***************************************************************************** * 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: St000814 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. For example, \$e_{22} = s_{1111} + s_{211} + s_{22}\$, so the statistic on the partition \$22\$ is \$3\$. ----------------------------------------------------------------------------- References: [1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. [[OEIS:A178718]] ----------------------------------------------------------------------------- Code: def statistic(mu): s = SymmetricFunctions(ZZ).s() e = SymmetricFunctions(ZZ).e() return sum(coeff for _, coeff in s(e(mu))) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 2 [1,1,1] => 4 [4] => 1 [3,1] => 2 [2,2] => 3 [2,1,1] => 5 [1,1,1,1] => 10 [5] => 1 [4,1] => 2 [3,2] => 3 [3,1,1] => 5 [2,2,1] => 7 [2,1,1,1] => 13 [1,1,1,1,1] => 26 [6] => 1 [5,1] => 2 [4,2] => 3 [4,1,1] => 5 [3,3] => 4 [3,2,1] => 8 [3,1,1,1] => 14 [2,2,2] => 11 [2,2,1,1] => 20 [2,1,1,1,1] => 38 [1,1,1,1,1,1] => 76 [7] => 1 [6,1] => 2 [5,2] => 3 [5,1,1] => 5 [4,3] => 4 [4,2,1] => 8 [4,1,1,1] => 14 [3,3,1] => 10 [3,2,2] => 13 [3,2,1,1] => 23 [3,1,1,1,1] => 42 [2,2,2,1] => 32 [2,2,1,1,1] => 60 [2,1,1,1,1,1] => 116 [1,1,1,1,1,1,1] => 232 [8] => 1 [7,1] => 2 [6,2] => 3 [6,1,1] => 5 [5,3] => 4 [5,2,1] => 8 [5,1,1,1] => 14 [4,4] => 5 [4,3,1] => 11 [4,2,2] => 14 [4,2,1,1] => 24 [4,1,1,1,1] => 43 [3,3,2] => 17 [3,3,1,1] => 30 [3,2,2,1] => 40 [3,2,1,1,1] => 73 [3,1,1,1,1,1] => 136 [2,2,2,2] => 56 [2,2,2,1,1] => 103 [2,2,1,1,1,1] => 196 [2,1,1,1,1,1,1] => 382 [1,1,1,1,1,1,1,1] => 764 [9] => 1 [8,1] => 2 [7,2] => 3 [7,1,1] => 5 [6,3] => 4 [6,2,1] => 8 [6,1,1,1] => 14 [5,4] => 5 [5,3,1] => 11 [5,2,2] => 14 [5,2,1,1] => 24 [5,1,1,1,1] => 43 [4,4,1] => 13 [4,3,2] => 19 [4,3,1,1] => 33 [4,2,2,1] => 43 [4,2,1,1,1] => 77 [4,1,1,1,1,1] => 141 [3,3,3] => 23 [3,3,2,1] => 53 [3,3,1,1,1] => 96 [3,2,2,2] => 72 [3,2,2,1,1] => 131 [3,2,1,1,1,1] => 244 [3,1,1,1,1,1,1] => 462 [2,2,2,2,1] => 184 [2,2,2,1,1,1] => 347 [2,2,1,1,1,1,1] => 668 [2,1,1,1,1,1,1,1] => 1310 [1,1,1,1,1,1,1,1,1] => 2620 [10] => 1 [9,1] => 2 [8,2] => 3 [8,1,1] => 5 [7,3] => 4 [7,2,1] => 8 [7,1,1,1] => 14 [6,4] => 5 [6,3,1] => 11 [6,2,2] => 14 [6,2,1,1] => 24 [6,1,1,1,1] => 43 [5,5] => 6 [5,4,1] => 14 [5,3,2] => 20 [5,3,1,1] => 34 [5,2,2,1] => 44 [5,2,1,1,1] => 78 [5,1,1,1,1,1] => 142 [4,4,2] => 23 [4,4,1,1] => 40 [4,3,3] => 27 [4,3,2,1] => 61 [4,3,1,1,1] => 109 [4,2,2,2] => 81 [4,2,2,1,1] => 145 [4,2,1,1,1,1] => 265 [4,1,1,1,1,1,1] => 492 [3,3,3,1] => 74 [3,3,2,2] => 100 [3,3,2,1,1] => 180 [3,3,1,1,1,1] => 332 [3,2,2,2,1] => 248 [3,2,2,1,1,1] => 460 [3,2,1,1,1,1,1] => 868 [3,1,1,1,1,1,1,1] => 1660 [2,2,2,2,2] => 348 [2,2,2,2,1,1] => 652 [2,2,2,1,1,1,1] => 1244 [2,2,1,1,1,1,1,1] => 2412 [2,1,1,1,1,1,1,1,1] => 4748 [1,1,1,1,1,1,1,1,1,1] => 9496 [11] => 1 [10,1] => 2 [9,2] => 3 [9,1,1] => 5 [8,3] => 4 [8,2,1] => 8 [8,1,1,1] => 14 [7,4] => 5 [7,3,1] => 11 [7,2,2] => 14 [7,2,1,1] => 24 [7,1,1,1,1] => 43 [6,5] => 6 [6,4,1] => 14 [6,3,2] => 20 [6,3,1,1] => 34 [6,2,2,1] => 44 [6,2,1,1,1] => 78 [6,1,1,1,1,1] => 142 [5,5,1] => 16 [5,4,2] => 25 [5,4,1,1] => 43 [5,3,3] => 29 [5,3,2,1] => 64 [5,3,1,1,1] => 113 [5,2,2,2] => 84 [5,2,2,1,1] => 149 [5,2,1,1,1,1] => 270 [5,1,1,1,1,1,1] => 498 [4,4,3] => 33 [4,4,2,1] => 74 [4,4,1,1,1] => 132 [4,3,3,1] => 88 [4,3,2,2] => 117 [4,3,2,1,1] => 209 [4,3,1,1,1,1] => 381 [4,2,2,2,1] => 282 [4,2,2,1,1,1] => 516 [4,2,1,1,1,1,1] => 958 [4,1,1,1,1,1,1,1] => 1800 [3,3,3,2] => 143 [3,3,3,1,1] => 256 [3,3,2,2,1] => 350 [3,3,2,1,1,1] => 644 [3,3,1,1,1,1,1] => 1204 [3,2,2,2,2] => 484 [3,2,2,2,1,1] => 897 [3,2,2,1,1,1,1] => 1688 [3,2,1,1,1,1,1,1] => 3218 [3,1,1,1,1,1,1,1,1] => 6204 [2,2,2,2,2,1] => 1268 [2,2,2,2,1,1,1] => 2408 [2,2,2,1,1,1,1,1] => 4636 [2,2,1,1,1,1,1,1,1] => 9040 [2,1,1,1,1,1,1,1,1,1] => 17848 [1,1,1,1,1,1,1,1,1,1,1] => 35696 [12] => 1 [11,1] => 2 [10,2] => 3 [10,1,1] => 5 [9,3] => 4 [9,2,1] => 8 [9,1,1,1] => 14 [8,4] => 5 [8,3,1] => 11 [8,2,2] => 14 [8,2,1,1] => 24 [8,1,1,1,1] => 43 [7,5] => 6 [7,4,1] => 14 [7,3,2] => 20 [7,3,1,1] => 34 [7,2,2,1] => 44 [7,2,1,1,1] => 78 [7,1,1,1,1,1] => 142 [6,6] => 7 [6,5,1] => 17 [6,4,2] => 26 [6,4,1,1] => 44 [6,3,3] => 30 [6,3,2,1] => 65 [6,3,1,1,1] => 114 [6,2,2,2] => 85 [6,2,2,1,1] => 150 [6,2,1,1,1,1] => 271 [6,1,1,1,1,1,1] => 499 [5,5,2] => 29 [5,5,1,1] => 50 [5,4,3] => 37 [5,4,2,1] => 82 [5,4,1,1,1] => 145 [5,3,3,1] => 96 [5,3,2,2] => 126 [5,3,2,1,1] => 223 [5,3,1,1,1,1] => 402 [5,2,2,2,1] => 297 [5,2,2,1,1,1] => 538 [5,2,1,1,1,1,1] => 989 [5,1,1,1,1,1,1,1] => 1842 [4,4,4] => 42 [4,4,3,1] => 110 [4,4,2,2] => 146 [4,4,2,1,1] => 259 [4,4,1,1,1,1] => 470 [4,3,3,2] => 175 [4,3,3,1,1] => 311 [4,3,2,2,1] => 420 [4,3,2,1,1,1] => 765 [4,3,1,1,1,1,1] => 1414 [4,2,2,2,2] => 572 [4,2,2,2,1,1] => 1046 [4,2,2,1,1,1,1] => 1941 [4,2,1,1,1,1,1,1] => 3643 [4,1,1,1,1,1,1,1,1] => 6904 [3,3,3,3] => 214 [3,3,3,2,1] => 517 [3,3,3,1,1,1] => 944 [3,3,2,2,2] => 710 [3,3,2,2,1,1] => 1306 [3,3,2,1,1,1,1] => 2436 [3,3,1,1,1,1,1,1] => 4600 [3,2,2,2,2,1] => 1824 [3,2,2,2,1,1,1] => 3425 [3,2,2,1,1,1,1,1] => 6508 [3,2,1,1,1,1,1,1,1] => 12498 [3,1,1,1,1,1,1,1,1,1] => 24232 [2,2,2,2,2,2] => 2578 [2,2,2,2,2,1,1] => 4876 [2,2,2,2,1,1,1,1] => 9340 [2,2,2,1,1,1,1,1,1] => 18092 [2,2,1,1,1,1,1,1,1,1] => 35420 [2,1,1,1,1,1,1,1,1,1,1] => 70076 [1,1,1,1,1,1,1,1,1,1,1,1] => 140152 ----------------------------------------------------------------------------- Created: May 20, 2017 at 17:44 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 31, 2017 at 08:10 by Martin Rubey