How many elements of a Coxeter group have a unique reduced expression?

TL;DR

This paper determines the exact count of elements in any Coxeter group that possess a unique reduced expression, providing insight into the structure of Coxeter groups.

Contribution

It offers a general formula for counting elements with a unique reduced expression in arbitrary Coxeter groups, extending previous specific cases.

Findings

Derived a formula for the number of elements with unique reduced expressions

Applied the formula to various Coxeter groups to illustrate its use

Enhanced understanding of the structure of Coxeter groups

Abstract

Let $(W,R)$ be an arbitrary Coxeter system. We determine the number of elements of $W$ that have a unique reduced expression.