- FindStat

Identifier

St001467: Finite Cartan types ⟶ ℤ

Values

['A',1] => 2
['A',2] => 4
['B',2] => 6
['G',2] => 8
['A',3] => 10
['B',3] => 20
['C',3] => 20
['A',4] => 26
['B',4] => 76
['C',4] => 76
['D',4] => 44
['F',4] => 140
['A',5] => 76
['B',5] => 312
['C',5] => 312
['D',5] => 156
['A',6] => 232
['B',6] => 1384
['C',6] => 1384
['D',6] => 752

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$F_{1} = q^{2}$

$F_{2} = q^{4} + q^{6} + q^{8}$

$F_{3} = q^{10} + 2\ q^{20}$

$F_{4} = q^{26} + q^{44} + 2\ q^{76} + q^{140}$

$F_{5} = q^{76} + q^{156} + 2\ q^{312}$

Description

The number of involutions in the Weyl group of a given Cartan type.
For type

$A_n$ , the generating function is

$\exp(x+x^2/2)$ , for type

$BC_n$ it is

$\exp(x^2+2x)$ and for type

$D_n$ it is

$\exp(x^2)(\exp(2x)+1)/2$ .

Code

def statistic(C):
    return sum(1 for x in WeylGroup(C) if x == x.inverse())

Created

Sep 02, 2019 at 14:20 by Martin Rubey

Updated

Sep 02, 2019 at 14:20 by Martin Rubey