• 1 Department of Mathematics [Lowell]
• 2 University of Connecticut

Many cyclic actions $τ$ on a finite set $\mathcal{S}$ ; of combinatorial objects, along with a natural statistic $f$ on $\mathcal{S}$, exhibit ``homomesy'': the average of $f$ over each $τ$-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 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.