*****************************************************************************
*       www.FindStat.org - The Combinatorial Statistic Finder               *
*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
*                                                                           *
*    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