***************************************************************************** * 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: St000813 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions. ----------------------------------------------------------------------------- References: [1] Number of different 0-1 matrices in which the number of 1's is n, with at least one 1 in each row and column. [[OEIS:A068313]] ----------------------------------------------------------------------------- Code: def statistic(mu): m = SymmetricFunctions(ZZ).m() e = SymmetricFunctions(ZZ).e() return sum(coeff for _, coeff in m(e(mu))) ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 3 [3] => 1 [2,1] => 4 [1,1,1] => 10 [4] => 1 [3,1] => 5 [2,2] => 9 [2,1,1] => 20 [1,1,1,1] => 47 [5] => 1 [4,1] => 6 [3,2] => 14 [3,1,1] => 30 [2,2,1] => 50 [2,1,1,1] => 110 [1,1,1,1,1] => 246 [6] => 1 [5,1] => 7 [4,2] => 20 [4,1,1] => 42 [3,3] => 29 [3,2,1] => 97 [3,1,1,1] => 206 [2,2,2] => 157 [2,2,1,1] => 338 [2,1,1,1,1] => 732 [1,1,1,1,1,1] => 1602 [7] => 1 [6,1] => 8 [5,2] => 27 [5,1,1] => 56 [4,3] => 49 [4,2,1] => 159 [4,1,1,1] => 332 [3,3,1] => 224 [3,2,2] => 353 [3,2,1,1] => 743 [3,1,1,1,1] => 1568 [2,2,2,1] => 1184 [2,2,1,1,1] => 2513 [2,1,1,1,1,1] => 5357 [1,1,1,1,1,1,1] => 11481 [8] => 1 [7,1] => 9 [6,2] => 35 [6,1,1] => 72 [5,3] => 76 [5,2,1] => 242 [5,1,1,1] => 500 [4,4] => 99 [4,3,1] => 436 [4,2,2] => 677 [4,2,1,1] => 1405 [4,1,1,1,1] => 2920 [3,3,2] => 943 [3,3,1,1] => 1965 [3,2,2,1] => 3078 [3,2,1,1,1] => 6437 [3,1,1,1,1,1] => 13487 [2,2,2,2] => 4845 [2,2,2,1,1] => 10163 [2,2,1,1,1,1] => 21378 [2,1,1,1,1,1,1] => 45104 [1,1,1,1,1,1,1,1] => 95503 [9] => 1 [8,1] => 10 [7,2] => 44 [7,1,1] => 90 [6,3] => 111 [6,2,1] => 349 [6,1,1,1] => 716 [5,4] => 175 [5,3,1] => 754 [5,2,2] => 1161 [5,2,1,1] => 2389 [5,1,1,1,1] => 4920 [4,4,1] => 972 [4,3,2] => 2059 [4,3,1,1] => 4249 [4,2,2,1] => 6577 [4,2,1,1,1] => 13599 [4,1,1,1,1,1] => 28147 [3,3,3] => 2836 [3,3,2,1] => 9095 [3,3,1,1,1] => 18843 [3,2,2,2] => 14139 [3,2,2,1,1] => 29338 [3,2,1,1,1,1] => 60962 [3,1,1,1,1,1,1] => 126866 [2,2,2,2,1] => 45796 [2,2,2,1,1,1] => 95359 [2,2,1,1,1,1,1] => 198920 [2,1,1,1,1,1,1,1] => 415781 [1,1,1,1,1,1,1,1,1] => 871030 [10] => 1 [9,1] => 11 [8,2] => 54 [8,1,1] => 110 [7,3] => 155 [7,2,1] => 483 [7,1,1,1] => 986 [6,4] => 286 [6,3,1] => 1214 [6,2,2] => 1859 [6,2,1,1] => 3803 [6,1,1,1,1] => 7784 [5,5] => 351 [5,4,1] => 1906 [5,3,2] => 3984 [5,3,1,1] => 8166 [5,2,2,1] => 12546 [5,2,1,1,1] => 25746 [5,1,1,1,1,1] => 52867 [4,4,2] => 5111 [4,4,1,1] => 10490 [4,3,3] => 6986 [4,3,2,1] => 22101 [4,3,1,1,1] => 45457 [4,2,2,2] => 34071 [4,2,2,1,1] => 70132 [4,2,1,1,1,1] => 144471 [4,1,1,1,1,1,1] => 297848 [3,3,3,1] => 30326 [3,3,2,2] => 46830 [3,3,2,1,1] => 96519 [3,3,1,1,1,1] => 199122 [3,2,2,2,1] => 149384 [3,2,2,1,1,1] => 308537 [3,2,1,1,1,1,1] => 637937 [3,1,1,1,1,1,1,1] => 1320510 [2,2,2,2,2] => 231571 [2,2,2,2,1,1] => 478940 [2,2,2,1,1,1,1] => 991743 [2,2,1,1,1,1,1,1] => 2056300 [2,1,1,1,1,1,1,1,1] => 4269680 [1,1,1,1,1,1,1,1,1,1] => 8879558 [11] => 1 [10,1] => 12 [9,2] => 65 [9,1,1] => 132 [8,3] => 209 [8,2,1] => 647 [8,1,1,1] => 1316 [7,4] => 441 [7,3,1] => 1852 [7,2,2] => 2825 [7,2,1,1] => 5755 [7,1,1,1,1] => 11728 [6,5] => 637 [6,4,1] => 3406 [6,3,2] => 7057 [6,3,1,1] => 14397 [6,2,2,1] => 22011 [6,2,1,1,1] => 44939 [6,1,1,1,1,1] => 91787 [5,5,1] => 4164 [5,4,2] => 11005 [5,4,1,1] => 22477 [5,3,3] => 14960 [5,3,2,1] => 46819 [5,3,1,1,1] => 95749 [5,2,2,2] => 71745 [5,2,2,1,1] => 146794 [5,2,1,1,1,1] => 300490 [5,1,1,1,1,1,1] => 615410 [4,4,3] => 19096 [4,4,2,1] => 59855 [4,4,1,1,1] => 122503 [4,3,3,1] => 81601 [4,3,2,2] => 125256 [4,3,2,1,1] => 256659 [4,3,1,1,1,1] => 526216 [4,2,2,2,1] => 394463 [4,2,2,1,1,1] => 809266 [4,2,1,1,1,1,1] => 1661295 [4,1,1,1,1,1,1,1] => 3412586 [3,3,3,2] => 171090 [3,3,3,1,1] => 350840 [3,3,2,2,1] => 539778 [3,3,2,1,1,1] => 1108391 [3,3,1,1,1,1,1] => 2277586 [3,2,2,2,2] => 831191 [3,2,2,2,1,1] => 1708151 [3,2,2,1,1,1,1] => 3512988 [3,2,1,1,1,1,1,1] => 7230549 [3,1,1,1,1,1,1,1,1] => 14894660 [2,2,2,2,2,1] => 2635257 [2,2,2,2,1,1,1] => 5424814 [2,2,2,1,1,1,1,1] => 11176952 [2,2,1,1,1,1,1,1,1] => 23049690 [2,1,1,1,1,1,1,1,1,1] => 47581727 [1,1,1,1,1,1,1,1,1,1,1] => 98329551 [12] => 1 [11,1] => 13 [10,2] => 77 [10,1,1] => 156 [9,3] => 274 [9,2,1] => 844 [9,1,1,1] => 1712 [8,4] => 650 [8,3,1] => 2708 [8,2,2] => 4119 [8,2,1,1] => 8365 [8,1,1,1,1] => 16992 [7,5] => 1078 [7,4,1] => 5699 [7,3,2] => 11734 [7,3,1,1] => 23856 [7,2,2,1] => 36346 [7,2,1,1,1] => 73932 [7,1,1,1,1,1] => 150427 [6,6] => 1275 [6,5,1] => 8213 [6,4,2] => 21483 [6,4,1,1] => 43716 [6,3,3] => 29094 [6,3,2,1] => 90344 [6,3,1,1,1] => 183999 [6,2,2,2] => 137868 [6,2,2,1,1] => 280874 [6,2,1,1,1,1] => 572393 [6,1,1,1,1,1,1] => 1166852 [5,5,2] => 26205 [5,5,1,1] => 53351 [5,4,3] => 45124 [5,4,2,1] => 140381 [5,4,1,1,1] => 286170 [5,3,3,1] => 190523 [5,3,2,2] => 291162 [5,3,2,1,1] => 593962 [5,3,1,1,1,1] => 1212100 [5,2,2,2,1] => 908387 [5,2,2,1,1,1] => 1854456 [5,2,1,1,1,1,1] => 3787285 [5,1,1,1,1,1,1,1] => 7737676 [4,4,4] => 57385 [4,4,3,1] => 242689 [4,4,2,2] => 371122 [4,4,2,1,1] => 757490 [4,4,1,1,1,1] => 1546726 [4,3,3,2] => 504587 [4,3,3,1,1] => 1030376 [4,3,2,2,1] => 1577875 [4,3,2,1,1,1] => 3224913 [4,3,1,1,1,1,1] => 6594182 [4,2,2,2,2] => 2417525 [4,2,2,2,1,1] => 4943420 [4,2,2,1,1,1,1] => 10113232 [4,2,1,1,1,1,1,1] => 20699788 [4,1,1,1,1,1,1,1,1] => 42390004 [3,3,3,3] => 686557 [3,3,3,2,1] => 2149402 [3,3,3,1,1,1] => 4395584 [3,3,2,2,2] => 3295560 [3,3,2,2,1,1] => 6743185 [3,3,2,1,1,1,1] => 13804634 [3,3,1,1,1,1,1,1] => 28276134 [3,2,2,2,2,1] => 10351471 [3,2,2,2,1,1,1] => 21204493 [3,2,2,1,1,1,1,1] => 43461404 [3,2,1,1,1,1,1,1,1] => 89133888 [3,1,1,1,1,1,1,1,1,1] => 182919233 [2,2,2,2,2,2] => 15902253 [2,2,2,2,2,1,1] => 32596683 [2,2,2,2,1,1,1,1] => 66858586 [2,2,2,1,1,1,1,1,1] => 137222522 [2,2,1,1,1,1,1,1,1,1] => 281835366 [2,1,1,1,1,1,1,1,1,1,1] => 579278088 [1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522 ----------------------------------------------------------------------------- Created: May 20, 2017 at 17:37 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 20, 2017 at 22:45 by Martin Rubey