***************************************************************************** * 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: St000818 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. For example, $QS_{31} = M_{1111} + M_{121} + M_{211} + M_{31}$, so the statistic on the composition $31$ is 4. Apparently, the sum over all compositions gives the sequence [[oeis:A138178]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): M = QuasiSymmetricFunctions(ZZ).M() QS = QuasiSymmetricFunctions(ZZ).QS() return sum(coeff for _, coeff in M(QS(mu))) ----------------------------------------------------------------------------- Statistic values: [1,1] => 1 [2] => 2 [1,1,1] => 1 [1,2] => 2 [2,1] => 2 [3] => 4 [1,1,1,1] => 1 [1,1,2] => 2 [1,2,1] => 2 [1,3] => 8 [2,1,1] => 2 [2,2] => 6 [3,1] => 4 [4] => 8 [1,1,1,1,1] => 1 [1,1,1,2] => 2 [1,1,2,1] => 2 [1,1,3] => 12 [1,2,1,1] => 2 [1,2,2] => 6 [1,3,1] => 8 [1,4] => 24 [2,1,1,1] => 2 [2,1,2] => 4 [2,2,1] => 6 [2,3] => 12 [3,1,1] => 4 [3,2] => 16 [4,1] => 8 [5] => 16 [1,1,1,1,1,1] => 1 [1,1,1,1,2] => 2 [1,1,1,2,1] => 2 [1,1,1,3] => 16 [1,1,2,1,1] => 2 [1,1,2,2] => 6 [1,1,3,1] => 12 [1,1,4] => 48 [1,2,1,1,1] => 2 [1,2,1,2] => 4 [1,2,2,1] => 6 [1,2,3] => 12 [1,3,1,1] => 8 [1,3,2] => 28 [1,4,1] => 24 [1,5] => 64 [2,1,1,1,1] => 2 [2,1,1,2] => 4 [2,1,2,1] => 4 [2,1,3] => 20 [2,2,1,1] => 6 [2,2,2] => 22 [2,3,1] => 12 [2,4] => 56 [3,1,1,1] => 4 [3,1,2] => 8 [3,2,1] => 16 [3,3] => 44 [4,1,1] => 8 [4,2] => 40 [5,1] => 16 [6] => 32 [1,1,1,1,1,1,1] => 1 [1,1,1,1,1,2] => 2 [1,1,1,1,2,1] => 2 [1,1,1,1,3] => 20 [1,1,1,2,1,1] => 2 [1,1,1,2,2] => 6 [1,1,1,3,1] => 16 [1,1,1,4] => 80 [1,1,2,1,1,1] => 2 [1,1,2,1,2] => 4 [1,1,2,2,1] => 6 [1,1,2,3] => 12 [1,1,3,1,1] => 12 [1,1,3,2] => 40 [1,1,4,1] => 48 [1,1,5] => 160 [1,2,1,1,1,1] => 2 [1,2,1,1,2] => 4 [1,2,1,2,1] => 4 [1,2,1,3] => 20 [1,2,2,1,1] => 6 [1,2,2,2] => 22 [1,2,3,1] => 12 [1,2,4] => 80 [1,3,1,1,1] => 8 [1,3,1,2] => 16 [1,3,2,1] => 28 [1,3,3] => 112 [1,4,1,1] => 24 [1,4,2] => 96 [1,5,1] => 64 [1,6] => 160 [2,1,1,1,1,1] => 2 [2,1,1,1,2] => 4 [2,1,1,2,1] => 4 [2,1,1,3] => 28 [2,1,2,1,1] => 4 [2,1,2,2] => 12 [2,1,3,1] => 20 [2,1,4] => 112 [2,2,1,1,1] => 6 [2,2,1,2] => 12 [2,2,2,1] => 22 [2,2,3] => 44 [2,3,1,1] => 12 [2,3,2] => 56 [2,4,1] => 56 [2,5] => 192 [3,1,1,1,1] => 4 [3,1,1,2] => 8 [3,1,2,1] => 8 [3,1,3] => 48 [3,2,1,1] => 16 [3,2,2] => 68 [3,3,1] => 44 [3,4] => 88 [4,1,1,1] => 8 [4,1,2] => 16 [4,2,1] => 40 [4,3] => 136 [5,1,1] => 16 [5,2] => 96 [6,1] => 32 [7] => 64 [1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,2] => 2 [1,1,1,1,1,2,1] => 2 [1,1,1,1,1,3] => 24 [1,1,1,1,2,1,1] => 2 [1,1,1,1,2,2] => 6 [1,1,1,1,3,1] => 20 [1,1,1,1,4] => 120 [1,1,1,2,1,1,1] => 2 [1,1,1,2,1,2] => 4 [1,1,1,2,2,1] => 6 [1,1,1,2,3] => 12 [1,1,1,3,1,1] => 16 [1,1,1,3,2] => 52 [1,1,1,4,1] => 80 [1,1,1,5] => 320 [1,1,2,1,1,1,1] => 2 [1,1,2,1,1,2] => 4 [1,1,2,1,2,1] => 4 [1,1,2,1,3] => 20 [1,1,2,2,1,1] => 6 [1,1,2,2,2] => 22 [1,1,2,3,1] => 12 [1,1,2,4] => 104 [1,1,3,1,1,1] => 12 [1,1,3,1,2] => 24 [1,1,3,2,1] => 40 [1,1,3,3] => 204 [1,1,4,1,1] => 48 [1,1,4,2] => 176 [1,1,5,1] => 160 [1,1,6] => 480 [1,2,1,1,1,1,1] => 2 [1,2,1,1,1,2] => 4 [1,2,1,1,2,1] => 4 [1,2,1,1,3] => 28 [1,2,1,2,1,1] => 4 [1,2,1,2,2] => 12 [1,2,1,3,1] => 20 [1,2,1,4] => 152 [1,2,2,1,1,1] => 6 [1,2,2,1,2] => 12 [1,2,2,2,1] => 22 [1,2,2,3] => 44 [1,2,3,1,1] => 12 [1,2,3,2] => 56 [1,2,4,1] => 80 [1,2,5] => 352 [1,3,1,1,1,1] => 8 [1,3,1,1,2] => 16 [1,3,1,2,1] => 16 [1,3,1,3] => 112 [1,3,2,1,1] => 28 [1,3,2,2] => 112 [1,3,3,1] => 112 [1,3,4] => 224 [1,4,1,1,1] => 24 [1,4,1,2] => 48 [1,4,2,1] => 96 [1,4,3] => 496 [1,5,1,1] => 64 [1,5,2] => 288 [1,6,1] => 160 [1,7] => 384 [2,1,1,1,1,1,1] => 2 [2,1,1,1,1,2] => 4 [2,1,1,1,2,1] => 4 [2,1,1,1,3] => 36 [2,1,1,2,1,1] => 4 [2,1,1,2,2] => 12 [2,1,1,3,1] => 28 [2,1,1,4] => 184 [2,1,2,1,1,1] => 4 [2,1,2,1,2] => 8 [2,1,2,2,1] => 12 [2,1,2,3] => 24 [2,1,3,1,1] => 20 [2,1,3,2] => 68 [2,1,4,1] => 112 [2,1,5] => 448 [2,2,1,1,1,1] => 6 [2,2,1,1,2] => 12 [2,2,1,2,1] => 12 [2,2,1,3] => 68 [2,2,2,1,1] => 22 [2,2,2,2] => 90 [2,2,3,1] => 44 [2,2,4] => 336 [2,3,1,1,1] => 12 [2,3,1,2] => 24 [2,3,2,1] => 56 [2,3,3] => 156 [2,4,1,1] => 56 [2,4,2] => 304 [2,5,1] => 192 [2,6] => 576 [3,1,1,1,1,1] => 4 [3,1,1,1,2] => 8 [3,1,1,2,1] => 8 [3,1,1,3] => 64 [3,1,2,1,1] => 8 [3,1,2,2] => 24 [3,1,3,1] => 48 [3,1,4] => 184 [3,2,1,1,1] => 16 [3,2,1,2] => 32 [3,2,2,1] => 68 [3,2,3] => 136 [3,3,1,1] => 44 [3,3,2] => 248 [3,4,1] => 88 [3,5] => 448 [4,1,1,1,1] => 8 [4,1,1,2] => 16 [4,1,2,1] => 16 [4,1,3] => 112 [4,2,1,1] => 40 [4,2,2] => 192 [4,3,1] => 136 [4,4] => 360 [5,1,1,1] => 16 [5,1,2] => 32 [5,2,1] => 96 [5,3] => 384 [6,1,1] => 32 [6,2] => 224 [7,1] => 64 [8] => 128 [1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,2] => 2 [1,1,1,1,1,1,2,1] => 2 [1,1,1,1,1,1,3] => 28 [1,1,1,1,1,2,1,1] => 2 [1,1,1,1,1,2,2] => 6 [1,1,1,1,1,3,1] => 24 [1,1,1,1,1,4] => 168 [1,1,1,1,2,1,1,1] => 2 [1,1,1,1,2,1,2] => 4 [1,1,1,1,2,2,1] => 6 [1,1,1,1,2,3] => 12 [1,1,1,1,3,1,1] => 20 [1,1,1,1,3,2] => 64 [1,1,1,1,4,1] => 120 [1,1,1,1,5] => 560 [1,1,1,2,1,1,1,1] => 2 [1,1,1,2,1,1,2] => 4 [1,1,1,2,1,2,1] => 4 [1,1,1,2,1,3] => 20 [1,1,1,2,2,1,1] => 6 [1,1,1,2,2,2] => 22 [1,1,1,2,3,1] => 12 [1,1,1,2,4] => 128 [1,1,1,3,1,1,1] => 16 [1,1,1,3,1,2] => 32 [1,1,1,3,2,1] => 52 [1,1,1,3,3] => 320 [1,1,1,4,1,1] => 80 [1,1,1,4,2] => 280 [1,1,1,5,1] => 320 [1,1,1,6] => 1120 [1,1,2,1,1,1,1,1] => 2 [1,1,2,1,1,1,2] => 4 [1,1,2,1,1,2,1] => 4 [1,1,2,1,1,3] => 28 [1,1,2,1,2,1,1] => 4 [1,1,2,1,2,2] => 12 [1,1,2,1,3,1] => 20 [1,1,2,1,4] => 192 [1,1,2,2,1,1,1] => 6 [1,1,2,2,1,2] => 12 [1,1,2,2,2,1] => 22 [1,1,2,2,3] => 44 [1,1,2,3,1,1] => 12 [1,1,2,3,2] => 56 [1,1,2,4,1] => 104 [1,1,2,5] => 560 [1,1,3,1,1,1,1] => 12 [1,1,3,1,1,2] => 24 [1,1,3,1,2,1] => 24 [1,1,3,1,3] => 192 [1,1,3,2,1,1] => 40 [1,1,3,2,2] => 156 [1,1,3,3,1] => 204 [1,1,3,4] => 408 [1,1,4,1,1,1] => 48 [1,1,4,1,2] => 96 [1,1,4,2,1] => 176 [1,1,4,3] => 1176 [1,1,5,1,1] => 160 [1,1,5,2] => 640 [1,1,6,1] => 480 [1,1,7] => 1344 [1,2,1,1,1,1,1,1] => 2 [1,2,1,1,1,1,2] => 4 [1,2,1,1,1,2,1] => 4 [1,2,1,1,1,3] => 36 [1,2,1,1,2,1,1] => 4 [1,2,1,1,2,2] => 12 [1,2,1,1,3,1] => 28 [1,2,1,1,4] => 240 [1,2,1,2,1,1,1] => 4 [1,2,1,2,1,2] => 8 [1,2,1,2,2,1] => 12 [1,2,1,2,3] => 24 [1,2,1,3,1,1] => 20 [1,2,1,3,2] => 68 [1,2,1,4,1] => 152 [1,2,1,5] => 752 [1,2,2,1,1,1,1] => 6 [1,2,2,1,1,2] => 12 [1,2,2,1,2,1] => 12 [1,2,2,1,3] => 68 [1,2,2,2,1,1] => 22 [1,2,2,2,2] => 90 [1,2,2,3,1] => 44 [1,2,2,4] => 424 [1,2,3,1,1,1] => 12 [1,2,3,1,2] => 24 [1,2,3,2,1] => 56 [1,2,3,3] => 156 [1,2,4,1,1] => 80 [1,2,4,2] => 416 [1,2,5,1] => 352 [1,2,6] => 1280 [1,3,1,1,1,1,1] => 8 [1,3,1,1,1,2] => 16 [1,3,1,1,2,1] => 16 [1,3,1,1,3] => 144 [1,3,1,2,1,1] => 16 [1,3,1,2,2] => 48 [1,3,1,3,1] => 112 [1,3,1,4] => 448 [1,3,2,1,1,1] => 28 [1,3,2,1,2] => 56 [1,3,2,2,1] => 112 [1,3,2,3] => 224 [1,3,3,1,1] => 112 [1,3,3,2] => 540 [1,3,4,1] => 224 [1,3,5] => 1440 [1,4,1,1,1,1] => 24 [1,4,1,1,2] => 48 [1,4,1,2,1] => 48 [1,4,1,3] => 400 [1,4,2,1,1] => 96 [1,4,2,2] => 416 [1,4,3,1] => 496 [1,4,4] => 1576 [1,5,1,1,1] => 64 [1,5,1,2] => 128 [1,5,2,1] => 288 [1,5,3] => 1760 [1,6,1,1] => 160 [1,6,2] => 800 [1,7,1] => 384 [1,8] => 896 [2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,2] => 4 [2,1,1,1,1,2,1] => 4 [2,1,1,1,1,3] => 44 [2,1,1,1,2,1,1] => 4 [2,1,1,1,2,2] => 12 [2,1,1,1,3,1] => 36 [2,1,1,1,4] => 272 [2,1,1,2,1,1,1] => 4 [2,1,1,2,1,2] => 8 [2,1,1,2,2,1] => 12 [2,1,1,2,3] => 24 [2,1,1,3,1,1] => 28 [2,1,1,3,2] => 92 [2,1,1,4,1] => 184 [2,1,1,5] => 848 [2,1,2,1,1,1,1] => 4 [2,1,2,1,1,2] => 8 [2,1,2,1,2,1] => 8 [2,1,2,1,3] => 40 [2,1,2,2,1,1] => 12 [2,1,2,2,2] => 44 [2,1,2,3,1] => 24 [2,1,2,4] => 232 [2,1,3,1,1,1] => 20 [2,1,3,1,2] => 40 [2,1,3,2,1] => 68 [2,1,3,3] => 316 [2,1,4,1,1] => 112 [2,1,4,2] => 408 [2,1,5,1] => 448 [2,1,6] => 1536 [2,2,1,1,1,1,1] => 6 [2,2,1,1,1,2] => 12 [2,2,1,1,2,1] => 12 [2,2,1,1,3] => 92 [2,2,1,2,1,1] => 12 [2,2,1,2,2] => 36 [2,2,1,3,1] => 68 [2,2,1,4] => 584 [2,2,2,1,1,1] => 22 [2,2,2,1,2] => 44 [2,2,2,2,1] => 90 [2,2,2,3] => 180 [2,2,3,1,1] => 44 [2,2,3,2] => 224 [2,2,4,1] => 336 [2,2,5] => 1664 [2,3,1,1,1,1] => 12 [2,3,1,1,2] => 24 [2,3,1,2,1] => 24 [2,3,1,3] => 160 [2,3,2,1,1] => 56 [2,3,2,2] => 260 [2,3,3,1] => 156 [2,3,4] => 312 [2,4,1,1,1] => 56 [2,4,1,2] => 112 [2,4,2,1] => 304 [2,4,3] => 944 [2,5,1,1] => 192 [2,5,2] => 1184 [2,6,1] => 576 [2,7] => 1600 [3,1,1,1,1,1,1] => 4 [3,1,1,1,1,2] => 8 [3,1,1,1,2,1] => 8 [3,1,1,1,3] => 80 [3,1,1,2,1,1] => 8 [3,1,1,2,2] => 24 [3,1,1,3,1] => 64 [3,1,1,4] => 312 [3,1,2,1,1,1] => 8 [3,1,2,1,2] => 16 [3,1,2,2,1] => 24 [3,1,2,3] => 48 [3,1,3,1,1] => 48 [3,1,3,2] => 160 [3,1,4,1] => 184 [3,1,5] => 1040 [3,2,1,1,1,1] => 16 [3,2,1,1,2] => 32 [3,2,1,2,1] => 32 [3,2,1,3] => 200 [3,2,2,1,1] => 68 [3,2,2,2] => 304 [3,2,3,1] => 136 [3,2,4] => 768 [3,3,1,1,1] => 44 [3,3,1,2] => 88 [3,3,2,1] => 248 [3,3,3] => 788 [3,4,1,1] => 88 [3,4,2] => 584 [3,5,1] => 448 [3,6] => 1664 [4,1,1,1,1,1] => 8 [4,1,1,1,2] => 16 [4,1,1,2,1] => 16 [4,1,1,3] => 144 [4,1,2,1,1] => 16 [4,1,2,2] => 48 [4,1,3,1] => 112 [4,1,4] => 496 [4,2,1,1,1] => 40 [4,2,1,2] => 80 [4,2,2,1] => 192 [4,2,3] => 384 [4,3,1,1] => 136 [4,3,2] => 880 [4,4,1] => 360 [4,5] => 720 [5,1,1,1,1] => 16 [5,1,1,2] => 32 [5,1,2,1] => 32 [5,1,3] => 256 [5,2,1,1] => 96 [5,2,2] => 512 [5,3,1] => 384 [5,4] => 1216 [6,1,1,1] => 32 [6,1,2] => 64 [6,2,1] => 224 [6,3] => 1024 [7,1,1] => 64 [7,2] => 512 [8,1] => 128 [9] => 256 ----------------------------------------------------------------------------- Created: May 20, 2017 at 21:59 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 20, 2017 at 22:26 by Martin Rubey