***************************************************************************** * 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: St000047 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux. ----------------------------------------------------------------------------- References: [1] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. The immaculate basis of the non-commutative symmetric functions [[arXiv:1303.4801]] [2] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions [[arXiv:1208.5191]] ----------------------------------------------------------------------------- Code: def statistic(mu): F = QuasiSymmetricFunctions(ZZ).F() dI = QuasiSymmetricFunctions(ZZ).dI() return sum(coeff for _, coeff in F(dI(mu))) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,1] => 1 [2] => 1 [1,1,1] => 1 [1,2] => 1 [2,1] => 2 [3] => 1 [1,1,1,1] => 1 [1,1,2] => 1 [1,2,1] => 2 [1,3] => 1 [2,1,1] => 3 [2,2] => 3 [3,1] => 3 [4] => 1 [1,1,1,1,1] => 1 [1,1,1,2] => 1 [1,1,2,1] => 2 [1,1,3] => 1 [1,2,1,1] => 3 [1,2,2] => 3 [1,3,1] => 3 [1,4] => 1 [2,1,1,1] => 4 [2,1,2] => 4 [2,2,1] => 8 [2,3] => 4 [3,1,1] => 6 [3,2] => 6 [4,1] => 4 [5] => 1 [1,1,1,1,1,1] => 1 [1,1,1,1,2] => 1 [1,1,1,2,1] => 2 [1,1,1,3] => 1 [1,1,2,1,1] => 3 [1,1,2,2] => 3 [1,1,3,1] => 3 [1,1,4] => 1 [1,2,1,1,1] => 4 [1,2,1,2] => 4 [1,2,2,1] => 8 [1,2,3] => 4 [1,3,1,1] => 6 [1,3,2] => 6 [1,4,1] => 4 [1,5] => 1 [2,1,1,1,1] => 5 [2,1,1,2] => 5 [2,1,2,1] => 10 [2,1,3] => 5 [2,2,1,1] => 15 [2,2,2] => 15 [2,3,1] => 15 [2,4] => 5 [3,1,1,1] => 10 [3,1,2] => 10 [3,2,1] => 20 [3,3] => 10 [4,1,1] => 10 [4,2] => 10 [5,1] => 5 [6] => 1 [1,1,1,1,1,1,1] => 1 [1,1,1,1,1,2] => 1 [1,1,1,1,2,1] => 2 [1,1,1,1,3] => 1 [1,1,1,2,1,1] => 3 [1,1,1,2,2] => 3 [1,1,1,3,1] => 3 [1,1,1,4] => 1 [1,1,2,1,1,1] => 4 [1,1,2,1,2] => 4 [1,1,2,2,1] => 8 [1,1,2,3] => 4 [1,1,3,1,1] => 6 [1,1,3,2] => 6 [1,1,4,1] => 4 [1,1,5] => 1 [1,2,1,1,1,1] => 5 [1,2,1,1,2] => 5 [1,2,1,2,1] => 10 [1,2,1,3] => 5 [1,2,2,1,1] => 15 [1,2,2,2] => 15 [1,2,3,1] => 15 [1,2,4] => 5 [1,3,1,1,1] => 10 [1,3,1,2] => 10 [1,3,2,1] => 20 [1,3,3] => 10 [1,4,1,1] => 10 [1,4,2] => 10 [1,5,1] => 5 [1,6] => 1 [2,1,1,1,1,1] => 6 [2,1,1,1,2] => 6 [2,1,1,2,1] => 12 [2,1,1,3] => 6 [2,1,2,1,1] => 18 [2,1,2,2] => 18 [2,1,3,1] => 18 [2,1,4] => 6 [2,2,1,1,1] => 24 [2,2,1,2] => 24 [2,2,2,1] => 48 [2,2,3] => 24 [2,3,1,1] => 36 [2,3,2] => 36 [2,4,1] => 24 [2,5] => 6 [3,1,1,1,1] => 15 [3,1,1,2] => 15 [3,1,2,1] => 30 [3,1,3] => 15 [3,2,1,1] => 45 [3,2,2] => 45 [3,3,1] => 45 [3,4] => 15 [4,1,1,1] => 20 [4,1,2] => 20 [4,2,1] => 40 [4,3] => 20 [5,1,1] => 15 [5,2] => 15 [6,1] => 6 [7] => 1 [1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,2] => 1 [1,1,1,1,1,2,1] => 2 [1,1,1,1,1,3] => 1 [1,1,1,1,2,1,1] => 3 [1,1,1,1,2,2] => 3 [1,1,1,1,3,1] => 3 [1,1,1,1,4] => 1 [1,1,1,2,1,1,1] => 4 [1,1,1,2,1,2] => 4 [1,1,1,2,2,1] => 8 [1,1,1,2,3] => 4 [1,1,1,3,1,1] => 6 [1,1,1,3,2] => 6 [1,1,1,4,1] => 4 [1,1,1,5] => 1 [1,1,2,1,1,1,1] => 5 [1,1,2,1,1,2] => 5 [1,1,2,1,2,1] => 10 [1,1,2,1,3] => 5 [1,1,2,2,1,1] => 15 [1,1,2,2,2] => 15 [1,1,2,3,1] => 15 [1,1,2,4] => 5 [1,1,3,1,1,1] => 10 [1,1,3,1,2] => 10 [1,1,3,2,1] => 20 [1,1,3,3] => 10 [1,1,4,1,1] => 10 [1,1,4,2] => 10 [1,1,5,1] => 5 [1,1,6] => 1 [1,2,1,1,1,1,1] => 6 [1,2,1,1,1,2] => 6 [1,2,1,1,2,1] => 12 [1,2,1,1,3] => 6 [1,2,1,2,1,1] => 18 [1,2,1,2,2] => 18 [1,2,1,3,1] => 18 [1,2,1,4] => 6 [1,2,2,1,1,1] => 24 [1,2,2,1,2] => 24 [1,2,2,2,1] => 48 [1,2,2,3] => 24 [1,2,3,1,1] => 36 [1,2,3,2] => 36 [1,2,4,1] => 24 [1,2,5] => 6 [1,3,1,1,1,1] => 15 [1,3,1,1,2] => 15 [1,3,1,2,1] => 30 [1,3,1,3] => 15 [1,3,2,1,1] => 45 [1,3,2,2] => 45 [1,3,3,1] => 45 [1,3,4] => 15 [1,4,1,1,1] => 20 [1,4,1,2] => 20 [1,4,2,1] => 40 [1,4,3] => 20 [1,5,1,1] => 15 [1,5,2] => 15 [1,6,1] => 6 [1,7] => 1 [2,1,1,1,1,1,1] => 7 [2,1,1,1,1,2] => 7 [2,1,1,1,2,1] => 14 [2,1,1,2,1,1] => 21 [2,1,1,2,2] => 21 [2,1,1,3,1] => 21 [2,1,2,1,1,1] => 28 [2,1,2,1,2] => 28 [2,1,2,2,1] => 56 [2,1,2,3] => 28 [2,1,3,1,1] => 42 [2,1,3,2] => 42 [2,1,4,1] => 28 [2,2,1,1,1,1] => 35 [2,2,1,1,2] => 35 [2,2,1,2,1] => 70 [2,2,1,3] => 35 [2,2,2,1,1] => 105 [2,2,2,2] => 105 [2,2,3,1] => 105 [2,2,4] => 35 [2,3,1,1,1] => 70 [2,3,1,2] => 70 [2,3,2,1] => 140 [2,3,3] => 70 [2,4,2] => 70 [2,5,1] => 35 [2,6] => 7 [3,1,1,1,1,1] => 21 [3,1,1,3] => 21 [3,1,2,1,1] => 63 [3,1,2,2] => 63 [3,1,3,1] => 63 [3,2,1,1,1] => 84 [3,2,1,2] => 84 [3,2,2,1] => 168 [3,2,3] => 84 [3,3,1,1] => 126 [3,3,2] => 126 [3,4,1] => 84 [3,5] => 21 [4,1,1,1,1] => 35 [4,1,2,1] => 70 [4,2,1,1] => 105 [4,2,2] => 105 [4,3,1] => 105 [4,4] => 35 [5,1,1,1] => 35 [5,2,1] => 70 [5,3] => 35 [6,1,1] => 21 [6,2] => 21 [7,1] => 7 [8] => 1 [1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,2] => 1 [1,1,1,1,1,1,2,1] => 2 [1,1,1,1,1,1,3] => 1 [1,1,1,1,1,2,1,1] => 3 [1,1,1,1,1,2,2] => 3 [1,1,1,1,1,3,1] => 3 [1,1,1,1,1,4] => 1 [1,1,1,1,2,1,1,1] => 4 [1,1,1,1,2,1,2] => 4 [1,1,1,1,2,3] => 4 [1,1,1,1,3,1,1] => 6 [1,1,1,1,4,1] => 4 [1,1,1,1,5] => 1 [1,1,1,2,1,1,1,1] => 5 [1,1,1,2,1,1,2] => 5 [1,1,1,2,2,2] => 15 [1,1,1,2,3,1] => 15 [1,1,1,2,4] => 5 [1,1,1,3,1,1,1] => 10 [1,1,1,3,3] => 10 [1,1,1,4,2] => 10 [1,1,1,6] => 1 [1,1,2,1,1,1,1,1] => 6 [1,1,2,1,1,1,2] => 6 [1,1,2,1,3,1] => 18 [1,1,2,1,4] => 6 [1,1,2,2,1,2] => 24 [1,1,2,2,3] => 24 [1,1,2,3,2] => 36 [1,1,2,5] => 6 [1,1,3,1,1,1,1] => 15 [1,1,3,1,1,2] => 15 [1,1,3,1,3] => 15 [1,1,3,2,2] => 45 [1,1,3,4] => 15 [1,1,4,1,2] => 20 [1,1,6,1] => 6 [1,1,7] => 1 [1,2,1,1,1,1,1,1] => 7 [1,2,1,1,1,1,2] => 7 [1,2,1,1,2,1,1] => 21 [1,2,1,1,2,2] => 21 [1,2,1,1,3,1] => 21 [1,2,1,1,4] => 7 [1,2,1,2,1,2] => 28 [1,2,1,2,2,1] => 56 [1,2,1,2,3] => 28 [1,2,1,3,2] => 42 [1,2,2,1,1,2] => 35 [1,2,2,1,2,1] => 70 [1,2,2,1,3] => 35 [1,2,2,2,2] => 105 [1,2,2,4] => 35 [1,2,3,1,2] => 70 [1,2,3,3] => 70 [1,2,4,2] => 70 [1,2,5,1] => 35 [1,2,6] => 7 [1,3,1,1,1,1,1] => 21 [1,3,1,1,1,2] => 21 [1,3,1,1,2,1] => 42 [1,3,1,1,3] => 21 [1,3,1,2,1,1] => 63 [1,3,1,2,2] => 63 [1,3,2,1,1,1] => 84 [1,3,2,1,2] => 84 [1,3,2,3] => 84 [1,3,3,1,1] => 126 [1,3,3,2] => 126 [1,3,4,1] => 84 [1,3,5] => 21 [1,4,1,1,1,1] => 35 [1,4,1,1,2] => 35 [1,4,1,3] => 35 [1,4,2,2] => 105 [1,4,3,1] => 105 [1,4,4] => 35 [1,5,1,2] => 35 [1,5,2,1] => 70 [1,5,3] => 35 [1,6,1,1] => 21 [1,6,2] => 21 [1,7,1] => 7 [1,8] => 1 [2,1,1,1,1,1,1,1] => 8 [2,1,1,2,1,1,1] => 32 [2,1,2,2,1,1] => 120 [2,1,5,1] => 40 [2,2,1,1,1,1,1] => 48 [2,2,1,2,1,1] => 144 [2,2,2,1,1,1] => 192 [2,2,2,2,1] => 384 [2,2,2,3] => 192 [2,2,3,2] => 288 [2,2,4,1] => 192 [2,2,5] => 48 [2,3,2,2] => 360 [2,3,3,1] => 360 [2,3,4] => 120 [2,4,3] => 160 [2,7] => 8 [3,1,1,1,1,1,1] => 28 [3,1,4,1] => 112 [3,2,1,1,1,1] => 140 [3,2,2,1,1] => 420 [3,2,2,2] => 420 [3,3,1,1,1] => 280 [3,3,2,1] => 560 [3,3,3] => 280 [3,4,2] => 280 [3,6] => 28 [4,1,1,1,1,1] => 56 [4,2,1,1,1] => 224 [4,2,2,1] => 448 [4,3,1,1] => 336 [4,3,2] => 336 [4,4,1] => 224 [4,5] => 56 [5,1,1,1,1] => 70 [5,2,1,1] => 210 [5,2,2] => 210 [5,3,1] => 210 [5,4] => 70 [6,1,1,1] => 56 [6,2,1] => 112 [6,3] => 56 [7,1,1] => 28 [7,2] => 28 [8,1] => 8 [9] => 1 [1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,2] => 1 [1,1,1,1,1,1,1,2,1] => 2 [1,1,1,1,1,1,1,3] => 1 [1,1,1,1,1,1,2,1,1] => 3 [1,1,1,1,1,1,2,2] => 3 [1,1,1,1,1,1,4] => 1 [1,1,1,1,1,2,1,1,1] => 4 [1,1,1,1,1,2,3] => 4 [1,1,1,1,1,5] => 1 [1,1,1,1,2,1,1,1,1] => 5 [1,1,1,1,2,1,1,2] => 5 [1,1,1,1,2,2,1,1] => 15 [1,1,1,1,2,2,2] => 15 [1,1,1,1,2,4] => 5 [1,1,1,1,3,3] => 10 [1,1,1,1,5,1] => 5 [1,1,1,1,6] => 1 [1,1,1,2,1,1,1,1,1] => 6 [1,1,1,2,1,1,1,2] => 6 [1,1,1,2,2,3] => 24 [1,1,1,2,5] => 6 [1,1,1,3,4] => 15 [1,1,1,7] => 1 [1,1,2,1,1,1,1,1,1] => 7 [1,1,2,1,1,1,1,2] => 7 [1,1,2,1,1,2,1,1] => 21 [1,1,2,1,2,3] => 28 [1,1,2,2,1,1,1,1] => 35 [1,1,2,2,2,2] => 105 [1,1,2,2,4] => 35 [1,1,2,3,3] => 70 [1,1,2,6] => 7 [1,1,3,1,1,1,2] => 21 [1,1,3,1,1,3] => 21 [1,1,3,2,1,2] => 84 [1,1,3,3,1,1] => 126 [1,1,3,5] => 21 [1,1,4,4] => 35 [1,1,7,1] => 7 [1,1,8] => 1 [1,2,1,1,1,1,1,1,1] => 8 [1,2,1,1,1,1,1,2] => 8 [1,2,1,1,4,1] => 32 [1,2,1,2,1,1,2] => 40 [1,2,2,1,1,1,2] => 48 [1,2,2,1,2,1,1] => 144 [1,2,2,2,2,1] => 384 [1,2,2,2,3] => 192 [1,2,2,3,2] => 288 [1,2,2,5] => 48 [1,2,3,1,1,2] => 120 [1,2,3,2,2] => 360 [1,2,3,3,1] => 360 [1,2,3,4] => 120 [1,2,4,3] => 160 [1,2,6,1] => 48 [1,2,7] => 8 [1,3,1,1,1,1,2] => 28 ----------------------------------------------------------------------------- Created: Mar 24, 2013 at 23:08 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Jun 07, 2022 at 20:40 by Martin Rubey