***************************************************************************** * 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: St000812 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. For example, $h_{11} = 2m_{11} + m_2$, so the statistic on the partition $11$ is 3. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): m = SymmetricFunctions(ZZ).m() h = SymmetricFunctions(ZZ).h() return sum(coeff for _, coeff in m(h(mu))) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 2 [1,1] => 3 [3] => 3 [2,1] => 6 [1,1,1] => 10 [4] => 5 [3,1] => 12 [2,2] => 16 [2,1,1] => 27 [1,1,1,1] => 47 [5] => 7 [4,1] => 20 [3,2] => 32 [3,1,1] => 56 [2,2,1] => 76 [2,1,1,1] => 136 [1,1,1,1,1] => 246 [6] => 11 [5,1] => 35 [4,2] => 65 [4,1,1] => 114 [3,3] => 79 [3,2,1] => 191 [3,1,1,1] => 344 [2,2,2] => 263 [2,2,1,1] => 476 [2,1,1,1,1] => 870 [1,1,1,1,1,1] => 1602 [7] => 15 [6,1] => 54 [5,2] => 113 [5,1,1] => 202 [4,3] => 160 [4,2,1] => 398 [4,1,1,1] => 727 [3,3,1] => 493 [3,2,2] => 685 [3,2,1,1] => 1261 [3,1,1,1,1] => 2335 [2,2,2,1] => 1765 [2,2,1,1,1] => 3280 [2,1,1,1,1,1] => 6124 [1,1,1,1,1,1,1] => 11481 [8] => 22 [7,1] => 86 [6,2] => 199 [6,1,1] => 357 [5,3] => 318 [5,2,1] => 800 [5,1,1,1] => 1468 [4,4] => 371 [4,3,1] => 1167 [4,2,2] => 1632 [4,2,1,1] => 3021 [4,1,1,1,1] => 5623 [3,3,2] => 2041 [3,3,1,1] => 3790 [3,2,2,1] => 5342 [3,2,1,1,1] => 9997 [3,1,1,1,1,1] => 18782 [2,2,2,2] => 7548 [2,2,2,1,1] => 14162 [2,2,1,1,1,1] => 26673 [2,1,1,1,1,1,1] => 50399 [1,1,1,1,1,1,1,1] => 95503 [9] => 30 [8,1] => 128 [7,2] => 323 [7,1,1] => 585 [6,3] => 573 [6,2,1] => 1462 [6,1,1,1] => 2702 [5,4] => 756 [5,3,1] => 2435 [5,2,2] => 3423 [5,2,1,1] => 6387 [5,1,1,1,1] => 11966 [4,4,1] => 2872 [4,3,2] => 5090 [4,3,1,1] => 9537 [4,2,2,1] => 13520 [4,2,1,1,1] => 25487 [4,1,1,1,1,1] => 48192 [3,3,3] => 6417 [3,3,2,1] => 17109 [3,3,1,1,1] => 32313 [3,2,2,2] => 24339 [3,2,2,1,1] => 46068 [3,2,1,1,1,1] => 87431 [3,1,1,1,1,1,1] => 166334 [2,2,2,2,1] => 65786 [2,2,2,1,1,1] => 125088 [2,2,1,1,1,1,1] => 238388 [2,1,1,1,1,1,1,1] => 455249 [1,1,1,1,1,1,1,1,1] => 871030 [10] => 42 [9,1] => 192 [8,2] => 523 [8,1,1] => 951 [7,3] => 1013 [7,2,1] => 2607 [7,1,1,1] => 4836 [6,4] => 1485 [6,3,1] => 4849 [6,2,2] => 6844 [6,2,1,1] => 12820 [6,1,1,1,1] => 24103 [5,5] => 1683 [5,4,1] => 6539 [5,3,2] => 11694 [5,3,1,1] => 22018 [5,2,2,1] => 31338 [5,2,1,1,1] => 59315 [5,1,1,1,1,1] => 112565 [4,4,2] => 13905 [4,4,1,1] => 26220 [4,3,3] => 17617 [4,3,2,1] => 47494 [4,3,1,1,1] => 90161 [4,2,2,2] => 67880 [4,2,2,1,1] => 129091 [4,2,1,1,1,1] => 246064 [4,1,1,1,1,1,1] => 469990 [3,3,3,1] => 60442 [3,3,2,2] => 86494 [3,3,2,1,1] => 164735 [3,3,1,1,1,1] => 314432 [3,2,2,2,1] => 236463 [3,2,2,1,1,1] => 452025 [3,2,1,1,1,1,1] => 865691 [3,1,1,1,1,1,1,1] => 1660708 [2,2,2,2,2] => 339857 [2,2,2,2,1,1] => 650606 [2,2,2,1,1,1,1] => 1247675 [2,2,1,1,1,1,1,1] => 2396498 [2,1,1,1,1,1,1,1,1] => 4609878 [1,1,1,1,1,1,1,1,1,1] => 8879558 [11] => 56 [10,1] => 275 [9,2] => 803 [9,1,1] => 1470 [8,3] => 1683 [8,2,1] => 4376 [8,1,1,1] => 8159 [7,4] => 2701 [7,3,1] => 8956 [7,2,2] => 12689 [7,2,1,1] => 23890 [7,1,1,1,1] => 45113 [6,5] => 3405 [6,4,1] => 13519 [6,3,2] => 24382 [6,3,1,1] => 46158 [6,2,2,1] => 65943 [6,2,1,1,1] => 125379 [6,1,1,1,1,1] => 238901 [5,5,1] => 15469 [5,4,2] => 33373 [5,4,1,1] => 63336 [5,3,3] => 42473 [5,3,2,1] => 115700 [5,3,1,1,1] => 220766 [5,2,2,2] => 166038 [5,2,2,1,1] => 317293 [5,2,1,1,1,1] => 607439 [5,1,1,1,1,1,1] => 1164833 [4,4,3] => 50801 [4,4,2,1] => 138695 [4,4,1,1,1] => 264934 [4,3,3,1] => 177326 [4,3,2,2] => 254989 [4,3,2,1,1] => 488372 [4,3,1,1,1,1] => 936927 [4,2,2,2,1] => 704026 [4,2,2,1,1,1] => 1352328 [4,2,1,1,1,1,1] => 2601382 [4,1,1,1,1,1,1,1] => 5010794 [3,3,3,2] => 326656 [3,3,3,1,1] => 626353 [3,3,2,2,1] => 903777 [3,3,2,1,1,1] => 1737801 [3,3,1,1,1,1,1] => 3346113 [3,2,2,2,2] => 1305408 [3,2,2,2,1,1] => 2512916 [3,2,2,1,1,1,1] => 4843739 [3,2,1,1,1,1,1,1] => 9347861 [3,1,1,1,1,1,1,1,1] => 18060757 [2,2,2,2,2,1] => 3637109 [2,2,2,2,1,1,1] => 7017789 [2,2,2,1,1,1,1,1] => 13556488 [2,2,1,1,1,1,1,1,1] => 26215787 [2,1,1,1,1,1,1,1,1,1] => 50747824 [1,1,1,1,1,1,1,1,1,1,1] => 98329551 [12] => 77 [11,1] => 399 [10,2] => 1237 [10,1,1] => 2270 [9,3] => 2776 [9,2,1] => 7255 [9,1,1,1] => 13558 [8,4] => 4822 [8,3,1] => 16125 [8,2,2] => 22903 [8,2,1,1] => 43226 [8,1,1,1,1] => 81810 [7,5] => 6662 [7,4,1] => 26806 [7,3,2] => 48618 [7,3,1,1] => 92304 [7,2,2,1] => 132198 [7,2,1,1,1] => 251974 [7,1,1,1,1,1] => 481218 [6,6] => 7413 [6,5,1] => 34359 [6,4,2] => 74874 [6,4,1,1] => 142597 [6,3,3] => 95581 [6,3,2,1] => 262004 [6,3,1,1,1] => 501393 [6,2,2,2] => 377032 [6,2,2,1,1] => 722486 [6,2,1,1,1,1] => 1386715 [6,1,1,1,1,1,1] => 2665558 [5,5,2] => 86219 [5,5,1,1] => 164371 [5,4,3] => 132211 [5,4,2,1] => 363756 [5,4,1,1,1] => 697388 [5,3,3,1] => 466577 [5,3,2,2] => 673104 [5,3,2,1,1] => 1293446 [5,3,1,1,1,1] => 2489112 [5,2,2,2,1] => 1870045 [5,2,2,1,1,1] => 3602603 [5,2,1,1,1,1,1] => 6949104 [5,1,1,1,1,1,1,1] => 13419859 [4,4,4] => 158810 [4,4,3,1] => 561917 [4,4,2,2] => 811289 [4,4,2,1,1] => 1560329 [4,4,1,1,1,1] => 3005156 [4,3,3,2] => 1043121 [4,3,3,1,1] => 2008165 [4,3,2,2,1] => 2907772 [4,3,2,1,1,1] => 5611045 [4,3,1,1,1,1,1] => 10840145 [4,2,2,2,2] => 4214013 [4,2,2,2,1,1] => 8139459 [4,2,2,1,1,1,1] => 15739048 [4,2,1,1,1,1,1,1] => 30465586 [4,1,1,1,1,1,1,1,1] => 59028086 [3,3,3,3] => 1342215 [3,3,3,2,1] => 3747889 [3,3,3,1,1,1] => 7238283 [3,3,2,2,2] => 5435521 [3,3,2,2,1,1] => 10507313 [3,3,2,1,1,1,1] => 20333176 [3,3,1,1,1,1,1,1] => 39386682 [3,2,2,2,2,1] => 15264278 [3,2,2,2,1,1,1] => 29563182 [3,2,2,1,1,1,1,1] => 57310817 [3,2,1,1,1,1,1,1,1] => 111200335 [3,1,1,1,1,1,1,1,1,1] => 215941579 [2,2,2,2,2,2] => 22190989 [2,2,2,2,2,1,1] => 43012769 [2,2,2,2,1,1,1,1] => 83446864 [2,2,2,1,1,1,1,1,1] => 162027834 [2,2,1,1,1,1,1,1,1,1] => 314857712 [2,1,1,1,1,1,1,1,1,1,1] => 612300434 [1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522 [13] => 101 [12,1] => 556 [10,3] => 4366 [9,2,2] => 39218 [8,5] => 12205 [8,4,1] => 49843 [8,3,2] => 90936 [8,3,1,1] => 173281 [7,6] => 14901 [7,5,1] => 70485 [7,4,2] => 155087 [7,3,3] => 198566 [6,6,1] => 79006 [6,5,2] => 201191 [6,5,1,1] => 385354 [6,4,3] => 310547 [6,4,2,1] => 860962 [6,3,2,2] => 1602689 [6,3,1,1,1,1] => 5965450 [6,2,2,1,1,1] => 8656230 [6,1,1,1,1,1,1,1] => 32417338 [5,5,3] => 359371 [5,4,4] => 433217 [5,4,3,1] => 1551710 [5,4,2,2] => 2248291 [5,4,2,1,1] => 4341108 [5,4,1,1,1,1] => 8391405 [5,3,3,2] => 2899935 [5,3,3,1,1] => 5603882 [5,3,2,2,1] => 8139289 [5,3,2,1,1,1] => 15758563 [5,3,1,1,1,1,1] => 30539286 [5,2,2,2,1,1] => 22923367 [5,2,2,1,1,1,1] => 44457921 [5,2,1,1,1,1,1,1] => 86295412 [4,4,4,1] => 1875822 [4,4,3,2] => 3510016 [4,4,3,1,1] => 6787502 [4,4,2,2,1] => 9864237 [4,3,3,3] => 4532834 [4,3,3,2,1] => 12756465 [3,3,3,3,1] => 16505959 [3,3,3,2,2] => 24033962 [3,3,2,2,2,1] => 68029106 [3,3,2,1,1,1,1,1] => 257335442 [3,2,2,2,2,2] => 99245511 [3,2,2,2,2,1,1] => 193076440 [2,2,2,2,2,2,1] => 282002663 [2,2,2,2,1,1,1,1,1] => 1070708800 ----------------------------------------------------------------------------- Created: May 20, 2017 at 17:54 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 29, 2023 at 11:12 by Martin Rubey