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On Enumeration Of Conjugacy Classes Of Coxeter Elements, Matthew Macauley, Henning S. Mortveit

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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 …