Some instances of Homomesy in product of two chains

Shahrzad Haddadan

Published 2014-10-17Version 1

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 \emph{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 got a lot of attention and a number of mathematicians are intrigued by it. While being 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 .