Some Instances of Homomesy Among Ideals of Posets

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 observed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). Propp and Roby studied homomesy in the set of order ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the order ideal. In this paper we extend their results to generalized rowmotion and promotion, together with a wider class of statistics in the product of two chains. Moreover, we derive similar results in other simply described posets. We believe that the framework we set up here can be fruitful in demonstrating homomesy results in order ideals of broader classes of posets.