Diameter of the commutation classes graph of a permutation

TL;DR

This paper introduces a new statistic on the graph of commutation classes of permutations, revealing a ranked poset structure and enabling the calculation of the graph's diameter for any permutation, including special cases.

Contribution

It defines a novel statistic that characterizes the commutation classes graph as a ranked poset, facilitating diameter computation for all permutations.

Findings

Graphs are ranked posets with a minimum and maximum.

Diameter can be computed for any permutation.

Results include special cases like longest and fully commutative permutations.

Abstract

We define a statistic on the graph of commutation classes of a permutation of the symmetric group which is used to show that these graphs are equipped with a ranked poset structure, with a minimum and maximum. This characterization also allows us to compute the diameter of the commutation graph for any permutation, from which the results for the longest permutation and for fully commutative permutations are recovered.