HOMOMESY: THEORY, APPLICATIONS, AND EXPLORATIONS

Abstract

Homomesy is a phenomenon that occurs in combinatorial structures when the average value of a statistic over each orbit is the same.But I extend the definition to allow for sets with infinite orbits. This thesis explores the theory of homomesy for arbitrary sets, functions, and statistics. I provide general results about homomesy and show how these can be used to solve problems in combinatorics more efficiently. Also, I also analyze some (subsets of) parking functions and apply functions and statistics from the FindStat database to them to see if they exhibit homomesic behavior. I don't prove homomesic behavior for these sets directly, but instead prove more general results that can be applied to these sets.