***************************************************************************** * 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: St000817 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. For example, $dI_{121} = 2M_{1111} + M_{112} + M_{121}$, so the statistic on the composition $121$ is 4. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(mu): M = QuasiSymmetricFunctions(ZZ).M() dI = QuasiSymmetricFunctions(ZZ).dI() return sum(coeff for _, coeff in M(dI(mu))) ----------------------------------------------------------------------------- Statistic values: [1,1] => 1 [2] => 2 [1,1,1] => 1 [1,2] => 2 [2,1] => 4 [3] => 4 [1,1,1,1] => 1 [1,1,2] => 2 [1,2,1] => 4 [1,3] => 4 [2,1,1] => 6 [2,2] => 10 [3,1] => 12 [4] => 8 [1,1,1,1,1] => 1 [1,1,1,2] => 2 [1,1,2,1] => 4 [1,1,3] => 4 [1,2,1,1] => 6 [1,2,2] => 10 [1,3,1] => 12 [1,4] => 8 [2,1,1,1] => 8 [2,1,2] => 14 [2,2,1] => 28 [2,3] => 24 [3,1,1] => 24 [3,2] => 36 [4,1] => 32 [5] => 16 [1,1,1,1,1,1] => 1 [1,1,1,1,2] => 2 [1,1,1,2,1] => 4 [1,1,1,3] => 4 [1,1,2,1,1] => 6 [1,1,2,2] => 10 [1,1,3,1] => 12 [1,1,4] => 8 [1,2,1,1,1] => 8 [1,2,1,2] => 14 [1,2,2,1] => 28 [1,2,3] => 24 [1,3,1,1] => 24 [1,3,2] => 36 [1,4,1] => 32 [1,5] => 16 [2,1,1,1,1] => 10 [2,1,1,2] => 18 [2,1,2,1] => 36 [2,1,3] => 32 [2,2,1,1] => 54 [2,2,2] => 82 [2,3,1] => 96 [2,4] => 56 [3,1,1,1] => 40 [3,1,2] => 64 [3,2,1] => 128 [3,3] => 100 [4,1,1] => 80 [4,2] => 112 [5,1] => 80 [6] => 32 [1,1,1,1,1,1,1] => 1 [1,1,1,1,1,2] => 2 [1,1,1,1,2,1] => 4 [1,1,1,1,3] => 4 [1,1,1,2,1,1] => 6 [1,1,1,2,2] => 10 [1,1,1,3,1] => 12 [1,1,1,4] => 8 [1,1,2,1,1,1] => 8 [1,1,2,1,2] => 14 [1,1,2,2,1] => 28 [1,1,2,3] => 24 [1,1,3,1,1] => 24 [1,1,3,2] => 36 [1,1,4,1] => 32 [1,1,5] => 16 [1,2,1,1,1,1] => 10 [1,2,1,1,2] => 18 [1,2,1,2,1] => 36 [1,2,1,3] => 32 [1,2,2,1,1] => 54 [1,2,2,2] => 82 [1,2,3,1] => 96 [1,2,4] => 56 [1,3,1,1,1] => 40 [1,3,1,2] => 64 [1,3,2,1] => 128 [1,3,3] => 100 [1,4,1,1] => 80 [1,4,2] => 112 [1,5,1] => 80 [1,6] => 32 [2,1,1,1,1,1] => 12 [2,1,1,1,2] => 22 [2,1,1,2,1] => 44 [2,1,1,3] => 40 [2,1,2,1,1] => 66 [2,1,2,2] => 102 [2,1,3,1] => 120 [2,1,4] => 72 [2,2,1,1,1] => 88 [2,2,1,2] => 142 [2,2,2,1] => 284 [2,2,3] => 224 [2,3,1,1] => 240 [2,3,2] => 336 [2,4,1] => 288 [2,5] => 128 [3,1,1,1,1] => 60 [3,1,1,2] => 100 [3,1,2,1] => 200 [3,1,3] => 164 [3,2,1,1] => 300 [3,2,2] => 428 [3,3,1] => 492 [3,4] => 264 [4,1,1,1] => 160 [4,1,2] => 240 [4,2,1] => 480 [4,3] => 352 [5,1,1] => 240 [5,2] => 320 [6,1] => 192 [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] => 4 [1,1,1,1,1,3] => 4 [1,1,1,1,2,1,1] => 6 [1,1,1,1,2,2] => 10 [1,1,1,1,3,1] => 12 [1,1,1,1,4] => 8 [1,1,1,2,1,1,1] => 8 [1,1,1,2,1,2] => 14 [1,1,1,2,2,1] => 28 [1,1,1,2,3] => 24 [1,1,1,3,1,1] => 24 [1,1,1,3,2] => 36 [1,1,1,4,1] => 32 [1,1,1,5] => 16 [1,1,2,1,1,1,1] => 10 [1,1,2,1,1,2] => 18 [1,1,2,1,2,1] => 36 [1,1,2,1,3] => 32 [1,1,2,2,1,1] => 54 [1,1,2,2,2] => 82 [1,1,2,3,1] => 96 [1,1,2,4] => 56 [1,1,3,1,1,1] => 40 [1,1,3,1,2] => 64 [1,1,3,2,1] => 128 [1,1,3,3] => 100 [1,1,4,1,1] => 80 [1,1,4,2] => 112 [1,1,5,1] => 80 [1,1,6] => 32 [1,2,1,1,1,1,1] => 12 [1,2,1,1,1,2] => 22 [1,2,1,1,2,1] => 44 [1,2,1,1,3] => 40 [1,2,1,2,1,1] => 66 [1,2,1,2,2] => 102 [1,2,1,3,1] => 120 [1,2,1,4] => 72 [1,2,2,1,1,1] => 88 [1,2,2,1,2] => 142 [1,2,2,2,1] => 284 [1,2,2,3] => 224 [1,2,3,1,1] => 240 [1,2,3,2] => 336 [1,2,4,1] => 288 [1,2,5] => 128 [1,3,1,1,1,1] => 60 [1,3,1,1,2] => 100 [1,3,1,2,1] => 200 [1,3,1,3] => 164 [1,3,2,1,1] => 300 [1,3,2,2] => 428 [1,3,3,1] => 492 [1,3,4] => 264 [1,4,1,1,1] => 160 [1,4,1,2] => 240 [1,4,2,1] => 480 [1,4,3] => 352 [1,5,1,1] => 240 [1,5,2] => 320 [1,6,1] => 192 [1,7] => 64 [2,1,1,1,1,1,1] => 14 [2,1,1,1,1,2] => 26 [2,1,1,1,2,1] => 52 [2,1,1,1,3] => 48 [2,1,1,2,1,1] => 78 [2,1,1,2,2] => 122 [2,1,1,3,1] => 144 [2,1,1,4] => 88 [2,1,2,1,1,1] => 104 [2,1,2,1,2] => 170 [2,1,2,2,1] => 340 [2,1,2,3] => 272 [2,1,3,1,1] => 288 [2,1,3,2] => 408 [2,1,4,1] => 352 [2,1,5] => 160 [2,2,1,1,1,1] => 130 [2,2,1,1,2] => 218 [2,2,1,2,1] => 436 [2,2,1,3] => 360 [2,2,2,1,1] => 654 [2,2,2,2] => 938 [2,2,3,1] => 1080 [2,2,4] => 584 [2,3,1,1,1] => 480 [2,3,1,2] => 720 [2,3,2,1] => 1440 [2,3,3] => 1056 [2,4,1,1] => 880 [2,4,2] => 1168 [2,5,1] => 800 [2,6] => 288 [3,1,1,1,1,1] => 84 [3,1,1,1,2] => 144 [3,1,1,2,1] => 288 [3,1,1,3] => 244 [3,1,2,1,1] => 432 [3,1,2,2] => 632 [3,1,3,1] => 732 [3,1,4] => 408 [3,2,1,1,1] => 576 [3,2,1,2] => 876 [3,2,2,1] => 1752 [3,2,3] => 1304 [3,3,1,1] => 1464 [3,3,2] => 1956 [3,4,1] => 1632 [3,5] => 672 [4,1,1,1,1] => 280 [4,1,1,2] => 440 [4,1,2,1] => 880 [4,1,3] => 680 [4,2,1,1] => 1320 [4,2,2] => 1800 [4,3,1] => 2040 [4,4] => 1032 [5,1,1,1] => 560 [5,1,2] => 800 [5,2,1] => 1600 [5,3] => 1120 [6,1,1] => 672 [6,2] => 864 [7,1] => 448 [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] => 4 [1,1,1,1,1,1,3] => 4 [1,1,1,1,1,2,1,1] => 6 [1,1,1,1,1,2,2] => 10 [1,1,1,1,1,3,1] => 12 [1,1,1,1,1,4] => 8 [1,1,1,1,2,1,1,1] => 8 [1,1,1,1,2,1,2] => 14 [1,1,1,1,2,2,1] => 28 [1,1,1,1,2,3] => 24 [1,1,1,1,3,1,1] => 24 [1,1,1,1,3,2] => 36 [1,1,1,1,4,1] => 32 [1,1,1,1,5] => 16 [1,1,1,2,1,1,1,1] => 10 [1,1,1,2,1,1,2] => 18 [1,1,1,2,1,2,1] => 36 [1,1,1,2,1,3] => 32 [1,1,1,2,2,1,1] => 54 [1,1,1,2,2,2] => 82 [1,1,1,2,3,1] => 96 [1,1,1,2,4] => 56 [1,1,1,3,1,1,1] => 40 [1,1,1,3,1,2] => 64 [1,1,1,3,2,1] => 128 [1,1,1,3,3] => 100 [1,1,1,4,1,1] => 80 [1,1,1,4,2] => 112 [1,1,1,5,1] => 80 [1,1,1,6] => 32 [1,1,2,1,1,1,1,1] => 12 [1,1,2,1,1,1,2] => 22 [1,1,2,1,1,2,1] => 44 [1,1,2,1,1,3] => 40 [1,1,2,1,2,1,1] => 66 [1,1,2,1,2,2] => 102 [1,1,2,1,3,1] => 120 [1,1,2,1,4] => 72 [1,1,2,2,1,1,1] => 88 [1,1,2,2,1,2] => 142 [1,1,2,2,2,1] => 284 [1,1,2,2,3] => 224 [1,1,2,3,1,1] => 240 [1,1,2,3,2] => 336 [1,1,2,4,1] => 288 [1,1,2,5] => 128 [1,1,3,1,1,1,1] => 60 [1,1,3,1,1,2] => 100 [1,1,3,1,2,1] => 200 [1,1,3,1,3] => 164 [1,1,3,2,1,1] => 300 [1,1,3,2,2] => 428 [1,1,3,3,1] => 492 [1,1,3,4] => 264 [1,1,4,1,1,1] => 160 [1,1,4,1,2] => 240 [1,1,4,2,1] => 480 [1,1,4,3] => 352 [1,1,5,1,1] => 240 [1,1,5,2] => 320 [1,1,6,1] => 192 [1,1,7] => 64 [1,2,1,1,1,1,1,1] => 14 [1,2,1,1,1,1,2] => 26 [1,2,1,1,1,2,1] => 52 [1,2,1,1,1,3] => 48 [1,2,1,1,2,1,1] => 78 [1,2,1,1,2,2] => 122 [1,2,1,1,3,1] => 144 [1,2,1,1,4] => 88 [1,2,1,2,1,1,1] => 104 [1,2,1,2,1,2] => 170 [1,2,1,2,2,1] => 340 [1,2,1,2,3] => 272 [1,2,1,3,1,1] => 288 [1,2,1,3,2] => 408 [1,2,1,4,1] => 352 [1,2,1,5] => 160 [1,2,2,1,1,1,1] => 130 [1,2,2,1,1,2] => 218 [1,2,2,1,2,1] => 436 [1,2,2,1,3] => 360 [1,2,2,2,1,1] => 654 [1,2,2,2,2] => 938 [1,2,2,3,1] => 1080 [1,2,2,4] => 584 [1,2,3,1,1,1] => 480 [1,2,3,1,2] => 720 [1,2,3,2,1] => 1440 [1,2,3,3] => 1056 [1,2,4,1,1] => 880 [1,2,4,2] => 1168 [1,2,5,1] => 800 [1,2,6] => 288 [1,3,1,1,1,1,1] => 84 [1,3,1,1,1,2] => 144 [1,3,1,1,2,1] => 288 [1,3,1,1,3] => 244 [1,3,1,2,1,1] => 432 [1,3,1,2,2] => 632 [1,3,1,3,1] => 732 [1,3,1,4] => 408 [1,3,2,1,1,1] => 576 [1,3,2,1,2] => 876 [1,3,2,2,1] => 1752 [1,3,2,3] => 1304 [1,3,3,1,1] => 1464 [1,3,3,2] => 1956 [1,3,4,1] => 1632 [1,3,5] => 672 [1,4,1,1,1,1] => 280 [1,4,1,1,2] => 440 [1,4,1,2,1] => 880 [1,4,1,3] => 680 [1,4,2,1,1] => 1320 [1,4,2,2] => 1800 [1,4,3,1] => 2040 [1,4,4] => 1032 [1,5,1,1,1] => 560 [1,5,1,2] => 800 [1,5,2,1] => 1600 [1,5,3] => 1120 [1,6,1,1] => 672 [1,6,2] => 864 [1,7,1] => 448 [1,8] => 128 [2,1,1,1,1,1,1,1] => 16 [2,1,1,1,1,1,2] => 30 [2,1,1,1,1,2,1] => 60 [2,1,1,1,1,3] => 56 [2,1,1,1,2,1,1] => 90 [2,1,1,1,2,2] => 142 [2,1,1,1,3,1] => 168 [2,1,1,1,4] => 104 [2,1,1,2,1,1,1] => 120 [2,1,1,2,1,2] => 198 [2,1,1,2,2,1] => 396 [2,1,1,2,3] => 320 [2,1,1,3,1,1] => 336 [2,1,1,3,2] => 480 [2,1,1,4,1] => 416 [2,1,1,5] => 192 [2,1,2,1,1,1,1] => 150 [2,1,2,1,1,2] => 254 [2,1,2,1,2,1] => 508 [2,1,2,1,3] => 424 [2,1,2,2,1,1] => 762 [2,1,2,2,2] => 1102 [2,1,2,3,1] => 1272 [2,1,2,4] => 696 [2,1,3,1,1,1] => 560 [2,1,3,1,2] => 848 [2,1,3,2,1] => 1696 [2,1,3,3] => 1256 [2,1,4,1,1] => 1040 [2,1,4,2] => 1392 [2,1,5,1] => 960 [2,1,6] => 352 [2,2,1,1,1,1,1] => 180 [2,2,1,1,1,2] => 310 [2,2,1,1,2,1] => 620 [2,2,1,1,3] => 528 [2,2,1,2,1,1] => 930 [2,2,1,2,2] => 1366 [2,2,1,3,1] => 1584 [2,2,1,4] => 888 [2,2,2,1,1,1] => 1240 [2,2,2,1,2] => 1894 [2,2,2,2,1] => 3788 [2,2,2,3] => 2832 [2,2,3,1,1] => 3168 [2,2,3,2] => 4248 [2,2,4,1] => 3552 [2,2,5] => 1472 [2,3,1,1,1,1] => 840 [2,3,1,1,2] => 1320 [2,3,1,2,1] => 2640 [2,3,1,3] => 2040 [2,3,2,1,1] => 3960 [2,3,2,2] => 5400 [2,3,3,1] => 6120 [2,3,4] => 3096 [2,4,1,1,1] => 2080 [2,4,1,2] => 2960 [2,4,2,1] => 5920 [2,4,3] => 4128 [2,5,1,1] => 2880 [2,5,2] => 3680 [2,6,1] => 2112 [2,7] => 640 [3,1,1,1,1,1,1] => 112 [3,1,1,1,1,2] => 196 [3,1,1,1,2,1] => 392 [3,1,1,1,3] => 340 [3,1,1,2,1,1] => 588 [3,1,1,2,2] => 876 [3,1,1,3,1] => 1020 [3,1,1,4] => 584 [3,1,2,1,1,1] => 784 [3,1,2,1,2] => 1216 [3,1,2,2,1] => 2432 [3,1,2,3] => 1848 [3,1,3,1,1] => 2040 [3,1,3,2] => 2772 [3,1,4,1] => 2336 [3,1,5] => 992 [3,2,1,1,1,1] => 980 [3,2,1,1,2] => 1556 [3,2,1,2,1] => 3112 [3,2,1,3] => 2432 [3,2,2,1,1] => 4668 [3,2,2,2] => 6420 [3,2,3,1] => 7296 [3,2,4] => 3736 [3,3,1,1,1] => 3400 [3,3,1,2] => 4864 [3,3,2,1] => 9728 [3,3,3] => 6820 [3,4,1,1] => 5840 [3,4,2] => 7472 [3,5,1] => 4960 [3,6] => 1664 [4,1,1,1,1,1] => 448 [4,1,1,1,2] => 728 [4,1,1,2,1] => 1456 [4,1,1,3] => 1168 [4,1,2,1,1] => 2184 [4,1,2,2] => 3064 [4,1,3,1] => 3504 [4,1,4] => 1848 [4,2,1,1,1] => 2912 [4,2,1,2] => 4232 [4,2,2,1] => 8464 [4,2,3] => 6032 [4,3,1,1] => 7008 [4,3,2] => 9048 [4,4,1] => 7392 [4,5] => 2880 [5,1,1,1,1] => 1120 [5,1,1,2] => 1680 [5,1,2,1] => 3360 [5,1,3] => 2480 [5,2,1,1] => 5040 [5,2,2] => 6640 [5,3,1] => 7440 [5,4] => 3600 [6,1,1,1] => 1792 [6,1,2] => 2464 [6,2,1] => 4928 [6,3] => 3328 [7,1,1] => 1792 [7,2] => 2240 [8,1] => 1024 [9] => 256 ----------------------------------------------------------------------------- Created: May 20, 2017 at 22:06 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 20, 2017 at 22:06 by Martin Rubey