Rowmotion on fences

TL;DR

This paper explores the dynamics of rowmotion on fences, a specific class of posets, revealing new phenomena like homometry and establishing homomesy results for antichains and ideals.

Contribution

It introduces the concept of homometry in rowmotion on fences and proves a general homomesy result for self-dual posets, advancing understanding of poset dynamics.

Findings

Visualization of antichain orbits using tilings

Homometry phenomenon where statistics are constant on same-sized orbits

Homomesy results for antichains and ideals in fences

Abstract

A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 < x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+1} < ... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a general homomesy result for all self-dual posets. We end with some conjectures and avenues for future research.