***************************************************************************** * 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: St000927 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The alternating sum of the coefficients of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. ----------------------------------------------------------------------------- References: [1] Garsia, A. M., Goupil, A. Character polynomials, their $q$-analogs and the Kronecker product [[MathSciNet:2576382]] ----------------------------------------------------------------------------- Code: def statistic(L): return L.character_polynomial()(*[-1]*sum(L)) ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 4 [3] => -1 [2,1] => -4 [1,1,1] => -7 [4] => 2 [3,1] => 4 [2,2] => 6 [2,1,1] => 9 [1,1,1,1] => 12 [5] => -2 [4,1] => -7 [3,2] => -10 [3,1,1] => -11 [2,2,1] => -17 [2,1,1,1] => -19 [1,1,1,1,1] => -19 [6] => 2 [5,1] => 9 [4,2] => 14 [4,1,1] => 22 [3,3] => 15 [3,2,1] => 36 [3,1,1,1] => 26 [2,2,2] => 15 [2,2,1,1] => 42 [2,1,1,1,1] => 34 [1,1,1,1,1,1] => 30 [7] => -2 [6,1] => -11 [5,2] => -22 [5,1,1] => -32 [4,3] => -29 [4,2,1] => -70 [4,1,1,1] => -53 [3,3,1] => -56 [3,2,2] => -48 [3,2,1,1] => -102 [3,1,1,1,1] => -53 [2,2,2,1] => -58 [2,2,1,1,1] => -83 [2,1,1,1,1,1] => -60 [1,1,1,1,1,1,1] => -45 [8] => 3 [7,1] => 12 [6,2] => 36 [6,1,1] => 41 [5,3] => 46 [5,2,1] => 125 [5,1,1,1] => 87 [4,4] => 39 [4,3,1] => 150 [4,2,2] => 127 [4,2,1,1] => 218 [4,1,1,1,1] => 116 [3,3,2] => 98 [3,3,1,1] => 174 [3,2,2,1] => 202 [3,2,1,1,1] => 233 [3,1,1,1,1,1] => 101 [2,2,2,2] => 68 [2,2,2,1,1] => 138 [2,2,1,1,1,1] => 158 [2,1,1,1,1,1,1] => 98 [1,1,1,1,1,1,1,1] => 67 [9] => -4 [8,1] => -15 [7,2] => -48 [7,1,1] => -52 [6,3] => -85 [6,2,1] => -202 [6,1,1,1] => -127 [5,4] => -84 [5,3,1] => -314 [5,2,2] => -260 [5,2,1,1] => -430 [5,1,1,1,1] => -206 [4,4,1] => -200 [4,3,2] => -371 [4,3,1,1] => -537 [4,2,2,1] => -552 [4,2,1,1,1] => -560 [4,1,1,1,1,1] => -231 [3,3,3] => -98 [3,3,2,1] => -472 [3,3,1,1,1] => -430 [3,2,2,2] => -274 [3,2,2,1,1] => -560 [3,2,1,1,1,1] => -484 [3,1,1,1,1,1,1] => -181 [2,2,2,2,1] => -212 [2,2,2,1,1,1] => -300 [2,2,1,1,1,1,1] => -275 [2,1,1,1,1,1,1,1] => -157 [1,1,1,1,1,1,1,1,1] => -97 [10] => 4 [9,1] => 20 [8,2] => 58 [8,1,1] => 71 [7,3] => 140 [7,2,1] => 300 [7,1,1,1] => 176 [6,4] => 157 [6,3,1] => 609 [6,2,2] => 447 [6,2,1,1] => 768 [6,1,1,1,1] => 326 [5,5] => 99 [5,4,1] => 594 [5,3,2] => 956 [5,3,1,1] => 1259 [5,2,2,1] => 1254 [5,2,1,1,1] => 1197 [5,1,1,1,1,1] => 446 [4,4,2] => 548 [4,4,1,1] => 775 [4,3,3] => 485 [4,3,2,1] => 1924 [4,3,1,1,1] => 1514 [4,2,2,2] => 803 [4,2,2,1,1] => 1696 [4,2,1,1,1,1] => 1256 [4,1,1,1,1,1,1] => 436 [3,3,3,1] => 554 [3,3,2,2] => 778 [3,3,2,1,1] => 1452 [3,3,1,1,1,1] => 960 [3,2,2,2,1] => 1041 [3,2,2,1,1,1] => 1322 [3,2,1,1,1,1,1] => 927 [3,1,1,1,1,1,1,1] => 309 [2,2,2,2,2] => 196 [2,2,2,2,1,1] => 539 [2,2,2,1,1,1,1] => 574 [2,2,1,1,1,1,1,1] => 465 [2,1,1,1,1,1,1,1,1] => 242 [1,1,1,1,1,1,1,1,1,1] => 139 [11] => -4 [10,1] => -24 [9,2] => -76 [9,1,1] => -94 [8,3] => -196 [8,2,1] => -428 [8,1,1,1] => -249 [7,4] => -299 [7,3,1] => -1056 [7,2,2] => -739 [7,2,1,1] => -1244 [7,1,1,1,1] => -497 [6,5] => -252 [6,4,1] => -1350 [6,3,2] => -2012 [6,3,1,1] => -2647 [6,2,2,1] => -2462 [6,2,1,1,1] => -2296 [6,1,1,1,1,1] => -765 [5,5,1] => -712 [5,4,2] => -2089 [5,4,1,1] => -2624 [5,3,3] => -1457 [5,3,2,1] => -5397 [5,3,1,1,1] => -3937 [5,2,2,2] => -2046 [5,2,2,1,1] => -4173 [5,2,1,1,1,1] => -2902 [5,1,1,1,1,1,1] => -895 [4,4,3] => -1026 [4,4,2,1] => -3240 [4,4,1,1,1] => -2338 [4,3,3,1] => -2970 [4,3,2,2] => -3503 [4,3,2,1,1] => -6576 [4,3,1,1,1,1] => -3711 [4,2,2,2,1] => -3530 [4,2,2,1,1,1] => -4292 [4,2,1,1,1,1,1] => -2598 [4,1,1,1,1,1,1,1] => -777 [3,3,3,2] => -1338 [3,3,3,1,1] => -1977 [3,3,2,2,1] => -3299 [3,3,2,1,1,1] => -3735 [3,3,1,1,1,1,1] => -1951 [3,2,2,2,2] => -1218 [3,2,2,2,1,1] => -2898 [3,2,2,1,1,1,1] => -2786 [3,2,1,1,1,1,1,1] => -1682 [3,1,1,1,1,1,1,1,1] => -509 [2,2,2,2,2,1] => -738 [2,2,2,2,1,1,1] => -1141 [2,2,2,1,1,1,1,1] => -1047 [2,2,1,1,1,1,1,1,1] => -751 [2,1,1,1,1,1,1,1,1,1] => -367 [1,1,1,1,1,1,1,1,1,1,1] => -195 [12] => 5 [11,1] => 27 [10,2] => 104 [10,1,1] => 115 [9,3] => 268 [9,2,1] => 596 [9,1,1,1] => 351 [8,4] => 507 [8,3,1] => 1670 [8,2,2] => 1195 [8,2,1,1] => 1904 [8,1,1,1,1] => 757 [7,5] => 531 [7,4,1] => 2715 [7,3,2] => 3782 [7,3,1,1] => 4979 [7,2,2,1] => 4434 [7,2,1,1,1] => 4033 [7,1,1,1,1,1] => 1250 [6,6] => 286 [6,5,1] => 2296 [6,4,2] => 5503 [6,4,1,1] => 6567 [6,3,3] => 3443 [6,3,2,1] => 12517 [6,3,1,1,1] => 8914 [6,2,2,2] => 4561 [6,2,2,1,1] => 8883 [6,2,1,1,1,1] => 5986 [6,1,1,1,1,1,1] => 1642 [5,5,2] => 2760 [5,5,1,1] => 3414 [5,4,3] => 4564 [5,4,2,1] => 13338 [5,4,1,1,1] => 8883 [5,3,3,1] => 9874 [5,3,2,2] => 10886 [5,3,2,1,1] => 20110 [5,3,1,1,1,1] => 10472 [5,2,2,2,1] => 9745 [5,2,2,1,1,1] => 11418 [5,2,1,1,1,1,1] => 6402 [5,1,1,1,1,1,1,1] => 1696 [4,4,4] => 1075 [4,4,3,1] => 7188 [4,4,2,2] => 6853 [4,4,2,1,1] => 12097 [4,4,1,1,1,1] => 6171 [4,3,3,2] => 7765 [4,3,3,1,1] => 11574 [4,3,2,2,1] => 16902 [4,3,2,1,1,1] => 18289 [4,3,1,1,1,1,1] => 8222 [4,2,2,2,2] => 4809 [4,2,2,2,1,1] => 10659 [4,2,2,1,1,1,1] => 9729 [4,2,1,1,1,1,1,1] => 5014 [4,1,1,1,1,1,1,1,1] => 1337 [3,3,3,3] => 1408 [3,3,3,2,1] => 6596 [3,3,3,1,1,1] => 5650 [3,3,2,2,2] => 4467 [3,3,2,2,1,1] => 10034 [3,3,2,1,1,1,1] => 8458 [3,3,1,1,1,1,1,1] => 3749 [3,2,2,2,2,1] => 4824 [3,2,2,2,1,1,1] => 6810 [3,2,2,1,1,1,1,1] => 5454 [3,2,1,1,1,1,1,1,1] => 2911 [3,1,1,1,1,1,1,1,1,1] => 814 [2,2,2,2,2,2] => 791 [2,2,2,2,2,1,1] => 1859 [2,2,2,2,1,1,1,1] => 2251 [2,2,2,1,1,1,1,1,1] => 1804 [2,2,1,1,1,1,1,1,1,1] => 1187 [2,1,1,1,1,1,1,1,1,1,1] => 541 [1,1,1,1,1,1,1,1,1,1,1,1] => 272 ----------------------------------------------------------------------------- Created: Aug 07, 2017 at 14:00 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Aug 07, 2017 at 14:00 by Christian Stump