Toggling Involutions and Homomesies for Maps on Finite Sets, Noncrossing Partitions, and Independent Sets of Path Graphs

This paper explores the orbit structure and homomesy properties of various actions on finite sets. The homomesy phenomenon, meaning constant averages over orbits, was proposed by Propp and Roby in 2011. For many of the known instances of homomesy, Reiner, Stanton, and White\u27s cyclic sieving phenomenon (CSP) is also present. However, we prove homomesy for several maps whose order is large relative to the size of the set, implying that a natural CSP is unlikely. Sometimes we can prove facts about the orbit sizes either as a corollary to the homomesy or by the technique used to prove homomesy. Many of the actions we describe are products of much simpler ones. Among these, we consider maps defined as products of simple toggling involutions. These come from the Striker\u27s theory of generalized toggle groups, an active area of research in dynamical algebraic combinatorics. Several known instances of homomesy have been discovered for elements of toggle groups. While the individual toggles have order two, the order of a composition of several toggles is more difficult to analyze. We also consider an action of whirling, due to Propp, that can be defined for any family of functions between finite sets. This action is also the composition of simpler ones