*****************************************************************************
*       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: 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