***************************************************************************** * 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: St001440 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. ----------------------------------------------------------------------------- References: [1] Ahlbach, C., Swanson, J. P. Cyclic sieving, necklaces, and branching rules related to Thrall's problem [[arXiv:1808.06043]] ----------------------------------------------------------------------------- Code: def statistic(P): n = P.size() return sum(Integer(1) for T in StandardTableaux(P) if T.standard_major_index() % n == 1) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 0 [3,1] => 1 [2,2] => 0 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 0 [4,1] => 1 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 1 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 1 [4,2] => 1 [4,1,1] => 2 [3,3] => 1 [3,2,1] => 3 [3,1,1,1] => 1 [2,2,2] => 0 [2,2,1,1] => 2 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 1 [5,2] => 2 [5,1,1] => 2 [4,3] => 2 [4,2,1] => 5 [4,1,1,1] => 3 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 5 [3,1,1,1,1] => 2 [2,2,2,1] => 2 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 1 [6,2] => 2 [6,1,1] => 3 [5,3] => 4 [5,2,1] => 8 [5,1,1,1] => 4 [4,4] => 1 [4,3,1] => 9 [4,2,2] => 6 [4,2,1,1] => 12 [4,1,1,1,1] => 4 [3,3,2] => 6 [3,3,1,1] => 6 [3,2,2,1] => 9 [3,2,1,1,1] => 8 [3,1,1,1,1,1] => 3 [2,2,2,2] => 1 [2,2,2,1,1] => 4 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 1 [7,2] => 3 [7,1,1] => 3 [6,3] => 5 [6,2,1] => 12 [6,1,1,1] => 6 [5,4] => 5 [5,3,1] => 18 [5,2,2] => 13 [5,2,1,1] => 21 [5,1,1,1,1] => 8 [4,4,1] => 9 [4,3,2] => 19 [4,3,1,1] => 24 [4,2,2,1] => 24 [4,2,1,1,1] => 21 [4,1,1,1,1,1] => 6 [3,3,3] => 4 [3,3,2,1] => 19 [3,3,1,1,1] => 13 [3,2,2,2] => 9 [3,2,2,1,1] => 18 [3,2,1,1,1,1] => 12 [3,1,1,1,1,1,1] => 3 [2,2,2,2,1] => 5 [2,2,2,1,1,1] => 5 [2,2,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 1 [8,2] => 3 [8,1,1] => 4 [7,3] => 8 [7,2,1] => 16 [7,1,1,1] => 8 [6,4] => 8 [6,3,1] => 32 [6,2,2] => 21 [6,2,1,1] => 36 [6,1,1,1,1] => 12 [5,5] => 5 [5,4,1] => 29 [5,3,2] => 46 [5,3,1,1] => 55 [5,2,2,1] => 53 [5,2,1,1,1] => 45 [5,1,1,1,1,1] => 13 [4,4,2] => 23 [4,4,1,1] => 32 [4,3,3] => 22 [4,3,2,1] => 77 [4,3,1,1,1] => 52 [4,2,2,2] => 28 [4,2,2,1,1] => 58 [4,2,1,1,1,1] => 34 [4,1,1,1,1,1,1] => 9 [3,3,3,1] => 20 [3,3,2,2] => 27 [3,3,2,1,1] => 44 [3,3,1,1,1,1] => 24 [3,2,2,2,1] => 29 [3,2,2,1,1,1] => 31 [3,2,1,1,1,1,1] => 16 [3,1,1,1,1,1,1,1] => 3 [2,2,2,2,2] => 3 [2,2,2,2,1,1] => 10 [2,2,2,1,1,1,1] => 7 [2,2,1,1,1,1,1,1] => 4 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 0 [5,4,2] => 90 [5,4,1,1] => 105 [5,3,3] => 60 [5,3,2,1] => 210 [5,3,1,1,1] => 140 [5,2,2,2] => 75 [5,2,2,1,1] => 140 [4,4,3] => 42 [4,4,2,1] => 120 [4,4,1,1,1] => 75 [4,3,3,1] => 108 [4,3,2,2] => 120 [4,3,2,1,1] => 210 [4,2,2,2,1] => 105 [3,3,3,2] => 42 [3,3,3,1,1] => 60 [3,3,2,2,1] => 90 [6,4,2] => 219 [5,4,3] => 177 [5,4,2,1] => 481 [5,4,1,1,1] => 294 [5,3,3,1] => 344 [5,3,2,2] => 375 [5,3,2,1,1] => 640 [5,2,2,2,1] => 294 [4,4,3,1] => 250 [4,4,2,2] => 214 [4,4,2,1,1] => 375 [4,3,3,2] => 250 [4,3,3,1,1] => 344 [4,3,2,2,1] => 481 [3,3,3,2,1] => 177 [3,3,2,2,1,1] => 219 [5,4,3,1] => 1155 [5,4,2,2] => 990 [5,4,2,1,1] => 1650 [5,3,3,2] => 891 [5,3,3,1,1] => 1232 [5,3,2,2,1] => 1650 [4,4,3,2] => 660 [4,4,3,1,1] => 891 [4,4,2,2,1] => 990 [4,3,3,2,1] => 1155 [5,4,3,2] => 3432 [5,4,3,1,1] => 4576 [5,4,2,2,1] => 4903 [5,3,3,2,1] => 4576 [4,4,3,2,1] => 3432 [5,4,3,2,1] => 19522 ----------------------------------------------------------------------------- Created: Jul 02, 2019 at 14:58 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jul 02, 2019 at 22:27 by Martin Rubey