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Statistic identifier: St000865

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Collection: Finite Cartan types

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Description: The number of Coxeter elements in the Weyl group of a finite Cartan type.

This is, the elements that are conjugate to the product of the simple generators in any order, or, equivalently, the elements that admit a primitive $h$-th root of unity as an eigenvalue where $h$ is the Coxeter number.

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References: [1]   Reiner, V., Ripoll, V., Stump, C. On non-conjugate Coxeter elements in well-generated reflection groups [[MathSciNet:3623739]]

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Code:
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return len(W.coxeter_elements())


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Statistic values:

['A',1] => 1
['A',2] => 2
['B',2] => 2
['G',2] => 2
['A',3] => 6
['B',3] => 8
['C',3] => 8
['A',4] => 24
['B',4] => 48
['C',4] => 48
['D',4] => 32
['F',4] => 96
['A',5] => 120
['B',5] => 384
['C',5] => 384
['D',5] => 240
['A',6] => 720
['B',6] => 3840
['C',6] => 3840
['D',6] => 2304
['E',6] => 4320
['A',7] => 5040
['B',7] => 46080
['C',7] => 46080
['D',7] => 26880
['E',7] => 161280
['A',8] => 40320
['B',8] => 645120

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Created: Jun 25, 2017 at 20:14 by Christian Stump

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Last Updated: Jun 26, 2017 at 08:34 by Christian Stump