Enumerations relating braid and commutation classes

TL;DR

This paper establishes bounds on the number of reduced words for permutations based on braid and commutation classes, classifies permutations that reach these bounds, and provides enumeration for these cases.

Contribution

It introduces bounds relating braid and commutation classes to reduced words and classifies permutations achieving these bounds, with enumeration of such cases.

Findings

Bounds for the number of reduced words in terms of braid and commutation classes

Classification of permutations that attain these bounds

Enumeration of permutations at the bounds

Abstract

We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these bounds, and enumerate both cases.