Maximum number of one-element commutation classes of a permutation

TL;DR

This paper establishes an upper bound on the count of one-element commutation classes of permutations and proves a related conjecture connecting reduced words and commutation classes.

Contribution

It introduces a new upper bound for the number of one-element commutation classes and confirms a conjecture linking reduced words to commutation classes.

Findings

Derived an upper bound for one-element commutation classes

Proved a conjecture relating reduced words and commutation classes

Enhanced understanding of permutation structure

Abstract

In this paper, we provide an upper bound for the number of one-element commutation classes of a permutation, that is, the number of reduced words in which no commutation can be applied. Using this upper bound, we prove a conjecture that relates the number of reduced words with the number of commutation classes of a permutation.