Homomesy in products of two chains

TL;DR

This paper explores the phenomenon of homomesy, where the average of a statistic over orbits of certain actions on combinatorial objects remains constant, and provides a framework with examples including products of two chains.

Contribution

It introduces a broad theoretical framework for homomesy, extending previous results and applying to various combinatorial and algebraic contexts, with detailed analysis on products of two chains.

Findings

Homomesy occurs in diverse combinatorial actions.

A unifying framework explains homomesy phenomena.

Detailed analysis of rowmotion on products of two chains.

Abstract

Many invertible actions $\tau$ on a set ${\mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${\mathcal{S}}$, exhibit the following property which we dub \textbf{homomesy}: the average of $f$ over each $\tau$-orbit in ${\mathcal{S}}$ is the same as the average of $f$ over the whole set ${\mathcal{S}}$. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind that applies more broadly, giving examples in a variety of contexts. These include linear actions on vector spaces, sandpile dynamics, Suter's action on certain subposets of Young's Lattice, Lyness 5-cycles, promotion of rectangular semi-standard Young tableaux, and the rowmotion and promotion actions on certain posets. We give a detailed description of the latter situation for products of two chains.