***************************************************************************** * 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: St001612 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. In particular, the value on the partition $(n)$ is the number of partitions of $n$, whereas the value on the partition $(1^n)$ is the number of permutations. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): h = SymmetricFunctions(QQ).h() F = species.PermutationSpecies().cycle_index_series() return F.coefficient(mu.size()).scalar(h(mu)) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 2 [1,1] => 2 [3] => 3 [2,1] => 4 [1,1,1] => 6 [4] => 5 [3,1] => 7 [2,2] => 10 [2,1,1] => 14 [1,1,1,1] => 24 [5] => 7 [4,1] => 12 [3,2] => 18 [3,1,1] => 28 [2,2,1] => 38 [2,1,1,1] => 66 [1,1,1,1,1] => 120 [6] => 11 [5,1] => 19 [4,2] => 34 [4,1,1] => 52 [3,3] => 38 [3,2,1] => 84 [3,1,1,1] => 150 [2,2,2] => 120 [2,2,1,1] => 208 [2,1,1,1,1] => 384 [1,1,1,1,1,1] => 720 [7] => 15 [6,1] => 30 [5,2] => 56 [5,1,1] => 90 [4,3] => 74 [4,2,1] => 170 [4,1,1,1] => 306 [3,3,1] => 206 [3,2,2] => 290 [3,2,1,1] => 526 [3,1,1,1,1] => 984 [2,2,2,1] => 744 [2,2,1,1,1] => 1392 [2,1,1,1,1,1] => 2640 [1,1,1,1,1,1,1] => 5040 [8] => 22 [7,1] => 45 [6,2] => 94 [6,1,1] => 150 [5,3] => 133 [5,2,1] => 316 [5,1,1,1] => 576 [4,4] => 158 [4,3,1] => 450 [4,2,2] => 644 [4,2,1,1] => 1172 [4,1,1,1,1] => 2208 [3,3,2] => 788 [3,3,1,1] => 1464 [3,2,2,1] => 2086 [3,2,1,1,1] => 3954 [3,1,1,1,1,1] => 7560 [2,2,2,2] => 3000 [2,2,2,1,1] => 5664 [2,2,1,1,1,1] => 10848 [2,1,1,1,1,1,1] => 20880 [1,1,1,1,1,1,1,1] => 40320 [9] => 30 [8,1] => 67 [7,2] => 146 [7,1,1] => 240 [6,3] => 233 [6,2,1] => 560 [6,1,1,1] => 1026 [5,4] => 297 [5,3,1] => 899 [5,2,2] => 1284 [5,2,1,1] => 2380 [5,1,1,1,1] => 4512 [4,4,1] => 1058 [4,3,2] => 1886 [4,3,1,1] => 3536 [4,2,2,1] => 5074 [4,2,1,1,1] => 9678 [4,1,1,1,1,1] => 18600 [3,3,3] => 2370 [3,3,2,1] => 6424 [3,3,1,1,1] => 12300 [3,2,2,2] => 9264 [3,2,2,1,1] => 17744 [3,2,1,1,1,1] => 34224 [3,1,1,1,1,1,1] => 66240 [2,2,2,2,1] => 25656 [2,2,2,1,1,1] => 49536 [2,2,1,1,1,1,1] => 96000 [2,1,1,1,1,1,1,1] => 186480 [1,1,1,1,1,1,1,1,1] => 362880 [10] => 42 [9,1] => 97 [8,2] => 228 [8,1,1] => 374 [7,3] => 385 [7,2,1] => 946 [7,1,1,1] => 1746 [6,4] => 550 [6,3,1] => 1692 [6,2,2] => 2434 [6,2,1,1] => 4526 [6,1,1,1,1] => 8616 [5,5] => 602 [5,4,1] => 2254 [5,3,2] => 4074 [5,3,1,1] => 7714 [5,2,2,1] => 11118 [5,2,1,1,1] => 21330 [5,1,1,1,1,1] => 41160 [4,4,2] => 4868 [4,4,1,1] => 9188 [4,3,3] => 6146 [4,3,2,1] => 16920 [4,3,1,1,1] => 32586 [4,2,2,2] => 24552 [4,2,2,1,1] => 47248 [4,2,1,1,1,1] => 91536 [4,1,1,1,1,1,1] => 177840 [3,3,3,1] => 21642 [3,3,2,2] => 31380 [3,3,2,1,1] => 60636 [3,3,1,1,1,1] => 117648 [3,2,2,2,1] => 88152 [3,2,2,1,1,1] => 171216 [3,2,1,1,1,1,1] => 333360 [3,1,1,1,1,1,1,1] => 650160 [2,2,2,2,2] => 128400 [2,2,2,2,1,1] => 249456 [2,2,2,1,1,1,1] => 486144 [2,2,1,1,1,1,1,1] => 948960 [2,1,1,1,1,1,1,1,1] => 1854720 [1,1,1,1,1,1,1,1,1,1] => 3628800 [11] => 56 [10,1] => 139 [9,2] => 340 [9,1,1] => 568 [8,3] => 623 [8,2,1] => 1548 [8,1,1,1] => 2868 [7,4] => 951 [7,3,1] => 3023 [7,2,2] => 4348 [7,2,1,1] => 8164 [7,1,1,1,1] => 15600 [6,5] => 1166 [6,4,1] => 4496 [6,3,2] => 8210 [6,3,1,1] => 15624 [6,2,2,1] => 22604 [6,2,1,1,1] => 43524 [6,1,1,1,1,1] => 84240 [5,5,1] => 5110 [5,4,2] => 11214 [5,4,1,1] => 21410 [5,3,3] => 14302 [5,3,2,1] => 39826 [5,3,1,1,1] => 77058 [5,2,2,2] => 57936 [5,2,2,1,1] => 112144 [5,2,1,1,1,1] => 218016 [5,1,1,1,1,1,1] => 424800 [4,4,3] => 17170 [4,4,2,1] => 47896 [4,4,1,1,1] => 92736 [4,3,3,1] => 61628 [4,3,2,2] => 89786 [4,3,2,1,1] => 174310 [4,3,1,1,1,1] => 339528 [4,2,2,2,1] => 254448 [4,2,2,1,1,1] => 496032 [4,2,1,1,1,1,1] => 968880 [4,1,1,1,1,1,1,1] => 1895040 [3,3,3,2] => 115788 [3,3,3,1,1] => 225192 [3,3,2,2,1] => 328956 [3,3,2,1,1,1] => 641988 [3,3,1,1,1,1,1] => 1254960 [3,2,2,2,2] => 481032 [3,2,2,2,1,1] => 939408 [3,2,2,1,1,1,1] => 1837728 [3,2,1,1,1,1,1,1] => 3599280 [3,1,1,1,1,1,1,1,1] => 7056000 [2,2,2,2,2,1] => 1375680 [2,2,2,2,1,1,1] => 2692944 [2,2,2,1,1,1,1,1] => 5277600 [2,2,1,1,1,1,1,1,1] => 10352160 [2,1,1,1,1,1,1,1,1,1] => 20321280 [1,1,1,1,1,1,1,1,1,1,1] => 39916800 [12] => 77 [11,1] => 195 [10,2] => 506 [10,1,1] => 846 [9,3] => 977 [9,2,1] => 2456 [9,1,1,1] => 4572 [8,4] => 1614 [8,3,1] => 5194 [8,2,2] => 7504 [8,2,1,1] => 14128 [8,1,1,1,1] => 27072 [7,5] => 2133 [7,4,1] => 8470 [7,3,2] => 15592 [7,3,1,1] => 29834 [7,2,2,1] => 43280 [7,2,1,1,1] => 83616 [7,1,1,1,1,1] => 162240 [6,6] => 2382 [6,5,1] => 10772 [6,4,2] => 24008 [6,4,1,1] => 46026 [6,3,3] => 30740 [6,3,2,1] => 86264 [6,3,1,1,1] => 167454 [6,2,2,2] => 125904 [6,2,2,1,1] => 244400 [6,2,1,1,1,1] => 476352 [6,1,1,1,1,1,1] => 930240 [5,5,2] => 27556 [5,5,1,1] => 53040 [5,4,3] => 42696 [5,4,2,1] => 120346 [5,4,1,1,1] => 234024 [5,3,3,1] => 155582 [5,3,2,2] => 227388 [5,3,2,1,1] => 443164 [5,3,1,1,1,1] => 865776 [5,2,2,2,1] => 648816 [5,2,2,1,1,1] => 1268496 [5,2,1,1,1,1,1] => 2483760 [5,1,1,1,1,1,1,1] => 4868640 [4,4,4] => 51630 [4,4,3,1] => 188322 [4,4,2,2] => 275572 [4,4,2,1,1] => 537148 [4,4,1,1,1,1] => 1050000 [4,3,3,2] => 356996 [4,3,3,1,1] => 697068 [4,3,2,2,1] => 1021810 [4,3,2,1,1,1] => 2000478 [4,3,1,1,1,1,1] => 3921480 [4,2,2,2,2] => 1499208 [4,2,2,2,1,1] => 2936400 [4,2,2,1,1,1,1] => 5759616 [4,2,1,1,1,1,1,1] => 11307600 [4,1,1,1,1,1,1,1,1] => 22216320 [3,3,3,3] => 463176 [3,3,3,2,1] => 1327848 [3,3,3,1,1,1] => 2601540 [3,3,2,2,2] => 1948944 [3,3,2,2,1,1] => 3820704 [3,3,2,1,1,1,1] => 7498368 [3,3,1,1,1,1,1,1] => 14728320 [3,2,2,2,2,1] => 5614344 [3,2,2,2,1,1,1] => 11024352 [3,2,2,1,1,1,1,1] => 21664800 [3,2,1,1,1,1,1,1,1] => 42603120 [3,1,1,1,1,1,1,1,1,1] => 83825280 [2,2,2,2,2,2] => 8254800 [2,2,2,2,2,1,1] => 16216080 [2,2,2,2,1,1,1,1] => 31882176 [2,2,2,1,1,1,1,1,1] => 62722080 [2,2,1,1,1,1,1,1,1,1] => 123459840 [2,1,1,1,1,1,1,1,1,1,1] => 243129600 [1,1,1,1,1,1,1,1,1,1,1,1] => 479001600 [13] => 101 [12,1] => 272 [11,2] => 730 [11,1,1] => 1236 [10,3] => 1501 [10,2,1] => 3808 [10,1,1,1] => 7110 [9,4] => 2627 [9,3,1] => 8627 [9,2,2] => 12468 [9,2,1,1] => 23612 [9,1,1,1,1] => 45360 [8,5] => 3775 [8,4,1] => 15278 [8,3,2] => 28310 [8,3,1,1] => 54350 [8,2,2,1] => 79040 [8,2,1,1,1] => 153072 [8,1,1,1,1,1] => 297600 [7,6] => 4551 [7,5,1] => 21375 [7,4,2] => 48126 [7,4,1,1] => 92800 [7,3,3] => 61940 [7,3,2,1] => 174970 [7,3,1,1,1] => 340572 [7,2,2,2] => 255840 [7,2,2,1,1] => 498192 [7,2,1,1,1,1] => 973056 [7,1,1,1,1,1,1] => 1903680 [6,6,1] => 23926 [6,5,2] => 62378 [6,5,1,1] => 120610 [6,4,3] => 97468 [6,4,2,1] => 276644 [6,4,1,1,1] => 539556 [6,3,3,1] => 358850 [6,3,2,2] => 525854 [6,3,2,1,1] => 1027546 [6,3,1,1,1,1] => 2011944 [6,2,2,2,1] => 1507920 [6,2,2,1,1,1] => 2954400 [6,2,1,1,1,1,1] => 5795760 [6,1,1,1,1,1,1,1] => 11380320 [5,5,3] => 112966 [5,5,2,1] => 321288 [5,5,1,1,1] => 627168 [5,4,4] => 137070 [5,4,3,1] => 506946 [5,4,2,2] => 743732 [5,4,2,1,1] => 1455028 [5,4,1,1,1,1] => 2851872 [5,3,3,2] => 967368 [5,3,3,1,1] => 1894560 [5,3,2,2,1] => 2784342 [5,3,2,1,1,1] => 5464242 [5,3,1,1,1,1,1] => 10734120 [5,2,2,2,2] => 4094616 [5,2,2,2,1,1] => 8039520 [5,2,2,1,1,1,1] => 15801120 [5,2,1,1,1,1,1,1] => 31078800 [5,1,1,1,1,1,1,1,1] => 61165440 [4,4,4,1] => 616596 [4,4,3,2] => 1177924 [4,4,3,1,1] => 2307928 [4,4,2,2,1] => 3393488 [4,4,2,1,1,1] => 6662400 [4,4,1,1,1,1,1] => 13092960 [4,3,3,3] => 1534176 [4,3,3,2,1] => 4425532 [4,3,3,1,1,1] => 8693700 [4,3,2,2,2] => 6513648 [4,3,2,2,1,1] => 12801680 [4,3,2,1,1,1,1] => 25181376 [4,3,1,1,1,1,1,1] => 49564800 [4,2,2,2,2,1] => 18859152 [4,2,2,2,1,1,1] => 37112400 [4,2,2,1,1,1,1,1] => 73078080 [4,2,1,1,1,1,1,1,1] => 143972640 [4,1,1,1,1,1,1,1,1,1] => 283772160 [3,3,3,3,1] => 5774568 [3,3,3,2,2] => 8502540 [3,3,3,2,1,1] => 16719348 [3,3,3,1,1,1,1] => 32901264 [3,3,2,2,2,1] => 24639744 [3,3,2,2,1,1,1] => 48507552 [3,3,2,1,1,1,1,1] => 95549760 [3,3,1,1,1,1,1,1,1] => 188304480 [3,2,2,2,2,2] => 36326640 [3,2,2,2,2,1,1] => 71542176 [3,2,2,2,1,1,1,1] => 140973984 [3,2,2,1,1,1,1,1,1] => 277917120 [3,2,1,1,1,1,1,1,1,1] => 548110080 [3,1,1,1,1,1,1,1,1,1,1] => 1081382400 [2,2,2,2,2,2,1] => 105551280 [2,2,2,2,2,1,1,1] => 208059120 [2,2,2,2,1,1,1,1,1] => 410299200 [2,2,2,1,1,1,1,1,1,1] => 809434080 [2,2,1,1,1,1,1,1,1,1,1] => 1597397760 [14] => 135 [13,1] => 373 [12,2] => 1050 [12,1,1] => 1780 [11,3] => 2255 [11,2,1] => 5774 [11,1,1,1] => 10818 [10,4] => 4202 [10,3,1] => 13936 [10,2,2] => 20198 [10,2,1,1] => 38338 [10,1,1,1,1] => 73800 [9,5] => 6437 [9,4,1] => 26532 [9,3,2] => 49430 [9,3,1,1] => 95216 [9,2,2,1] => 138732 [9,2,1,1,1] => 269268 [9,1,1,1,1,1] => 524400 [8,6] => 8424 [8,5,1] => 40428 [8,4,2] => 91902 [8,4,1,1] => 177706 [8,3,3] => 118588 [8,3,2,1] => 336670 [8,3,1,1,1] => 656694 [8,2,2,2] => 493296 [8,2,2,1,1] => 962416 [8,2,1,1,1,1] => 1882944 [8,1,1,1,1,1,1] => 3689280 [7,7] => 9142 [7,6,1] => 49852 [7,5,2] => 131924 [7,5,1,1] => 256160 [7,4,3] => 207560 [7,4,2,1] => 592540 [7,4,1,1,1] => 1158528 [7,3,3,1] => 770730 [7,3,2,2] => 1131670 [7,3,2,1,1] => 2216250 [7,3,1,1,1,1] => 4347288 [7,2,2,2,1] => 3258336 [7,2,2,1,1,1] => 6395088 [7,2,1,1,1,1,1] => 12564720 [7,1,1,1,1,1,1,1] => 24706080 [6,6,2] => 148908 [6,6,1,1] => 289072 [6,5,3] => 272844 [6,5,2,1] => 780920 [6,5,1,1,1] => 1528554 [6,4,4] => 332306 [6,4,3,1] => 1239908 [6,4,2,2] => 1823414 [6,4,2,1,1] => 3575418 [6,4,1,1,1,1] => 7022040 [6,3,3,2] => 2377952 [6,3,3,1,1] => 4667352 [6,3,2,2,1] => 6873208 [6,3,2,1,1,1] => 13513224 [6,3,1,1,1,1,1] => 26589600 [6,2,2,2,2] => 10127304 [6,2,2,2,1,1] => 19918560 [6,2,2,1,1,1,1] => 39210240 [6,2,1,1,1,1,1,1] => 77233680 [6,1,1,1,1,1,1,1,1] => 152208000 [5,5,4] => 387146 [5,5,3,1] => 1448146 [5,5,2,2] => 2130104 [5,5,2,1,1] => 4179800 [5,5,1,1,1,1] => 8212416 [5,4,4,1] => 1767558 [5,4,3,2] => 3396002 [5,4,3,1,1] => 6671408 [5,4,2,2,1] => 9831618 [5,4,2,1,1,1] => 19343598 [5,4,1,1,1,1,1] => 38086440 [5,3,3,3] => 4436310 [5,3,3,2,1] => 12856144 [5,3,3,1,1,1] => 25305864 [5,3,2,2,2] => 18961344 [5,3,2,2,1,1] => 37338368 [5,3,2,1,1,1,1] => 73573584 [5,3,1,1,1,1,1,1] => 145048320 [5,2,2,2,2,1] => 55111848 [5,2,2,2,1,1,1] => 108634320 [5,2,2,1,1,1,1,1] => 214241280 [5,2,1,1,1,1,1,1,1] => 422689680 [5,1,1,1,1,1,1,1,1,1] => 834261120 [4,4,4,2] => 4151112 [4,4,4,1,1] => 8156976 [4,4,3,3] => 5424220 [4,4,3,2,1] => 15730404 [4,4,3,1,1,1] => 30973584 [4,4,2,2,2] => 23209152 [4,4,2,2,1,1] => 45715136 [4,4,2,1,1,1,1] => 90105312 [4,4,1,1,1,1,1,1] => 177687360 [4,3,3,3,1] => 20585340 [4,3,3,2,2] => 30380928 [4,3,3,2,1,1] => 59867472 [4,3,3,1,1,1,1] => 118038816 [4,3,2,2,2,1] => 88417584 [4,3,2,2,1,1,1] => 174387744 [4,3,2,1,1,1,1,1] => 344099520 [4,3,1,1,1,1,1,1,1] => 679230720 [4,2,2,2,2,2] => 130625040 [4,2,2,2,2,1,1] => 257710080 [4,2,2,2,1,1,1,1] => 508657824 [4,2,2,1,1,1,1,1,1] => 1004330880 [4,2,1,1,1,1,1,1,1,1] => 1983663360 [3,3,3,3,2] => 39784752 [3,3,3,3,1,1] => 78426528 [3,3,3,2,2,1] => 115860468 [3,3,3,2,1,1,1] => 228581964 [3,3,3,1,1,1,1,1] => 451155600 [3,3,2,2,2,2] => 171212304 [3,3,2,2,2,1,1] => 337886496 [3,3,2,2,1,1,1,1] => 667077696 [3,3,2,1,1,1,1,1,1] => 1317437280 [3,2,2,2,2,2,1] => 499588800 [3,2,2,2,2,1,1,1] => 986581584 [3,2,2,2,1,1,1,1,1] => 1948920480 [2,2,2,2,2,2,2] => 738864000 [2,2,2,2,2,2,1,1] => 1459457280 ----------------------------------------------------------------------------- Created: Sep 27, 2020 at 18:40 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 27, 2020 at 18:40 by Martin Rubey