Some Instances of Homomesy Among Ideals of Posets

TL;DR

This paper extends the study of homomesy, a phenomenon where a statistic has the same average over all orbits of a permutation, to more general settings involving posets and various statistics.

Contribution

It generalizes previous homomesy results from product of chains to broader classes of posets and permutations, introducing a versatile framework.

Findings

Homomesy holds for generalized rowmotion and promotion in product of two chains.

New homomesy results are established for posets of type A and B.

The framework suggests potential for broader applications in poset theory.

Abstract

Given a permutation $\tau$ defined on a set of combinatorial objects $S$, together with some statistic $f:S\rightarrow \mathbb{R}$, we say that the triple $\langle S, \tau,f \rangle$ exhibits homomesy if $f$ has the same average along all orbits of $\tau$ in $S$. This phenomenon was noticed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). After Propp and Roby's paper, homomesy has received a lot of attention and a number of mathematicians are intrigued by it. While seeming ubiquitous, homomesy is often surprisingly non-trivial to prove. Propp and Roby studied homomesy in the set of ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the ideal. In this paper we extend their results to generalized rowmotion and promotion together with a wider class of statistics in product of two chains . Moreover, we derive some homomesy results in posets of type A and B. We believe that the framework that we set up can be used to prove similar results in wider classes of posets.