***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * 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: St001753 ----------------------------------------------------------------------------- Collection: Finite Cartan types ----------------------------------------------------------------------------- Description: The number of subsets of the positive roots that form a basis of the associated vector space. For the group $W$ and an associated set of positive roots $\Phi^+ \subseteq V$ this counts the number of subsets $S \subseteq \Phi^+$ that form a basis of $V$. This is also the number of subsets of the reflections $R \subseteq W$ that form a minimal set of generators of a reflection subgroup of full rank. The Coxeter permutahedron can be defined as the Minkowski sum of the line segments $[- \frac{\alpha}{2}, \frac{\alpha}{2}]$ for $\alpha \in \Phi^+$. As a zonotope this polytope can be decomposed into a (disjoint) union of (half-open) parallel epipeds [1]. This also counts the number of full dimensional parallel epipeds among this decomposition. ----------------------------------------------------------------------------- References: [1] Shephard, G. C. Combinatorial properties of associated zonotopes [[MathSciNet:0362054]] [[DOI:10.4153/CJM-1974-032-5]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ['A',1] => 1 ['A',2] => 3 ['B',2] => 6 ['G',2] => 15 ['A',3] => 16 ['B',3] => 68 ['C',3] => 68 ['A',4] => 125 ['B',4] => 1138 ['C',4] => 1138 ['D',4] => 315 ['F',4] => 7560 ['A',5] => 1296 ['B',5] => 25218 ['C',5] => 25218 ['D',5] => 7712 ['A',6] => 16807 ['B',6] => 695860 ['C',6] => 695860 ['D',6] => 228055 ['E',6] => 846720 ['A',7] => 262144 ['B',7] => 22985512 ['C',7] => 22985512 ['D',7] => 7932816 ['E',7] => 221714415 ['A',8] => 4782969 ['C',2] => 6 ----------------------------------------------------------------------------- Created: Dec 13, 2021 at 13:33 by Dennis Jahn ----------------------------------------------------------------------------- Last Updated: May 04, 2022 at 12:03 by Dennis Jahn