- FindStat

Identifier

St001680: Finite Cartan types ⟶ ℤ

Values

=>
                            Cc0022;cc-rep
                            
                            
                            

                        ['A',1]=>1
['A',2]=>5
['B',2]=>10
['G',2]=>26
['A',3]=>14
['B',3]=>35
['C',3]=>35
['A',4]=>30
['B',4]=>84
['C',4]=>84
['D',4]=>44
['F',4]=>196
['A',5]=>55
['B',5]=>165
['C',5]=>165
['D',5]=>100
['A',6]=>91
['B',6]=>286
['C',6]=>286
['D',6]=>190
['E',6]=>276
['A',7]=>140
['B',7]=>455
['C',7]=>455
['D',7]=>322
['E',7]=>735
['A',8]=>204
['B',8]=>680
['C',8]=>680
['D',8]=>504
['E',8]=>2360
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The sum of the squares of the exponents of the Weyl group of the finite Cartan type.
According to Suter [1], this equals $\frac{1}{6}n(h^2 + \gamma - h)$, where $n$ is the rank, $h$ is the Coxeter number and $\gamma$ the gamma number.

References

[1] Suter, R. Coxeter and dual Coxeter numbers MathSciNet:1600666

Code

def statistic(ct):
    return sum((d-1)^2 for d in WeylGroup(ct).degrees())

Created

Feb 06, 2021 at 23:20 by Martin Rubey

Updated

Feb 06, 2021 at 23:20 by Martin Rubey