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arXiv · 2307.08520

Toggling, rowmotion, and homomesy on interval-closed sets

Jennifer Elder·Nadia Lafrenière·Erin McNicholas·Jessica Striker·Amanda Welch
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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.

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Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, Amanda Welch. 2023-07-17. Toggling, rowmotion, and homomesy on interval-closed sets. https://arxiv.org/abs/2307.08520

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