***************************************************************************** * 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: St000810 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): m = SymmetricFunctions(ZZ).m() p = SymmetricFunctions(ZZ).p() return sum(coeff for _, coeff in m(p(mu))) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 3 [3] => 1 [2,1] => 2 [1,1,1] => 10 [4] => 1 [3,1] => 2 [2,2] => 3 [2,1,1] => 7 [1,1,1,1] => 47 [5] => 1 [4,1] => 2 [3,2] => 2 [3,1,1] => 6 [2,2,1] => 6 [2,1,1,1] => 26 [1,1,1,1,1] => 246 [6] => 1 [5,1] => 2 [4,2] => 2 [4,1,1] => 6 [3,3] => 3 [3,2,1] => 6 [3,1,1,1] => 24 [2,2,2] => 10 [2,2,1,1] => 26 [2,1,1,1,1] => 138 [1,1,1,1,1,1] => 1602 [7] => 1 [6,1] => 2 [5,2] => 2 [5,1,1] => 6 [4,3] => 2 [4,2,1] => 5 [4,1,1,1] => 23 [3,3,1] => 6 [3,2,2] => 6 [3,2,1,1] => 20 [3,1,1,1,1] => 114 [2,2,2,1] => 23 [2,2,1,1,1] => 105 [2,1,1,1,1,1] => 767 [1,1,1,1,1,1,1] => 11481 [8] => 1 [7,1] => 2 [6,2] => 2 [6,1,1] => 6 [5,3] => 2 [5,2,1] => 5 [5,1,1,1] => 23 [4,4] => 3 [4,3,1] => 6 [4,2,2] => 7 [4,2,1,1] => 19 [4,1,1,1,1] => 111 [3,3,2] => 6 [3,3,1,1] => 22 [3,2,2,1] => 20 [3,2,1,1,1] => 90 [3,1,1,1,1,1] => 652 [2,2,2,2] => 47 [2,2,2,1,1] => 111 [2,2,1,1,1,1] => 599 [2,1,1,1,1,1,1] => 5295 [1,1,1,1,1,1,1,1] => 95503 [9] => 1 [8,1] => 2 [7,2] => 2 [7,1,1] => 6 [6,3] => 2 [6,2,1] => 5 [6,1,1,1] => 23 [5,4] => 2 [5,3,1] => 5 [5,2,2] => 6 [5,2,1,1] => 18 [5,1,1,1,1] => 110 [4,4,1] => 6 [4,3,2] => 5 [4,3,1,1] => 19 [4,2,2,1] => 18 [4,2,1,1,1] => 80 [4,1,1,1,1,1] => 622 [3,3,3] => 10 [3,3,2,1] => 21 [3,3,1,1,1] => 91 [3,2,2,2] => 23 [3,2,2,1,1] => 85 [3,2,1,1,1,1] => 471 [3,1,1,1,1,1,1] => 4285 [2,2,2,2,1] => 110 [2,2,2,1,1,1] => 512 [2,2,1,1,1,1,1] => 3586 [2,1,1,1,1,1,1,1] => 39468 [1,1,1,1,1,1,1,1,1] => 871030 [10] => 1 [9,1] => 2 [8,2] => 2 [8,1,1] => 6 [7,3] => 2 [7,2,1] => 5 [7,1,1,1] => 23 [6,4] => 2 [6,3,1] => 5 [6,2,2] => 6 [6,2,1,1] => 18 [6,1,1,1,1] => 110 [5,5] => 3 [5,4,1] => 6 [5,3,2] => 6 [5,3,1,1] => 18 [5,2,2,1] => 18 [5,2,1,1,1] => 78 [5,1,1,1,1,1] => 618 [4,4,2] => 6 [4,4,1,1] => 22 [4,3,3] => 6 [4,3,2,1] => 18 [4,3,1,1,1] => 84 [4,2,2,2] => 26 [4,2,2,1,1] => 78 [4,2,1,1,1,1] => 426 [4,1,1,1,1,1,1] => 4086 [3,3,3,1] => 23 [3,3,2,2] => 22 [3,3,2,1,1] => 82 [3,3,1,1,1,1] => 470 [3,2,2,2,1] => 86 [3,2,2,1,1,1] => 412 [3,2,1,1,1,1,1] => 2866 [3,1,1,1,1,1,1,1] => 32048 [2,2,2,2,2] => 246 [2,2,2,2,1,1] => 582 [2,2,2,1,1,1,1] => 3134 [2,2,1,1,1,1,1,1] => 26038 [2,1,1,1,1,1,1,1,1] => 340198 [1,1,1,1,1,1,1,1,1,1] => 8879558 [11] => 1 [10,1] => 2 [9,2] => 2 [9,1,1] => 6 [8,3] => 2 [8,2,1] => 5 [8,1,1,1] => 23 [7,4] => 2 [7,3,1] => 5 [7,2,2] => 6 [7,2,1,1] => 18 [7,1,1,1,1] => 110 [6,5] => 2 [6,4,1] => 5 [6,3,2] => 5 [6,3,1,1] => 17 [6,2,2,1] => 17 [6,2,1,1,1] => 77 [6,1,1,1,1,1] => 617 [5,5,1] => 6 [5,4,2] => 5 [5,4,1,1] => 19 [5,3,3] => 6 [5,3,2,1] => 17 [5,3,1,1,1] => 75 [5,2,2,2] => 23 [5,2,2,1,1] => 73 [5,2,1,1,1,1] => 407 [5,1,1,1,1,1,1] => 4041 [4,4,3] => 6 [4,4,2,1] => 17 [4,4,1,1,1] => 87 [4,3,3,1] => 19 [4,3,2,2] => 18 [4,3,2,1,1] => 70 [4,3,1,1,1,1] => 418 [4,2,2,2,1] => 77 [4,2,2,1,1,1] => 363 [4,2,1,1,1,1,1] => 2545 [4,1,1,1,1,1,1,1] => 30319 [3,3,3,2] => 23 [3,3,3,1,1] => 87 [3,3,2,2,1] => 79 [3,3,2,1,1,1] => 383 [3,3,1,1,1,1,1] => 2751 [3,2,2,2,2] => 110 [3,2,2,2,1,1] => 398 [3,2,2,1,1,1,1] => 2326 [3,2,1,1,1,1,1,1] => 19742 [3,1,1,1,1,1,1,1,1] => 268350 [2,2,2,2,2,1] => 617 [2,2,2,2,1,1,1] => 2855 [2,2,2,1,1,1,1,1] => 19853 [2,2,1,1,1,1,1,1,1] => 201403 [2,1,1,1,1,1,1,1,1,1] => 3166097 [1,1,1,1,1,1,1,1,1,1,1] => 98329551 [12] => 1 [11,1] => 2 [10,2] => 2 [10,1,1] => 6 [9,3] => 2 [9,2,1] => 5 [9,1,1,1] => 23 [8,4] => 2 [8,3,1] => 5 [8,2,2] => 6 [8,2,1,1] => 18 [8,1,1,1,1] => 110 [7,5] => 2 [7,4,1] => 5 [7,3,2] => 5 [7,3,1,1] => 17 [7,2,2,1] => 17 [7,2,1,1,1] => 77 [7,1,1,1,1,1] => 617 [6,6] => 3 [6,5,1] => 6 [6,4,2] => 6 [6,4,1,1] => 18 [6,3,3] => 7 [6,3,2,1] => 17 [6,3,1,1,1] => 73 [6,2,2,2] => 24 [6,2,2,1,1] => 72 [6,2,1,1,1,1] => 404 [6,1,1,1,1,1,1] => 4036 [5,5,2] => 6 [5,5,1,1] => 22 [5,4,3] => 5 [5,4,2,1] => 17 [5,4,1,1,1] => 83 [5,3,3,1] => 18 [5,3,2,2] => 19 [5,3,2,1,1] => 69 [5,3,1,1,1,1] => 387 [5,2,2,2,1] => 75 [5,2,2,1,1,1] => 349 [5,2,1,1,1,1,1] => 2451 [5,1,1,1,1,1,1,1] => 29997 [4,4,4] => 10 [4,4,3,1] => 21 [4,4,2,2] => 26 [4,4,2,1,1] => 74 [4,4,1,1,1,1] => 450 [4,3,3,2] => 18 [4,3,3,1,1] => 78 [4,3,2,2,1] => 73 [4,3,2,1,1,1] => 345 [4,3,1,1,1,1,1] => 2505 [4,2,2,2,2] => 138 [4,2,2,2,1,1] => 378 [4,2,2,1,1,1,1] => 2098 [4,2,1,1,1,1,1,1] => 17682 [4,1,1,1,1,1,1,1,1] => 253818 [3,3,3,3] => 47 [3,3,3,2,1] => 95 [3,3,3,1,1,1] => 419 [3,3,2,2,2] => 88 [3,3,2,2,1,1] => 392 [3,3,2,1,1,1,1] => 2204 [3,3,1,1,1,1,1,1] => 18908 [3,2,2,2,2,1] => 437 [3,2,2,2,1,1,1] => 2233 [3,2,2,1,1,1,1,1] => 15389 [3,2,1,1,1,1,1,1,1] => 154649 [3,1,1,1,1,1,1,1,1,1] => 2501957 [2,2,2,2,2,2] => 1602 [2,2,2,2,2,1,1] => 3546 [2,2,2,2,1,1,1,1] => 19274 [2,2,2,1,1,1,1,1,1] => 154210 [2,2,1,1,1,1,1,1,1,1] => 1807634 [2,1,1,1,1,1,1,1,1,1,1] => 33022346 [1,1,1,1,1,1,1,1,1,1,1,1] => 1191578522 ----------------------------------------------------------------------------- Created: May 20, 2017 at 17:57 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 31, 2017 at 08:07 by Martin Rubey