***************************************************************************** * 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: St001053 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The second positive Fuss-Catalan number of a finite Cartan type. The positive Fuss-Catalan numbers of a finite Cartan type are given by $$\frac{1}{|W|}\prod (d_i+mh-2) = \prod \frac{d_i+mh-2}{d_i}$$ where the products run over all degrees of homoneneous fundamenal invariants of the Weyl group of a Cartan type. For the second Fuss-Catalan numbers see [[St000852]] and for the positive Fuss-Catalan numbers see [[St000140]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(ct): return ReflectionGroup(ct).fuss_catalan_number(m=2, positive=True) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 7 ['B',2] => 10 ['G',2] => 16 ['A',3] => 30 ['B',3] => 56 ['C',3] => 56 ['A',4] => 143 ['B',4] => 330 ['C',4] => 330 ['D',4] => 196 ['F',4] => 595 ['A',5] => 728 ['B',5] => 2002 ['C',5] => 2002 ['D',5] => 1254 ['A',6] => 3876 ['B',6] => 12376 ['C',6] => 12376 ['D',6] => 8008 ['E',6] => 11067 ['A',7] => 21318 ['B',7] => 77520 ['C',7] => 77520 ['D',7] => 51272 ['E',7] => 105248 ['A',8] => 120175 ['B',8] => 490314 ['C',8] => 490314 ['D',8] => 329460 ['E',8] => 1225367 ['A',9] => 690690 ['B',9] => 3124550 ['C',9] => 3124550 ['D',9] => 2124694 ['A',10] => 4032015 ['B',10] => 20030010 ['C',10] => 20030010 ['D',10] => 13748020 ----------------------------------------------------------------------------- Created: Nov 21, 2017 at 09:31 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Nov 21, 2017 at 09:31 by Christian Stump