Homomesy and Toggling Actions on Parking Functions

Cyrus Young
Graduate Student
University of Wisconsin-Milwaukee

Say you have a finite set, a permutation on your set, and a discrete statistic usually counting something within the elements of your set. The homomesy phenomenon occurs when the average value of the statistic over each orbit of the permutation is the same. Since the formalization of the homomesy phenomenon by James Propp in 2013, there have been many results regarding the appearance of homomesy in a wide variety of contexts. In this talk, we will primarily investigate work from Dowling and Lafrenière, who proved in 2023 a collection of homomesies on permutations obtained from toggling actions, which are a class of group actions related to rotation. It turns out we can generalize these results to parking functions, and this generalization further motivates ideas about the study of collections of combinatorial objects and the appearance of homomesy in nature.