Homomesy in products of two chains

Abstract

Many cyclic actions τ on a finite set S of combinatorial objects, along with a natural statistic f on S, exhibit "homomesy": The average of f over each τ -orbit in S is the same as the average of f over the whole set S. This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. We describe a theoretical framework for results of this kind and discuss old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains. © 2013 Discrete Mathematics and Theoretical Computer Science (DMTCS).