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Stanley's formula for the number of reduced expressions of a permutationregarded as a Coxeter group element raises the question of how to enumerate thereduced expressions of an arbitrary Coxeter group element. We provide aframework for answering this question by constructing combinatorial objectsthat represent the inversion set and the reduced expressions for an arbitraryCoxeter group element. The framework also provides a formula for the length ofan element formed by deleting a generator from a Coxeter group element. Fan andHagiwara, et al$.$ showed that for certain Coxeter groups, the short-braidavoiding elements characterize those elements that give reduced expressionswhen any generator is deleted from a reduced expression. We provide acharacterization that holds in all Coxeter groups. Lastly, we give applicationsto the freely braided elements introduced by Green and Losonczy, generalizingsome of their results that hold in simply-laced Coxeter groups to the arbitraryCoxeter group setting.

Some combinatorial models for reduced expressions in Coxeter groups