**Identifier**

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

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..

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**searching the database

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