Orbitmesy and promotion on self-dual posets

TL;DR

This paper introduces orbitmesy, a new concept related to homomesy, and studies its occurrence under promotion actions on certain self-dual posets, providing classifications and general results.

Contribution

It defines orbitmesy, explores its relation to homomesy, classifies orbitmesic promotion orbits on zig-zag posets, and proves general theorems for self-dual posets.

Findings

Classified all orbitmesic promotion orbits for 4-element zig-zag posets.

Established methods to find orbitmesic orbits using homomesy of different actions.

Proved general results for infinite families of orbitmesic orbits in self-dual posets.

Abstract

We introduce the notion of orbitmesy, which is related to homomesy, a central phenomenon in dynamical algebraic combinatorics. An orbit $O$ is said to be orbitmesic with respect to a statistic if the orbit's average statistic value is equal to the global average. We particularly focus on the action of promotion on increasing labelings of certain fence posets called zig-zag posets, and two statistics, the antipodal sum statistic and the total sum statistic. We classify all of the orbitmesic promotion orbits for the zig-zag poset with four elements. Along the way, we investigate how homomesy of one action can be used to find orbitmesic orbits for another action, for the same fixed statistic. We prove several general results which can be used to find infinite families of orbitmesic orbits for any self-dual poset.