***************************************************************************** * 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: St000620 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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 odd. The case of an even minimum is [[St000621]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(L): n = sum(L) return sum( 1 for SYT in StandardTableaux(L) if is_odd(min( SYT.standard_descents() + [n] )) ) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 1 [3] => 1 [2,1] => 1 [1,1,1] => 1 [4] => 0 [3,1] => 2 [2,2] => 1 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 1 [4,1] => 2 [3,2] => 3 [3,1,1] => 4 [2,2,1] => 3 [2,1,1,1] => 3 [1,1,1,1,1] => 1 [6] => 0 [5,1] => 3 [4,2] => 5 [4,1,1] => 6 [3,3] => 3 [3,2,1] => 10 [3,1,1,1] => 7 [2,2,2] => 3 [2,2,1,1] => 6 [2,1,1,1,1] => 4 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 3 [5,2] => 8 [5,1,1] => 9 [4,3] => 8 [4,2,1] => 21 [4,1,1,1] => 13 [3,3,1] => 13 [3,2,2] => 13 [3,2,1,1] => 23 [3,1,1,1,1] => 11 [2,2,2,1] => 9 [2,2,1,1,1] => 10 [2,1,1,1,1,1] => 5 [1,1,1,1,1,1,1] => 1 [8] => 0 [7,1] => 4 [6,2] => 11 [6,1,1] => 12 [5,3] => 16 [5,2,1] => 38 [5,1,1,1] => 22 [4,4] => 8 [4,3,1] => 42 [4,2,2] => 34 [4,2,1,1] => 57 [4,1,1,1,1] => 24 [3,3,2] => 26 [3,3,1,1] => 36 [3,2,2,1] => 45 [3,2,1,1,1] => 44 [3,1,1,1,1,1] => 16 [2,2,2,2] => 9 [2,2,2,1,1] => 19 [2,2,1,1,1,1] => 15 [2,1,1,1,1,1,1] => 6 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 4 [7,2] => 15 [7,1,1] => 16 [6,3] => 27 [6,2,1] => 61 [6,1,1,1] => 34 [5,4] => 24 [5,3,1] => 96 [5,2,2] => 72 [5,2,1,1] => 117 [5,1,1,1,1] => 46 [4,4,1] => 50 [4,3,2] => 102 [4,3,1,1] => 135 [4,2,2,1] => 136 [4,2,1,1,1] => 125 [4,1,1,1,1,1] => 40 [3,3,3] => 26 [3,3,2,1] => 107 [3,3,1,1,1] => 80 [3,2,2,2] => 54 [3,2,2,1,1] => 108 [3,2,1,1,1,1] => 75 [3,1,1,1,1,1,1] => 22 [2,2,2,2,1] => 28 [2,2,2,1,1,1] => 34 [2,2,1,1,1,1,1] => 21 [2,1,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1,1] => 1 [10] => 0 [9,1] => 5 [8,2] => 19 [8,1,1] => 20 [7,3] => 42 [7,2,1] => 92 [7,1,1,1] => 50 [6,4] => 51 [6,3,1] => 184 [6,2,2] => 133 [6,2,1,1] => 212 [6,1,1,1,1] => 80 [5,5] => 24 [5,4,1] => 170 [5,3,2] => 270 [5,3,1,1] => 348 [5,2,2,1] => 325 [5,2,1,1,1] => 288 [5,1,1,1,1,1] => 86 [4,4,2] => 152 [4,4,1,1] => 185 [4,3,3] => 128 [4,3,2,1] => 480 [4,3,1,1,1] => 340 [4,2,2,2] => 190 [4,2,2,1,1] => 369 [4,2,1,1,1,1] => 240 [4,1,1,1,1,1,1] => 62 [3,3,3,1] => 133 [3,3,2,2] => 161 [3,3,2,1,1] => 295 [3,3,1,1,1,1] => 155 [3,2,2,2,1] => 190 [3,2,2,1,1,1] => 217 [3,2,1,1,1,1,1] => 118 [3,1,1,1,1,1,1,1] => 29 [2,2,2,2,2] => 28 [2,2,2,2,1,1] => 62 [2,2,2,1,1,1,1] => 55 [2,2,1,1,1,1,1,1] => 28 [2,1,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 5 [9,2] => 24 [9,1,1] => 25 [8,3] => 61 [8,2,1] => 131 [8,1,1,1] => 70 [7,4] => 93 [7,3,1] => 318 [7,2,2] => 225 [7,2,1,1] => 354 [7,1,1,1,1] => 130 [6,5] => 75 [6,4,1] => 405 [6,3,2] => 587 [6,3,1,1] => 744 [6,2,2,1] => 670 [6,2,1,1,1] => 580 [6,1,1,1,1,1] => 166 [5,5,1] => 194 [5,4,2] => 592 [5,4,1,1] => 703 [5,3,3] => 398 [5,3,2,1] => 1423 [5,3,1,1,1] => 976 [5,2,2,2] => 515 [5,2,2,1,1] => 982 [5,2,1,1,1,1] => 614 [5,1,1,1,1,1,1] => 148 [4,4,3] => 280 [4,4,2,1] => 817 [4,4,1,1,1] => 525 [4,3,3,1] => 741 [4,3,2,2] => 831 [4,3,2,1,1] => 1484 [4,3,1,1,1,1] => 735 [4,2,2,2,1] => 749 [4,2,2,1,1,1] => 826 [4,2,1,1,1,1,1] => 420 [4,1,1,1,1,1,1,1] => 91 [3,3,3,2] => 294 [3,3,3,1,1] => 428 [3,3,2,2,1] => 646 [3,3,2,1,1,1] => 667 [3,3,1,1,1,1,1] => 273 [3,2,2,2,2] => 218 [3,2,2,2,1,1] => 469 [3,2,2,1,1,1,1] => 390 [3,2,1,1,1,1,1,1] => 175 [3,1,1,1,1,1,1,1,1] => 37 [2,2,2,2,2,1] => 90 [2,2,2,2,1,1,1] => 117 [2,2,2,1,1,1,1,1] => 83 [2,2,1,1,1,1,1,1,1] => 36 [2,1,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 0 [11,1] => 6 [10,2] => 29 [10,1,1] => 30 [9,3] => 85 [9,2,1] => 180 [9,1,1,1] => 95 [8,4] => 154 [8,3,1] => 510 [8,2,2] => 356 [8,2,1,1] => 555 [8,1,1,1,1] => 200 [7,5] => 168 [7,4,1] => 816 [7,3,2] => 1130 [7,3,1,1] => 1416 [7,2,2,1] => 1249 [7,2,1,1,1] => 1064 [7,1,1,1,1,1] => 296 [6,6] => 75 [6,5,1] => 674 [6,4,2] => 1584 [6,4,1,1] => 1852 [6,3,3] => 985 [6,3,2,1] => 3424 [6,3,1,1,1] => 2300 [6,2,2,2] => 1185 [6,2,2,1,1] => 2232 [6,2,1,1,1,1] => 1360 [6,1,1,1,1,1,1] => 314 [5,5,2] => 786 [5,5,1,1] => 897 [5,4,3] => 1270 [5,4,2,1] => 3535 [5,4,1,1,1] => 2204 [5,3,3,1] => 2562 [5,3,2,2] => 2769 [5,3,2,1,1] => 4865 [5,3,1,1,1,1] => 2325 [5,2,2,2,1] => 2246 [5,2,2,1,1,1] => 2422 [5,2,1,1,1,1,1] => 1182 [5,1,1,1,1,1,1,1] => 239 [4,4,4] => 280 [4,4,3,1] => 1838 [4,4,2,2] => 1648 [4,4,2,1,1] => 2826 [4,4,1,1,1,1] => 1260 [4,3,3,2] => 1866 [4,3,3,1,1] => 2653 [4,3,2,2,1] => 3710 [4,3,2,1,1,1] => 3712 [4,3,1,1,1,1,1] => 1428 [4,2,2,2,2] => 967 [4,2,2,2,1,1] => 2044 [4,2,2,1,1,1,1] => 1636 [4,2,1,1,1,1,1,1] => 686 [4,1,1,1,1,1,1,1,1] => 128 [3,3,3,3] => 294 [3,3,3,2,1] => 1368 [3,3,3,1,1,1] => 1095 [3,3,2,2,2] => 864 [3,3,2,2,1,1] => 1782 [3,3,2,1,1,1,1] => 1330 [3,3,1,1,1,1,1,1] => 448 [3,2,2,2,2,1] => 777 [3,2,2,2,1,1,1] => 976 [3,2,2,1,1,1,1,1] => 648 [3,2,1,1,1,1,1,1,1] => 248 [3,1,1,1,1,1,1,1,1,1] => 46 [2,2,2,2,2,2] => 90 [2,2,2,2,2,1,1] => 207 [2,2,2,2,1,1,1,1] => 200 [2,2,2,1,1,1,1,1,1] => 119 [2,2,1,1,1,1,1,1,1,1] => 45 [2,1,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Oct 12, 2016 at 15:26 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 12, 2016 at 15:34 by Christian Stump