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Identifier
Values
=>
Cc0022;cc-rep
['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
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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