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.
@article{arxiv.2601.09395,
title = {Maximum number of one-element commutation classes of a permutation},
author = {Ricardo Mamede and José Luis Santos and Diogo Soares},
journal= {arXiv preprint arXiv:2601.09395},
year = {2026}
}