edit this statistic or download as text // json
Identifier
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
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 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 $A_2$.
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
Code
def statistic(cartan_type):
    W = ReflectionGroup(cartan_type)
    return factorial(W.rank()) * W.coxeter_number()**W.rank() / W.cardinality()
Created
Jun 25, 2017 at 10:00 by Christian Stump
Updated
Jun 25, 2017 at 10:00 by Christian Stump