] => 10 [1,1] => 6 [3] => 20 [2,1] => 20 [1,1,1] => 4 [4] => 35 [3,1] => 45 [2,2] => 20 [2,1,1] => 15 [1,1,1,1] => 1 [5] => 56 [4,1] => 84 [3,2] => 60 [3,1,1] => 36 [2,2,1] => 20 [2,1,1,1] => 4 [1,1,1,1,1] => 0 [6] => 84 [5,1] => 140 [4,2] => 126 [4,1,1] => 70 [3,3] => 50 [3,2,1] => 64 [3,1,1,1] => 10 [2,2,2] => 10 [2,2,1,1] => 6 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 120 [6,1] => 216 [5,2] => 224 [5,1,1] => 120 [4,3] => 140 [4,2,1] => 140 [4,1,1,1] => 20 [3,3,1] => 60 [3,2,2] => 36 [3,2,1,1] => 20 [3,1,1,1,1] => 0 [2,2,2,1] => 4 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 0 [8] => 165 [7,1] => 315 [6,2] => 360 [6,1,1] => 189 [5,3] => 280 [5,2,1] => 256 [5,1,1,1] => 35 [4,4] => 105 [4,3,1] => 175 [4,2,2] => 84 [4,2,1,1] => 45 [4,1,1,1,1] => 0 [3,3,2] => 45 [3,3,1,1] => 20 [3,2,2,1] => 15 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 0 [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] => 220 [8,1] => 440 [7,2] => 540 [7,1,1] => 280 [6,3] => 480 [6,2,1] => 420 [6,1,1,1] => 56 [5,4] => 280 [5,3,1] => 360 [5,2,2] => 160 [5,2,1,1] => 84 [5,1,1,1,1] => 0 [4,4,1] => 140 [4,3,2] => 140 [4,3,1,1] => 60 [4,2,2,1] => 36 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 0 [3,3,3] => 20 [3,3,2,1] => 20 [3,3,1,1,1] => 0 [3,2,2,2] => 4 [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] => 286 [9,1] => 594 [8,2] => 770 [8,1,1] => 396 [7,3] => 750 [7,2,1] => 640 [7,1,1,1] => 84 [6,4] => 540 [6,3,1] => 630 [6,2,2] => 270 [6,2,1,1] => 140 [6,1,1,1,1] => 0 [5,5] => 196 [5,4,1] => 384 [5,3,2] => 300 [5,3,1,1] => 126 [5,2,2,1] => 70 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => 0 [4,4,2] => 126 [4,4,1,1] => 50 [4,3,3] => 70 [4,3,2,1] => 64 [4,3,1,1,1] => 0 [4,2,2,2] => 10 [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] => 10 [3,3,2,2] => 6 [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] => 364 [10,1] => 780 [9,2] => 1056 [9,1,1] => 540 [8,3] => 1100 [8,2,1] => 924 [8,1,1,1] => 120 [7,4] => 900 [7,3,1] => 1000 [7,2,2] => 420 [7,2,1,1] => 216 [7,1,1,1,1] => 0 [6,5] => 504 [6,4,1] => 756 [6,3,2] => 540 [6,3,1,1] => 224 [6,2,2,1] => 120 [6,2,1,1,1] => 0 [6,1,1,1,1,1] => 0 [5,5,1] => 280 [5,4,2] => 360 [5,4,1,1] => 140 [5,3,3] => 160 [5,3,2,1] => 140 [5,3,1,1,1] => 0 [5,2,2,2] => 20 [5,2,2,1,1] => 0 [5,2,1,1,1,1] => 0 [5,1,1,1,1,1,1] => 0 [4,4,3] => 84 [4,4,2,1] => 60 [4,4,1,1,1] => 0 [4,3,3,1] => 36 [4,3,2,2] => 20 [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] => 4 [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] => 455 [11,1] => 1001 [10,2] => 1404 [10,1,1] => 715 [9,3] => 1540 [9,2,1] => 1280 [9,1,1,1] => 165 [8,4] => 1375 [8,3,1] => 1485 [8,2,2] => 616 [8,2,1,1] => 315 [8,1,1,1,1] => 0 [7,5] => 945 [7,4,1] => 1280 [7,3,2] => 875 [7,3,1,1] => 360 [7,2,2,1] => 189 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 336 [6,5,1] => 735 [6,4,2] => 729 [6,4,1,1] => 280 [6,3,3] => 300 [6,3,2,1] => 256 [6,3,1,1,1] => 0 [6,2,2,2] => 35 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => 0 [5,5,2] => 280 [5,5,1,1] => 105 [5,4,3] => 256 [5,4,2,1] => 175 [5,4,1,1,1] => 0 [5,3,3,1] => 84 [5,3,2,2] => 45 [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] => 35 [4,4,3,1] => 45 [4,4,2,2] => 20 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 15 [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] => 1 [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 07:35 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 21, 2017 at 07:35 by Martin Rubey