On Enumeration of Conjugacy Classes of Coxeter Elements

TL;DR

This paper investigates the enumeration of conjugacy classes of Coxeter elements by analyzing acyclic orientations of graphs and their transformations, providing recursive formulas and connections to the Tutte polynomial.

Contribution

It introduces a recursive method for counting equivalence classes of acyclic orientations related to Coxeter elements using graph operations.

Findings

Recursive formula for equivalence classes count

Connection to Tutte polynomial evaluation

Bijections with known acyclic orientation classes

Abstract

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.