edit this statistic or download as text // json
Identifier
Values
['A',1] => 1
['A',2] => 3
['B',2] => 5
['G',2] => 9
['A',3] => 13
['B',3] => 35
['C',3] => 35
['A',4] => 71
['B',4] => 309
['C',4] => 309
['D',4] => 135
['F',4] => 1057
['A',5] => 461
['B',5] => 3287
['C',5] => 3287
['D',5] => 1537
['A',6] => 3447
['B',6] => 41005
['C',6] => 41005
['D',6] => 19811
['E',6] => 47527
['A',7] => 29093
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 connected elements in the Coxeter group corresponding to a finite Cartan type.
Let $(W, S)$ be a Coxeter system. Then, according to [1], the connectivity set of $w\in W$ is the cardinality of $S \setminus S(w)$, where $S(w)$ is the set of generators appearing in any reduced word for $w$.
For $A_n$, this is [2], for $B_n$ this is [3] and for $D_n$ this is [4].
References
[1] Bergeron, N., Hohlweg, C., Zabrocki, M. Posets related to the connectivity set of Coxeter groups arXiv:math/0509271
[2] Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations. OEIS:A003319
[3] Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections. OEIS:A109253
[4] Number of elements of a Weyl group of order 2^n-1 n! of type D for which a reduced word contains all of the simple reflections. OEIS:A112225
Code
def connected(ct):
    W = CoxeterGroup(ct)
    I = set(W.index_set())
    return sum(1 for w in W if not I.difference(w.reduced_word()))
Created
Feb 08, 2023 at 18:00 by Martin Rubey
Updated
Feb 08, 2023 at 18:00 by Martin Rubey