***************************************************************************** * 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: St000140 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The positive Catalan number of an irreducible finite Cartan type. The positive Catalan number of an irreducible finite Cartan type is defined as the product $$ Cat^+(W) = \prod_{i=1}^n \frac{d_i-2+h}{d_i} = \prod_{i=1}^n \frac{d^*_i+h}{d_i}$$ 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 $W$, * $d^*_1 \geq d^*_2 \geq \ldots \geq d^*_n$ are the codegrees for $W$, see [Wiki], and * $h = d_n$ is the corresponding Coxeter number. The positive Catalan number $Cat^+(W)$ counts various combinatorial objects, among which are * noncrossing partitions of full Coxeter support inside $W$, * antichains not containing simple roots in the root poset, * bounded regions within the fundamental chamber in the Shi arrangement. 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(positive=True) ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 2 ['B',2] => 3 ['G',2] => 5 ['A',3] => 5 ['B',3] => 10 ['C',3] => 10 ['A',4] => 14 ['B',4] => 35 ['C',4] => 35 ['D',4] => 20 ['F',4] => 66 ['A',5] => 42 ['B',5] => 126 ['C',5] => 126 ['D',5] => 77 ['A',6] => 132 ['B',6] => 462 ['C',6] => 462 ['D',6] => 294 ['E',6] => 418 ['A',7] => 429 ['B',7] => 1716 ['C',7] => 1716 ['D',7] => 1122 ['E',7] => 2431 ['A',8] => 1430 ['B',8] => 6435 ['C',8] => 6435 ['D',8] => 4290 ['E',8] => 17342 ['A',9] => 4862 ['B',9] => 24310 ['C',9] => 24310 ['D',9] => 16445 ['A',10] => 16796 ['B',10] => 92378 ['C',10] => 92378 ['D',10] => 63206 ----------------------------------------------------------------------------- Created: Jun 24, 2013 at 21:32 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Nov 21, 2017 at 09:22 by Christian Stump