An inversion metric for reduced words

TL;DR

This paper introduces a new inversion metric on reduced words in the symmetric group, creating a ranked graph structure that extends to balanced tableaux and recovers key enumerative results.

Contribution

It defines a novel inversion metric on reduced words, establishing a ranked graph structure and extending to balanced tableaux, linking combinatorial structures with enumeration.

Findings

The graph on reduced words becomes ranked with a unique maximal element.

The metric extends naturally to balanced tableaux.

The approach recovers known enumerative results by Edelman, Greene, Reiner, and Roichman.

Abstract

We study the graph on reduced words with edges given by the Coxeter relations for the symmetric group. We define a metric on reduced words for a given permutation, analogous to Coxeter length for permutations, for which the graph becomes ranked with unique maximal element. We show this metric extends naturally to balanced tableaux, and use it to recover enumerative results of Edelman and Greene and of Reiner and Roichman.