***************************************************************************** * 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: St000621 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even. This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1]. The case of an odd minimum is [[St000620]]. ----------------------------------------------------------------------------- References: [1] Reiner, V., Webb, P. The combinatorics of the bar resolution in group cohomology [[MathSciNet:2043333]] [2] Athanasiadis, C. A. The symmetric group action on rank-selected posets of injective words [[arXiv:1606.03829]] ----------------------------------------------------------------------------- Code: def statistic(L): n = sum(L) return sum(1 for SYT in StandardTableaux(L) if is_even(min( SYT.standard_descents() + [n] )) ) ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 0 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 1 [3,1] => 1 [2,2] => 1 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 0 [4,1] => 2 [3,2] => 2 [3,1,1] => 2 [2,2,1] => 2 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 1 [5,1] => 2 [4,2] => 4 [4,1,1] => 4 [3,3] => 2 [3,2,1] => 6 [3,1,1,1] => 3 [2,2,2] => 2 [2,2,1,1] => 3 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 3 [5,2] => 6 [5,1,1] => 6 [4,3] => 6 [4,2,1] => 14 [4,1,1,1] => 7 [3,3,1] => 8 [3,2,2] => 8 [3,2,1,1] => 12 [3,1,1,1,1] => 4 [2,2,2,1] => 5 [2,2,1,1,1] => 4 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 0 [8] => 1 [7,1] => 3 [6,2] => 9 [6,1,1] => 9 [5,3] => 12 [5,2,1] => 26 [5,1,1,1] => 13 [4,4] => 6 [4,3,1] => 28 [4,2,2] => 22 [4,2,1,1] => 33 [4,1,1,1,1] => 11 [3,3,2] => 16 [3,3,1,1] => 20 [3,2,2,1] => 25 [3,2,1,1,1] => 20 [3,1,1,1,1,1] => 5 [2,2,2,2] => 5 [2,2,2,1,1] => 9 [2,2,1,1,1,1] => 5 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 4 [7,2] => 12 [7,1,1] => 12 [6,3] => 21 [6,2,1] => 44 [6,1,1,1] => 22 [5,4] => 18 [5,3,1] => 66 [5,2,2] => 48 [5,2,1,1] => 72 [5,1,1,1,1] => 24 [4,4,1] => 34 [4,3,2] => 66 [4,3,1,1] => 81 [4,2,2,1] => 80 [4,2,1,1,1] => 64 [4,1,1,1,1,1] => 16 [3,3,3] => 16 [3,3,2,1] => 61 [3,3,1,1,1] => 40 [3,2,2,2] => 30 [3,2,2,1,1] => 54 [3,2,1,1,1,1] => 30 [3,1,1,1,1,1,1] => 6 [2,2,2,2,1] => 14 [2,2,2,1,1,1] => 14 [2,2,1,1,1,1,1] => 6 [2,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1] => 0 [10] => 1 [9,1] => 4 [8,2] => 16 [8,1,1] => 16 [7,3] => 33 [7,2,1] => 68 [7,1,1,1] => 34 [6,4] => 39 [6,3,1] => 131 [6,2,2] => 92 [6,2,1,1] => 138 [6,1,1,1,1] => 46 [5,5] => 18 [5,4,1] => 118 [5,3,2] => 180 [5,3,1,1] => 219 [5,2,2,1] => 200 [5,2,1,1,1] => 160 [5,1,1,1,1,1] => 40 [4,4,2] => 100 [4,4,1,1] => 115 [4,3,3] => 82 [4,3,2,1] => 288 [4,3,1,1,1] => 185 [4,2,2,2] => 110 [4,2,2,1,1] => 198 [4,2,1,1,1,1] => 110 [4,1,1,1,1,1,1] => 22 [3,3,3,1] => 77 [3,3,2,2] => 91 [3,3,2,1,1] => 155 [3,3,1,1,1,1] => 70 [3,2,2,2,1] => 98 [3,2,2,1,1,1] => 98 [3,2,1,1,1,1,1] => 42 [3,1,1,1,1,1,1,1] => 7 [2,2,2,2,2] => 14 [2,2,2,2,1,1] => 28 [2,2,2,1,1,1,1] => 20 [2,2,1,1,1,1,1,1] => 7 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 0 [11] => 0 [10,1] => 5 [9,2] => 20 [9,1,1] => 20 [8,3] => 49 [8,2,1] => 100 [8,1,1,1] => 50 [7,4] => 72 [7,3,1] => 232 [7,2,2] => 160 [7,2,1,1] => 240 [7,1,1,1,1] => 80 [6,5] => 57 [6,4,1] => 288 [6,3,2] => 403 [6,3,1,1] => 488 [6,2,2,1] => 430 [6,2,1,1,1] => 344 [6,1,1,1,1,1] => 86 [5,5,1] => 136 [5,4,2] => 398 [5,4,1,1] => 452 [5,3,3] => 262 [5,3,2,1] => 887 [5,3,1,1,1] => 564 [5,2,2,2] => 310 [5,2,2,1,1] => 558 [5,2,1,1,1,1] => 310 [5,1,1,1,1,1,1] => 62 [4,4,3] => 182 [4,4,2,1] => 503 [4,4,1,1,1] => 300 [4,3,3,1] => 447 [4,3,2,2] => 489 [4,3,2,1,1] => 826 [4,3,1,1,1,1] => 365 [4,2,2,2,1] => 406 [4,2,2,1,1,1] => 406 [4,2,1,1,1,1,1] => 174 [4,1,1,1,1,1,1,1] => 29 [3,3,3,2] => 168 [3,3,3,1,1] => 232 [3,3,2,2,1] => 344 [3,3,2,1,1,1] => 323 [3,3,1,1,1,1,1] => 112 [3,2,2,2,2] => 112 [3,2,2,2,1,1] => 224 [3,2,2,1,1,1,1] => 160 [3,2,1,1,1,1,1,1] => 56 [3,1,1,1,1,1,1,1,1] => 8 [2,2,2,2,2,1] => 42 [2,2,2,2,1,1,1] => 48 [2,2,2,1,1,1,1,1] => 27 [2,2,1,1,1,1,1,1,1] => 8 [2,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 1 [11,1] => 5 [10,2] => 25 [10,1,1] => 25 [9,3] => 69 [9,2,1] => 140 [9,1,1,1] => 70 [8,4] => 121 [8,3,1] => 381 [8,2,2] => 260 [8,2,1,1] => 390 [8,1,1,1,1] => 130 [7,5] => 129 [7,4,1] => 592 [7,3,2] => 795 [7,3,1,1] => 960 [7,2,2,1] => 830 [7,2,1,1,1] => 664 [7,1,1,1,1,1] => 166 [6,6] => 57 [6,5,1] => 481 [6,4,2] => 1089 [6,4,1,1] => 1228 [6,3,3] => 665 [6,3,2,1] => 2208 [6,3,1,1,1] => 1396 [6,2,2,2] => 740 [6,2,2,1,1] => 1332 [6,2,1,1,1,1] => 740 [6,1,1,1,1,1,1] => 148 [5,5,2] => 534 [5,5,1,1] => 588 [5,4,3] => 842 [5,4,2,1] => 2240 [5,4,1,1,1] => 1316 [5,3,3,1] => 1596 [5,3,2,2] => 1686 [5,3,2,1,1] => 2835 [5,3,1,1,1,1] => 1239 [5,2,2,2,1] => 1274 [5,2,2,1,1,1] => 1274 [5,2,1,1,1,1,1] => 546 [5,1,1,1,1,1,1,1] => 91 [4,4,4] => 182 [4,4,3,1] => 1132 [4,4,2,2] => 992 [4,4,2,1,1] => 1629 [4,4,1,1,1,1] => 665 [4,3,3,2] => 1104 [4,3,3,1,1] => 1505 [4,3,2,2,1] => 2065 [4,3,2,1,1,1] => 1920 [4,3,1,1,1,1,1] => 651 [4,2,2,2,2] => 518 [4,2,2,2,1,1] => 1036 [4,2,2,1,1,1,1] => 740 [4,2,1,1,1,1,1,1] => 259 [4,1,1,1,1,1,1,1,1] => 37 [3,3,3,3] => 168 [3,3,3,2,1] => 744 [3,3,3,1,1,1] => 555 [3,3,2,2,2] => 456 [3,3,2,2,1,1] => 891 [3,3,2,1,1,1,1] => 595 [3,3,1,1,1,1,1,1] => 168 [3,2,2,2,2,1] => 378 [3,2,2,2,1,1,1] => 432 [3,2,2,1,1,1,1,1] => 243 [3,2,1,1,1,1,1,1,1] => 72 [3,1,1,1,1,1,1,1,1,1] => 9 [2,2,2,2,2,2] => 42 [2,2,2,2,2,1,1] => 90 [2,2,2,2,1,1,1,1] => 75 [2,2,2,1,1,1,1,1,1] => 35 [2,2,1,1,1,1,1,1,1,1] => 9 [2,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Oct 12, 2016 at 15:26 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 12, 2016 at 15:35 by Christian Stump