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Identifier
Values
['A',1] => 3
['A',2] => 15
['B',2] => 21
['G',2] => 33
['A',3] => 91
['B',3] => 165
['C',3] => 165
['A',4] => 612
['B',4] => 1365
['C',4] => 1365
['D',4] => 825
['F',4] => 2415
['A',5] => 4389
['B',5] => 11628
['C',5] => 11628
['D',5] => 7371
['A',6] => 32890
['B',6] => 100947
['C',6] => 100947
['D',6] => 65892
['E',6] => 89999
['A',7] => 254475
['B',7] => 888030
['C',7] => 888030
['D',7] => 591261
['E',7] => 1186680
['A',8] => 2017356
['B',8] => 7888725
['C',8] => 7888725
['D',8] => 5328180
['E',8] => 19137240
['A',9] => 16301164
['B',9] => 70607460
['C',9] => 70607460
['D',9] => 48208875
['A',10] => 133767543
['B',10] => 635745396
['C',10] => 635745396
['D',10] => 437766252
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Description
The third 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 third Fuss-Catalan numbers see St000851The third Fuss-Catalan number of a finite Cartan type. and for the positive Fuss-Catalan numbers see St000140The positive Catalan number of an irreducible finite Cartan type..
Code
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return W.fuss_catalan_number(m=3, positive=True)
Created
Nov 21, 2017 at 09:32 by Christian Stump
Updated
Nov 21, 2017 at 09:32 by Christian Stump