*****************************************************************************
*       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: St000851

-----------------------------------------------------------------------------
Collection: Finite Cartan types

-----------------------------------------------------------------------------
Description: The third Fuss-Catalan number of a finite Cartan type.

The Fuss-Catalan numbers of a finite Cartan type are given by
$$\frac{1}{|W|}\prod (d_i+mh) = \prod \frac{d_i+mh}{d_i}$$
where the products run over all degrees of homoneneous fundamenal invariants of the Weyl group of a Cartan type.

-----------------------------------------------------------------------------
References: 

-----------------------------------------------------------------------------
Code:
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return W.fuss_catalan_number(3)


-----------------------------------------------------------------------------
Statistic values:

['A',1] => 4
['A',2] => 22
['B',2] => 28
['G',2] => 40
['A',3] => 140
['B',3] => 220
['C',3] => 220
['A',4] => 969
['B',4] => 1820
['C',4] => 1820
['D',4] => 1210
['F',4] => 2926
['A',5] => 7084
['B',5] => 15504
['C',5] => 15504
['D',5] => 10556
['A',6] => 53820
['B',6] => 134596
['C',6] => 134596
['D',6] => 93024
['E',6] => 119966
['A',7] => 420732
['B',7] => 1184040
['C',7] => 1184040
['D',7] => 826804
['E',7] => 1484032
['A',8] => 3362260
['B',8] => 10518300
['C',8] => 10518300
['D',8] => 7400250
['E',8] => 22309287

-----------------------------------------------------------------------------
Created: Jun 25, 2017 at 10:11 by Christian Stump

-----------------------------------------------------------------------------
Last Updated: Jun 25, 2017 at 10:11 by Christian Stump