On Enumeration of Conjugacy Classes of Coxeter Elements

Details On Enumeration of Conjugacy Classes of Coxeter Elements On Enumeration of Conjugacy Classes of Coxeter Elements

2007-11-07

arxiv.org/abs/0711.1140v2

Proc. Amer. Math. Soc. 136 (2008), 4157-4165.

Mathematics

Combinatorics

Added a few results about connections to the Tutte polynomial

Scientific paper

10.1090/S0002-9939-08-09543-9

In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T(Y,1,0), and we provide bijections to two other classes of acyclic orientations that are known to be counted in the same way. A transversal of the set of equivalence classes is given.

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