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Statistic identifier: St000858

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Collection: Finite Cartan types

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Description: The number of factorizations of any Coxeter element into reflections of a finite Cartan type.

The number of such factorizations is given by $n!h^n / |W|$ where $n$ is the rank, $h$ is the Coxeter number and $W$ is the Weyl group of the given Cartan type.

This was originally proven in a letter from Deligne to Looijenga in the 1970s, and then recovered in [2, Theorem 3.6].

As an example, consider the three ($=2!3^2/6$) factorizations of the Coxeter element
$$(1,2,3) = (1,2)(2,3) = (1,3)(1,2) = (2,3)(1,3)$$
in type $A_2$.

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References: [1] Letter from Deligne to Looijenga  [[http://homepage.univie.ac.at/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf]]
[2]   Reading, N. Chains in the noncrossing partition lattice [[MathSciNet:2424827]]

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Code:


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Statistic values:

['A',1] => 1
['A',2] => 3
['B',2] => 4
['G',2] => 6
['A',3] => 16
['B',3] => 27
['C',3] => 27
['A',4] => 125
['B',4] => 256
['C',4] => 256
['D',4] => 162
['F',4] => 432
['A',5] => 1296
['B',5] => 3125
['C',5] => 3125
['D',5] => 2048
['A',6] => 16807
['B',6] => 46656
['C',6] => 46656
['D',6] => 31250
['E',6] => 41472
['A',7] => 262144
['B',7] => 823543
['C',7] => 823543
['D',7] => 559872
['E',7] => 1062882
['A',8] => 4782969
['B',8] => 16777216
['C',8] => 16777216
['D',8] => 11529602
['E',8] => 37968750

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Created: Jun 25, 2017 at 10:00 by Christian Stump

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Last Updated: Jun 25, 2017 at 10:00 by Christian Stump