Toggling, rowmotion, and homomesy on interval-closed sets

TL;DR

This paper explores the combinatorial dynamics of interval-closed sets in finite posets, introducing a generalized toggle group, characterizing rowmotion, and revealing homomesy phenomena through enumeration and orbit analysis.

Contribution

It develops a comprehensive framework for rowmotion on interval-closed sets, extending toggle group concepts, and provides new enumeration and homomesy results for specific poset classes.

Findings

Characterization of rowmotion as toggle product

Enumeration of interval-closed sets in specific posets

Homomesy results involving signed cardinality

Abstract

Interval-closed sets of a poset are a natural superset of order ideals. We initiate the study of interval-closed sets of finite posets from enumerative and dynamical perspectives. In particular, we use the generalized toggle group to define rowmotion on interval-closed sets as a product of these toggles. Our main theorem is an intricate global characterization of rowmotion on interval-closed sets, which we show is equivalent to the toggling definition. We also study specific posets; we enumerate interval-closed sets of ordinal sums of antichains, completely describe their rowmotion orbits, and prove a homomesy result involving the signed cardinality statistic. Finally, we study interval-closed sets of product of chains posets, proving further results about enumeration and homomesy.