Interval-closed set rowmotion and homomesy on products of two chains

TL;DR

This paper investigates the dynamics of rowmotion on interval-closed sets, simplifies its global definition, characterizes orbits on specific posets, and proves a homomesy conjecture related to signed cardinality.

Contribution

It provides a simplified definition of interval-closed set rowmotion and completely describes its orbits on products of two chains, advancing understanding of its combinatorial properties.

Findings

Simplified the global definition of interval-closed set rowmotion.

Described all orbits of rowmotion on $[2]\times[n]$.

Proved a homomesy conjecture involving signed cardinality.

Abstract

We study rowmotion dynamics on interval-closed sets. Our first main result proves a simplification of the global definition of interval-closed set rowmotion from (Elder, Lafreni\`ere, McNicholas, Striker, and Welch 2024). We then completely describe the orbits of interval-closed set rowmotion on products of two chains $[2]\times[n]$ and use this understanding to prove a homomesy conjecture from (ELMSW 2024) involving the signed cardinality statistic.