  Stanley's formula for the number of reduced expressions of a permutation
regarded as a Coxeter group element raises the question of how to enumerate the
reduced expressions of an arbitrary Coxeter group element. We provide a
framework for answering this question by constructing combinatorial objects
that represent the inversion set and the reduced expressions for an arbitrary
Coxeter group element. The framework also provides a formula for the length of
an element formed by deleting a generator from a Coxeter group element. Fan and
Hagiwara, et al$.$ showed that for certain Coxeter groups, the short-braid
avoiding elements characterize those elements that give reduced expressions
when any generator is deleted from a reduced expression. We provide a
characterization that holds in all Coxeter groups. Lastly, we give applications
to the freely braided elements introduced by Green and Losonczy, generalizing
some of their results that hold in simply-laced Coxeter groups to the arbitrary
Coxeter group setting.
