Maximizing the Edelman-Greene statistic

TL;DR

This paper establishes the maximum Edelman-Greene statistic value for permutations, equating it to the number of involutions in the symmetric group, and provides explicit combinatorial mappings.

Contribution

It proves the maximum of the Edelman-Greene statistic equals the involution count in S_n and describes the permutations achieving this maximum.

Findings

Maximum Edelman-Greene statistic equals the number of involutions in S_n.

Explicit description of permutations attaining the maximum.

Constructs a combinatorial injection confirming a recent conjecture.

Abstract

The $\textit{Edelman-Greene statistic}$ of S. Billey-B. Pawlowski measures the "shortness" of the Schur expansion of a Stanley symmetric function. We show that the maximum value of this statistic on permutations of Coxeter length $n$ is the number of involutions in the symmetric group $S_n$, and explicitly describe the permutations that attain this maximum. Our proof confirms a recent conjecture of C. Monical, B. Pankow, and A. Yong: we give an explicit combinatorial injection between a certain collections of Edelman-Greene tableaux and standard Young tableaux.