One-element commutation classes

TL;DR

This paper characterizes reduced words of permutations that are their own commutation class, revealing specific counts for the long element in symmetric groups.

Contribution

It provides a complete characterization of reduced words that form their own commutation class for any permutation, including explicit counts for the longest element.

Findings

Exactly four such words for the long element when n ≥ 4

Complete characterization of reduced words self-contained in their commutation class

Insight into the structure of permutation reduced words

Abstract

For any permutation w, we characterize the reduced words of w that are their own commutation class. When w is the long element n(n-1)...321 and n \ge 4, there are exactly four such words.