Diameters of Graphs on Reduced Words of 12 and 21-Inflations

TL;DR

This paper investigates the diameters of graphs formed by reduced words of permutations, providing inductive formulas for certain inflation classes, recursive formulas for the longest permutation, and progress on diameter bounds conjectures.

Contribution

It introduces inductive formulas for graph diameters of 12- and many 21-inflations, extends results to related classes, and advances understanding of diameter bounds in permutation graphs.

Findings

Derived inductive formulas for diameters of inflation graphs.

Established recursive formulas for the longest permutation.

Identified permutation families achieving diameter bounds.

Abstract

It is a classical result that any permutation in the symmetric group can be generated by a sequence of adjacent transpositions. The sequences of minimal length are called reduced words, and in this paper we study the graphs of these reduced words, with edges determined by relations in the underlying Coxeter group. Recently, the diameter has been calculated for the longest permutation $n\ldots 21$ by Reiner and Roichman as well as Assaf. In this paper we find inductive formulas for the diameter of the graphs of 12-inflations and many 21-inflations. These results extend to the associated graphs on commutation and long braid classes. Also, these results give a recursive formula for the diameter of the longest permutation, which matches that of Reiner, Roichman and Assaf. Lastly, We make progress on conjectured bounds of the diameter by Reiner and Roichman, which are based on the underlying hyperplane arrangement, and find families of permutations that achieve the upper bound and lower bound of the conjecture. In particular permutations that avoid 312 or 231 have graphs that achieve the upper bound.