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Statistic identifier: St001680

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Collection: Finite Cartan types

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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.

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

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


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Statistic values:

['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

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Created: Feb 06, 2021 at 23:20 by Martin Rubey

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Last Updated: Feb 06, 2021 at 23:20 by Martin Rubey