*****************************************************************************
*       www.FindStat.org - The Combinatorial Statistic Finder               *
*                                                                           *
*       Copyright (C) 2019 The FindStatCrew <info@findstat.org>             *
*                                                                           *
*    This information is distributed in the hope that it will be useful,    *
*    but WITHOUT ANY WARRANTY; without even the implied warranty of         *
*    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.                   *
*****************************************************************************

-----------------------------------------------------------------------------
Statistic identifier: St001888

-----------------------------------------------------------------------------
Collection: Finite Cartan types

-----------------------------------------------------------------------------
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()))


-----------------------------------------------------------------------------
Statistic 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

-----------------------------------------------------------------------------
Created: Feb 08, 2023 at 18:00 by Martin Rubey

-----------------------------------------------------------------------------
Last Updated: Feb 08, 2023 at 18:00 by Martin Rubey