***************************************************************************** * 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: St001602 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): s = SymmetricFunctions(QQ).s() A = CombinatorialSpecies() X = species.SingletonSpecies() E = species.SetSpecies() A.define(X*E(A)) F = species.PermutationSpecies()(A).cycle_index_series() return F.coefficient(mu.size()).scalar(s(mu)) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 3 [1,1] => 1 [3] => 7 [2,1] => 8 [1,1,1] => 4 [4] => 19 [3,1] => 33 [2,2] => 24 [2,1,1] => 27 [1,1,1,1] => 9 [5] => 47 [4,1] => 128 [3,2] => 141 [3,1,1] => 151 [2,2,1] => 119 [2,1,1,1] => 85 [1,1,1,1,1] => 20 [6] => 130 [5,1] => 441 [4,2] => 699 [4,1,1] => 695 [3,3] => 344 [3,2,1] => 1016 [3,1,1,1] => 587 [2,2,2] => 309 [2,2,1,1] => 486 [2,1,1,1,1] => 254 [1,1,1,1,1,1] => 45 [7] => 343 [6,1] => 1495 [5,2] => 2952 [5,1,1] => 2931 [4,3] => 2674 [4,2,1] => 6113 [4,1,1,1] => 3216 [3,3,1] => 3455 [3,2,2] => 3318 [3,2,1,1] => 5216 [3,1,1,1,1] => 2058 [2,2,2,1] => 1963 [2,2,1,1,1] => 1830 [2,1,1,1,1,1] => 725 [1,1,1,1,1,1,1] => 111 [8] => 951 [7,1] => 4883 [6,2] => 11756 [6,1,1] => 11530 [5,3] => 14623 [5,2,1] => 30987 [5,1,1,1] => 15764 [4,4] => 7017 [4,3,1] => 31419 [4,2,2] => 24189 [4,2,1,1] => 36892 [4,1,1,1,1] => 13285 [3,3,2] => 17176 [3,3,1,1] => 22029 [3,2,2,1] => 26440 [3,2,1,1,1] => 22787 [3,1,1,1,1,1] => 6874 [2,2,2,2] => 4976 [2,2,2,1,1] => 9408 [2,2,1,1,1,1] => 6345 [2,1,1,1,1,1,1] => 2039 [1,1,1,1,1,1,1,1] => 276 [9] => 2615 [8,1] => 15748 [7,2] => 44317 [7,1,1] => 43508 [6,3] => 69637 [6,2,1] => 142113 [6,1,1,1] => 70942 [5,4] => 56799 [5,3,1] => 200488 [5,2,2] => 142907 [5,2,1,1] => 215676 [5,1,1,1,1] => 74188 [4,4,1] => 99619 [4,3,2] => 187636 [4,3,1,1] => 231863 [4,2,2,1] => 223852 [4,2,1,1,1] => 185526 [4,1,1,1,1,1] => 50814 [3,3,3] => 44148 [3,3,2,1] => 167380 [3,3,1,1,1] => 113503 [3,2,2,2] => 79703 [3,2,2,1,1] => 148266 [3,2,1,1,1,1] => 90369 [3,1,1,1,1,1,1] => 22238 [2,2,2,2,1] => 36103 [2,2,2,1,1,1] => 39407 [2,2,1,1,1,1,1] => 20783 [2,1,1,1,1,1,1,1] => 5702 [1,1,1,1,1,1,1,1,1] => 666 [10] => 7318 [9,1] => 50054 [8,2] => 161848 [8,1,1] => 158530 [7,3] => 303642 [7,2,1] => 609037 [7,1,1,1] => 301251 [6,4] => 335433 [6,3,1] => 1084007 [6,2,2] => 747377 [6,2,1,1] => 1118689 [6,1,1,1,1] => 376160 [5,5] => 149827 [5,4,1] => 934739 [5,3,2] => 1383719 [5,3,1,1] => 1682891 [5,2,2,1] => 1507586 [5,2,1,1,1] => 1224236 [5,1,1,1,1,1] => 319364 [4,4,2] => 747207 [4,4,1,1] => 857886 [4,3,3] => 600389 [4,3,2,1] => 2090631 [4,3,1,1,1] => 1362268 [4,2,2,2] => 779984 [4,2,2,1,1] => 1426827 [4,2,1,1,1,1] => 831473 [4,1,1,1,1,1,1] => 184396 [3,3,3,1] => 543502 [3,3,2,2] => 632355 [3,3,2,1,1] => 1095900 [3,3,1,1,1,1] => 517200 [3,2,2,2,1] => 671697 [3,2,2,1,1,1] => 704295 [3,2,1,1,1,1,1] => 335763 [3,1,1,1,1,1,1,1] => 69889 [2,2,2,2,2] => 91548 [2,2,2,2,1,1] => 190161 [2,2,2,1,1,1,1] => 151099 [2,2,1,1,1,1,1,1] => 65952 [2,1,1,1,1,1,1,1,1] => 15782 [1,1,1,1,1,1,1,1,1,1] => 1618 [11] => 20491 [10,1] => 157780 [9,2] => 575068 [9,1,1] => 563741 [8,3] => 1251488 [8,2,1] => 2484790 [8,1,1,1] => 1222372 [7,4] => 1707296 [7,3,1] => 5299381 [7,2,2] => 3588664 [7,2,1,1] => 5349917 [7,1,1,1,1] => 1774629 [6,5] => 1291703 [6,4,1] => 6218244 [6,3,2] => 8456839 [6,3,1,1] => 10187557 [6,2,2,1] => 8822458 [6,2,1,1,1] => 7079072 [6,1,1,1,1,1] => 1798803 [5,5,1] => 2858639 [5,4,2] => 8036094 [5,4,1,1] => 9082522 [5,3,3] => 5188260 [5,3,2,1] => 17352513 [5,3,1,1,1] => 11067920 [5,2,2,2] => 5933784 [5,2,2,1,1] => 10757584 [5,2,1,1,1,1] => 6112049 [5,1,1,1,1,1,1] => 1287996 [4,4,3] => 3515188 [4,4,2,1] => 9601456 [4,4,1,1,1] => 5744585 [4,3,3,1] => 8383548 [4,3,2,2] => 9057507 [4,3,2,1,1] => 15409122 [4,3,1,1,1,1] => 6955992 [4,2,2,2,1] => 7398235 [4,2,2,1,1,1] => 7586236 [4,2,1,1,1,1,1] => 3441883 [4,1,1,1,1,1,1,1] => 643234 [3,3,3,2] => 3030590 [3,3,3,1,1] => 4213456 [3,3,2,2,1] => 6156194 [3,3,2,1,1,1] => 5922497 [3,3,1,1,1,1,1] => 2168628 [3,2,2,2,2] => 1949331 [3,2,2,2,1,1] => 3990844 [3,2,2,1,1,1,1] => 3021837 [3,2,1,1,1,1,1,1] => 1189785 [3,1,1,1,1,1,1,1,1] => 214585 [2,2,2,2,2,1] => 714378 [2,2,2,2,1,1,1] => 862836 [2,2,2,1,1,1,1,1] => 546107 [2,2,1,1,1,1,1,1,1] => 204611 [2,1,1,1,1,1,1,1,1,1] => 43259 [1,1,1,1,1,1,1,1,1,1,1] => 4017 [12] => 57903 [11,1] => 493441 [10,2] => 2003665 [10,1,1] => 1964179 [9,3] => 4945437 [9,2,1] => 9764801 [9,1,1,1] => 4790326 [8,4] => 7960553 [8,3,1] => 24166972 [8,2,2] => 16203849 [8,2,1,1] => 24087817 [8,1,1,1,1] => 7928086 [7,5] => 8032446 [7,4,1] => 35192705 [7,3,2] => 45986266 [7,3,1,1] => 55094809 [7,2,2,1] => 46853100 [7,2,1,1,1] => 37323659 [7,1,1,1,1,1] => 9334812 [6,6] => 3454759 [6,5,1] => 27509562 [6,4,2] => 59982316 [6,4,1,1] => 67132743 [6,3,3] => 35953378 [6,3,2,1] => 117669543 [6,3,1,1,1] => 74117760 [6,2,2,2] => 38619079 [6,2,2,1,1] => 69591781 [6,2,1,1,1,1] => 38956533 [6,1,1,1,1,1,1] => 7973068 [5,5,2] => 28714329 [5,5,1,1] => 31397152 [5,4,3] => 43981477 [5,4,2,1] => 115386199 [5,4,1,1,1] => 67554399 [5,3,3,1] => 80799343 [5,3,2,2] => 84325120 [5,3,2,1,1] => 141990915 [5,3,1,1,1,1] => 62507326 [5,2,2,2,1] => 62478917 [5,2,2,1,1,1] => 63223877 [5,2,1,1,1,1,1] => 27890959 [5,1,1,1,1,1,1,1] => 4938240 [4,4,4] => 9210566 [4,4,3,1] => 56069180 [4,4,2,2] => 48565468 [4,4,2,1,1] => 79855194 [4,4,1,1,1,1] => 32840479 [4,3,3,2] => 53172292 [4,3,3,1,1] => 72585160 [4,3,2,2,1] => 98351648 [4,3,2,1,1,1] => 92505243 [4,3,1,1,1,1,1] => 32248843 [4,2,2,2,2] => 24093179 [4,2,2,2,1,1] => 48798953 [4,2,2,1,1,1,1] => 35997249 [4,2,1,1,1,1,1,1] => 13444088 [4,1,1,1,1,1,1,1,1] => 2173367 [3,3,3,3] => 7844047 [3,3,3,2,1] => 34570311 [3,3,3,1,1,1] => 26068121 [3,3,2,2,2] => 20851166 [3,3,2,2,1,1] => 41261745 [3,3,2,1,1,1,1] => 28424977 [3,3,1,1,1,1,1,1] => 8548160 [3,2,2,2,2,1] => 17027562 [3,2,2,2,1,1,1] => 20095913 [3,2,2,1,1,1,1,1] => 12088852 [3,2,1,1,1,1,1,1,1] => 4066989 [3,1,1,1,1,1,1,1,1,1] => 647678 [2,2,2,2,2,2] => 1820231 [2,2,2,2,2,1,1] => 4007963 [2,2,2,2,1,1,1,1] => 3569980 [2,2,2,1,1,1,1,1,1] => 1893192 [2,2,1,1,1,1,1,1,1,1] => 622202 [2,1,1,1,1,1,1,1,1,1,1] => 117912 [1,1,1,1,1,1,1,1,1,1,1,1] => 10006 [13] => 163898 [12,1] => 1534884 [11,2] => 6866042 [11,1,1] => 6735856 [10,3] => 18932370 [10,2,1] => 37257908 [10,1,1,1] => 18248596 [9,4] => 34926525 [9,3,1] => 104689964 [9,2,2] => 69765576 [9,2,1,1] => 103548408 [9,1,1,1,1] => 33915591 [8,5] => 43083435 [8,4,1] => 180311081 [8,3,2] => 230557039 [8,3,1,1] => 275262177 [8,2,2,1] => 231632829 [8,2,1,1,1] => 183679631 [8,1,1,1,1,1] => 45491926 [7,6] => 30862502 [7,5,1] => 188805866 [7,4,2] => 375258494 [7,4,1,1] => 417527639 [7,3,3] => 217400912 [7,3,2,1] => 702439122 [7,3,1,1,1] => 439019063 [7,2,2,2] => 225396305 [7,2,2,1,1] => 404637998 [7,2,1,1,1,1] => 224393866 [7,1,1,1,1,1,1] => 45118576 [6,6,1] => 82507937 [6,5,2] => 308145050 [6,5,1,1] => 333478433 [6,4,3] => 369974178 [6,4,2,1] => 949445830 [6,4,1,1,1] => 548727730 [6,3,3,1] => 616847240 [6,3,2,2] => 631958558 [6,3,2,1,1] => 1057274716 [6,3,1,1,1,1] => 458458454 [6,2,2,2,1] => 446236187 [6,2,2,1,1,1] => 447781496 [6,2,1,1,1,1,1] => 194186707 [6,1,1,1,1,1,1,1] => 33323070 [5,5,3] => 191912104 [5,5,2,1] => 461889786 [5,5,1,1,1] => 259632180 [5,4,4] => 140170118 [5,4,3,1] => 777627957 [5,4,2,2] => 650540673 [5,4,2,1,1] => 1058536010 [5,4,1,1,1,1] => 424405370 [5,3,3,2] => 571013146 [5,3,3,1,1] => 771132672 [5,3,2,2,1] => 1009017695 [5,3,2,1,1,1] => 936261132 [5,3,1,1,1,1,1] => 317412076 [5,2,2,2,2] => 224884082 [5,2,2,2,1,1] => 452474815 [5,2,2,1,1,1,1] => 328469438 [5,2,1,1,1,1,1,1] => 119017787 [5,1,1,1,1,1,1,1,1] => 18187641 [4,4,4,1] => 170939928 [4,4,3,2] => 412137147 [4,4,3,1,1] => 543527557 [4,4,2,2,1] => 590137101 [4,4,2,1,1,1] => 532551999 [4,4,1,1,1,1,1] => 168209289 [4,3,3,3] => 158953807 [4,3,3,2,1] => 672100179 [4,3,3,1,1,1] => 495345910 [4,3,2,2,2] => 371401367 [4,3,2,2,1,1] => 726807599 [4,3,2,1,1,1,1] => 487819474 [4,3,1,1,1,1,1,1] => 139227587 [4,2,2,2,2,1] => 231939400 [4,2,2,2,1,1,1] => 269892707 [4,2,2,1,1,1,1,1] => 157718591 [4,2,1,1,1,1,1,1,1] => 50214314 [4,1,1,1,1,1,1,1,1,1] => 7156167 [3,3,3,3,1] => 109855195 [3,3,3,2,2] => 143320388 [3,3,3,2,1,1] => 263057036 [3,3,3,1,1,1,1] => 140222034 [3,3,2,2,2,1] => 203842565 [3,3,2,2,1,1,1] => 230689740 [3,3,2,1,1,1,1,1] => 125590365 [3,3,1,1,1,1,1,1,1] => 32136763 [3,2,2,2,2,2] => 48368371 [3,2,2,2,2,1,1] => 105389293 [3,2,2,2,1,1,1,1] => 91340349 [3,2,2,1,1,1,1,1,1] => 45923314 [3,2,1,1,1,1,1,1,1,1] => 13512735 [3,1,1,1,1,1,1,1,1,1,1] => 1928348 [2,2,2,2,2,2,1] => 14871610 [2,2,2,2,2,1,1,1] => 19295410 [2,2,2,2,1,1,1,1,1] => 13863944 [2,2,2,1,1,1,1,1,1,1] => 6357081 [2,2,1,1,1,1,1,1,1,1,1] => 1861237 [2,1,1,1,1,1,1,1,1,1,1,1] => 320098 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 24844 [14] => 466199 [13,1] => 4751067 [12,2] => 23220889 [12,1,1] => 22792381 [11,3] => 70660963 [11,2,1] => 138798812 [11,1,1,1] => 67934060 [10,4] => 146533592 [10,3,1] => 435779847 [10,2,2] => 289312657 [10,2,1,1] => 428975159 [10,1,1,1,1] => 140094715 [9,5] => 211206424 [9,4,1] => 861831551 [9,3,2] => 1087926899 [9,3,1,1] => 1296040948 [9,2,2,1] => 1083506723 [9,2,1,1,1] => 856696497 [9,1,1,1,1,1] => 210846304 [8,6] => 199590706 [8,5,1] => 1106181430 [8,4,2] => 2102116342 [8,3,3] => 1195738351 [8,2,2,2] => 1213429834 [8,2,1,1,1,1] => 1197685752 [8,1,1,1,1,1,1] => 238088967 [7,7] => 83164062 [7,6,1] => 809114769 [7,2,1,1,1,1,1] => 1212899049 [7,1,1,1,1,1,1,1] => 204140465 [6,6,2] => 1040046726 [6,6,1,1] => 1105345887 [6,4,4] => 1351682301 [6,2,2,2,2] => 1755726690 [6,2,1,1,1,1,1,1] => 898485923 [6,1,1,1,1,1,1,1,1] => 132869758 [5,5,4] => 878598627 [5,5,1,1,1,1] => 1791894547 [5,3,3,3] => 1913295926 [5,3,1,1,1,1,1,1] => 1488995568 [5,2,2,1,1,1,1,1] => 1563455526 [5,2,1,1,1,1,1,1,1] => 482026626 [5,1,1,1,1,1,1,1,1,1] => 64842108 [4,4,4,2] => 1528488191 [4,4,4,1,1] => 1869834480 [4,4,3,3] => 1494808371 [4,4,1,1,1,1,1,1] => 794247984 [4,3,1,1,1,1,1,1,1] => 568895541 [4,2,2,2,2,2] => 724482057 [4,2,2,2,2,1,1] => 1567164433 [4,2,2,2,1,1,1,1] => 1335796407 [4,2,2,1,1,1,1,1,1] => 651000095 [4,2,1,1,1,1,1,1,1,1] => 180957588 [4,1,1,1,1,1,1,1,1,1,1] => 23072233 [3,3,3,3,2] => 657082503 [3,3,3,3,1,1] => 966400608 [3,3,3,2,2,1] => 1579308058 [3,3,3,2,1,1,1] => 1637308947 [3,3,3,1,1,1,1,1] => 684228704 [3,3,2,2,2,2] => 651027342 [3,3,2,2,2,1,1] => 1391798860 [3,3,2,2,1,1,1,1] => 1150806320 [3,3,2,1,1,1,1,1,1] => 521658141 [3,3,1,1,1,1,1,1,1,1] => 116403383 [3,2,2,2,2,2,1] => 433351822 [3,2,2,2,2,1,1,1] => 554077940 [3,2,2,2,1,1,1,1,1] => 386453490 [3,2,2,1,1,1,1,1,1,1] => 167590962 [3,2,1,1,1,1,1,1,1,1,1] => 43873472 [3,1,1,1,1,1,1,1,1,1,1,1] => 5673705 [2,2,2,2,2,2,2] => 38064059 [2,2,2,2,2,2,1,1] => 87325799 [2,2,2,2,2,1,1,1,1] => 84558244 [2,2,2,2,1,1,1,1,1,1] => 51392991 [2,2,2,1,1,1,1,1,1,1,1] => 20806755 [2,2,1,1,1,1,1,1,1,1,1,1] => 5497254 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 865394 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 61924 [15] => 1328993 [14,1] => 14650472 [13,2] => 77656094 [13,1,1] => 76277462 [12,3] => 258353178 [12,2,1] => 506949725 [12,1,1,1] => 248060238 [11,4] => 593679559 [11,3,1] => 1756905162 [11,2,2] => 1163556871 [11,2,1,1] => 1724327614 [11,1,1,1,1] => 562130089 [10,5] => 972970999 [10,1,1,1,1,1] => 938066335 [9,6] => 1113418862 [9,1,1,1,1,1,1] => 1187996973 [8,7] => 764820279 [8,1,1,1,1,1,1,1] => 1159374758 [7,1,1,1,1,1,1,1,1] => 875914505 [6,1,1,1,1,1,1,1,1,1] => 509785347 [5,2,1,1,1,1,1,1,1,1] => 1871942391 [5,1,1,1,1,1,1,1,1,1,1] => 225027951 [4,2,1,1,1,1,1,1,1,1,1] => 633304011 [4,1,1,1,1,1,1,1,1,1,1,1] => 73088079 [3,3,3,3,3] => 1709646319 [3,3,2,1,1,1,1,1,1,1] => 2064913017 [3,3,1,1,1,1,1,1,1,1,1] => 409065407 [3,2,2,2,2,2,2] => 1214078549 [3,2,2,2,1,1,1,1,1,1] => 1549474924 [3,2,2,1,1,1,1,1,1,1,1] => 592236331 [3,2,1,1,1,1,1,1,1,1,1,1] => 139757331 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 16525365 [2,2,2,2,2,2,2,1] => 321418259 [2,2,2,2,2,2,1,1,1] => 439916729 [2,2,2,2,2,1,1,1,1,1] => 347096922 [2,2,2,2,1,1,1,1,1,1,1] => 183750367 [2,2,2,1,1,1,1,1,1,1,1,1] => 66693596 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 16066799 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2331217 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 155123 [16] => 3799624 [15,1] => 45025792 [14,2] => 257276727 [14,1,1] => 252865940 [13,3] => 928500885 [13,2,1] => 1821037070 [13,1,1,1] => 891128587 [8,8] => 2073523096 [6,1,1,1,1,1,1,1,1,1,1] => 1894170931 [5,1,1,1,1,1,1,1,1,1,1,1] => 763364397 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 228069815 [3,3,1,1,1,1,1,1,1,1,1,1] => 1401797177 [3,2,2,1,1,1,1,1,1,1,1,1] => 2038283551 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 438097447 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 47724923 [2,2,2,2,2,2,2,2] => 825736331 [2,2,2,2,2,2,2,1,1] => 1954038143 [2,2,2,2,2,2,1,1,1,1] => 2016436391 [2,2,2,2,2,1,1,1,1,1,1] => 1356704514 [2,2,2,2,1,1,1,1,1,1,1,1] => 638107591 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 210114264 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 46531170 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 6262169 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 389289 [17] => 10884049 [16,1] => 137993681 [15,2] => 845474608 [15,1,1] => 831521798 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 702522095 ----------------------------------------------------------------------------- Created: Sep 26, 2020 at 23:55 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 26, 2020 at 23:55 by Martin Rubey