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Statistic identifier: St001157

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Collection: Finite Cartan types

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Description: The exponent of the Weyl group of given type.

This is the least common multiple of the orders of the elements of the group.

In a comment to [4], see also [5], it is asked whether this is the same as the least common multiple of the degrees of the Weyl group.


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References: [1]   [[groupprops:Exponent_of_a_group]]
[2]   Least common multiple (or LCM) of 1, 2, ..., n for n >= 1, a(0) = 1. [[OEIS:A003418]]
[3]   Least common multiple of 2, 4, 6, ..., 2n. [[OEIS:A051426]]
[4]   user66288 The maximal order of an element in a Coxeter group [[MathOverflow:343118]]
[5]   user66288 The least common multiple of all degrees of a finite Coxeter group and indecomposable elements in the generalized cycle decomposition [[MathOverflow:345908]]

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Code:
def statistic(C):
    return WeylGroup(C).as_permutation_group().exponent()


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Statistic values:

['A',1]  => 2
['A',2]  => 6
['B',2]  => 4
['G',2]  => 6
['A',3]  => 12
['B',3]  => 12
['C',3]  => 12
['A',4]  => 60
['B',4]  => 24
['C',4]  => 24
['D',4]  => 12
['F',4]  => 24
['A',5]  => 60
['B',5]  => 120
['C',5]  => 120
['D',5]  => 120
['A',6]  => 420
['B',6]  => 120
['C',6]  => 120
['D',6]  => 120
['E',6]  => 360
['A',7]  => 840
['B',7]  => 840
['C',7]  => 840
['D',7]  => 840
['E',7]  => 2520
['A',8]  => 2520
['B',8]  => 1680
['C',8]  => 1680
['D',8]  => 840
['E',8]  => 2520
['D',9]  => 5040
['D',10] => 5040
['D',11] => 55440

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Created: Apr 22, 2018 at 15:19 by Martin Rubey

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Last Updated: Nov 13, 2019 at 10:37 by Martin Rubey