- FindStat

Identifier

St000851: Finite Cartan types ⟶ ℤ

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

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$F_{1} = q^{4}$

$F_{2} = q^{22} + q^{28} + q^{40}$

$F_{3} = q^{140} + 2\ q^{220}$

$F_{4} = q^{969} + q^{1210} + 2\ q^{1820} + q^{2926}$

$F_{5} = q^{7084} + q^{10556} + 2\ q^{15504}$

$F_{6} = q^{53820} + q^{93024} + q^{119966} + 2\ q^{134596}$

$F_{7} = q^{420732} + q^{826804} + 2\ q^{1184040} + q^{1484032}$

$F_{8} = q^{3362260} + q^{7400250} + 2\ q^{10518300} + q^{22309287}$

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.

Code

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

Created

Jun 25, 2017 at 10:11 by Christian Stump

Updated

Jun 25, 2017 at 10:11 by Christian Stump