edit this statistic or download as text // json
Identifier
Values
['A',1] => 2
['A',2] => 6
['B',2] => 8
['G',2] => 8
['A',3] => 24
['B',3] => 38
['C',3] => 38
['A',4] => 115
['B',4] => 237
['C',4] => 237
['D',4] => 26
['F',4] => 297
['A',5] => 631
['B',5] => 1813
['C',5] => 1813
['D',5] => 229
['A',6] => 3907
['B',6] => 16981
['C',6] => 16981
['D',6] => 4143
['E',6] => 10690
['E',7] => 159368
['E',8] => 4854344
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 proper elements in the Weyl group of a finite Cartan type.
References
[1] Balogh, József, Brewster, D., Hodges, R. Proper elements of Coxeter Groups arXiv:2111.15105
Code
@cached_function
def statistic(ct):
    return sum(1 for w in WeylGroup(ct) if is_proper(ct, w))

def is_proper(ct, w):
    ct = CartanType(ct)
    n = ct.rank()
    l = w.length()
    d = len(w.descents(side="left"))
    if ct.type() == "A":
        return l <= n + binomial(d + 1, 2)
    if ct.type() in ["B", "C"]:
        return l <= n + d^2
    if ct.type() == "D":
        if d > 3:
            return l <= n + d^2 - d
        return l <= binomial(d + 1, 2)
    if ct.type() == "E":
        m = [0, 1, 3, 6, 12, 20, 36, 63, 120]
        return l <= n + m[d]
    if ct.type() == "F":
        m = [0, 1, 4, 9, 24]
        return l <= n + m[d]
    if ct.type() == "G": # = I_2(6)
        if d == 2:
            return l <= 6 + n
        if d < 2:
            return l <= 2 + d
Created
Dec 02, 2021 at 12:26 by Martin Rubey
Updated
Dec 02, 2021 at 12:26 by Martin Rubey