***************************************************************************** * 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: St001054 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The third 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 third Fuss-Catalan numbers see [[St000851]] and for the positive Fuss-Catalan numbers see [[St000140]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ['A',1] => 3 ['A',2] => 15 ['B',2] => 21 ['G',2] => 33 ['A',3] => 91 ['B',3] => 165 ['C',3] => 165 ['A',4] => 612 ['B',4] => 1365 ['C',4] => 1365 ['D',4] => 825 ['F',4] => 2415 ['A',5] => 4389 ['B',5] => 11628 ['C',5] => 11628 ['D',5] => 7371 ['A',6] => 32890 ['B',6] => 100947 ['C',6] => 100947 ['D',6] => 65892 ['E',6] => 89999 ['A',7] => 254475 ['B',7] => 888030 ['C',7] => 888030 ['D',7] => 591261 ['E',7] => 1186680 ['A',8] => 2017356 ['B',8] => 7888725 ['C',8] => 7888725 ['D',8] => 5328180 ['E',8] => 19137240 ['A',9] => 16301164 ['B',9] => 70607460 ['C',9] => 70607460 ['D',9] => 48208875 ['A',10] => 133767543 ['B',10] => 635745396 ['C',10] => 635745396 ['D',10] => 437766252 ----------------------------------------------------------------------------- Created: Nov 21, 2017 at 09:32 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Nov 21, 2017 at 09:32 by Christian Stump