***************************************************************************** * 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: St000138 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The Catalan number of an irreducible finite Cartan type. The Catalan number of an irreducible finite Cartan type is defined as the product $$Cat(W) = \prod_{i=1}^n \frac{d_i+h}{d_i}$$ where *$W$ is the Weyl group of the given Cartan type, * $n$ is the rank of $W$, * $d_1 \leq d_2 \leq \ldots \leq d_n$ are the degrees of the fundamental invariants of $W$, and * $h = d_n$ is the corresponding Coxeter number. The Catalan number $Cat(W)$ counts various combinatorial objects, among which are * noncrossing partitions inside $W$, * antichains in the root poset, * regions within the fundamental chamber in the Shi arrangement, * dimensions of several modules in the context of the '''diagonal coininvariant ring''' and of '''rational Cherednik algebras'''. For a detailed treatment and further references, see [1]. ----------------------------------------------------------------------------- References: [1] Armstrong, D. Generalized noncrossing partitions and combinatorics of Coxeter groups [[MathSciNet:2561274]] [[arXiv:math/0611106]] [2] [[wikipedia:Complex reflection group]] ----------------------------------------------------------------------------- Code: def statistic(ct): return ReflectionGroup(ct).catalan_number() ----------------------------------------------------------------------------- Statistic values: ['A',1] => 2 ['A',2] => 5 ['B',2] => 6 ['G',2] => 8 ['A',3] => 14 ['B',3] => 20 ['C',3] => 20 ['A',4] => 42 ['B',4] => 70 ['C',4] => 70 ['D',4] => 50 ['F',4] => 105 ['A',5] => 132 ['B',5] => 252 ['C',5] => 252 ['D',5] => 182 ['A',6] => 429 ['B',6] => 924 ['C',6] => 924 ['D',6] => 672 ['E',6] => 833 ['A',7] => 1430 ['B',7] => 3432 ['C',7] => 3432 ['D',7] => 2508 ['E',7] => 4160 ['A',8] => 4862 ['B',8] => 12870 ['C',8] => 12870 ['D',8] => 9438 ['E',8] => 25080 ['A',9] => 16796 ['B',9] => 48620 ['C',9] => 48620 ['D',9] => 35750 ['A',10] => 58786 ['B',10] => 184756 ['C',10] => 184756 ['D',10] => 136136 ----------------------------------------------------------------------------- Created: Jun 23, 2013 at 12:31 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Nov 21, 2017 at 09:21 by Christian Stump