***************************************************************************** * 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: St000208 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]]. ----------------------------------------------------------------------------- References: [1] De Loera, JesÃºs A., McAllister, T. B. Vertices of Gelfand-Tsetlin polytopes [[MathSciNet:2096742]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 2 [1,1] => 1 [3] => 3 [2,1] => 2 [1,1,1] => 1 [4] => 5 [3,1] => 4 [2,2] => 3 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 7 [4,1] => 6 [3,2] => 4 [3,1,1] => 4 [2,2,1] => 2 [2,1,1,1] => 2 [1,1,1,1,1] => 1 [6] => 11 [5,1] => 10 [4,2] => 8 [4,1,1] => 7 [3,3] => 6 [3,2,1] => 4 [3,1,1,1] => 4 [2,2,2] => 3 [2,2,1,1] => 2 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 15 [6,1] => 14 [5,2] => 12 [5,1,1] => 11 [4,3] => 8 [4,2,1] => 6 [4,1,1,1] => 7 [3,3,1] => 5 [3,2,2] => 5 [3,2,1,1] => 4 [3,1,1,1,1] => 4 [2,2,2,1] => 2 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 1 [8] => 22 [7,1] => 21 [6,2] => 19 [6,1,1] => 17 [5,3] => 15 [5,2,1] => 9 [5,1,1,1] => 12 [4,4] => 12 [4,3,1] => 9 [4,2,2] => 9 [4,2,1,1] => 7 [4,1,1,1,1] => 7 [3,3,2] => 5 [3,3,1,1] => 5 [3,2,2,1] => 3 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 4 [2,2,2,2] => 3 [2,2,2,1,1] => 2 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: May 19, 2014 at 11:36 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: May 29, 2015 at 17:10 by Martin Rubey