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

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, Lafrenière, 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.

42 pages