$P$-strict promotion and $Q$-partition rowmotion: the graded case

TL;DR

This paper explores the interplay between promotion and rowmotion actions on graded posets, introducing new results and bijections that generalize classical tableau theory and reveal homomesy and resonance phenomena.

Contribution

It introduces $P$-strict labelings and establishes their equivariant bijection with $Q$-partitions, extending classical combinatorial actions to a broader graded poset setting.

Findings

New homomesy results for promotion and rowmotion

Order results for $P$-strict labelings and $Q$-partitions

Resonance phenomena in promotion and rowmotion actions

Abstract

Promotion and rowmotion are intriguing actions in dynamical algebraic combinatorics which have inspired much work in recent years. In this paper, we study $P$-strict labelings of a finite, graded poset $P$ of rank $n$ and labels at most $q$, which generalize semistandard Young tableaux with $n$ rows and entries at most $q$, under promotion. These $P$-strict labelings are in equivariant bijection with $Q$-partitions under rowmotion, where $Q$ equals the product of $P$ and a chain of $q-n-1$ elements. We study the case where $P$ equals the product of chains in detail, yielding new homomesy and order results in the realm of tableaux and beyond. Furthermore, we apply the bijection to the cases in which $P$ is a minuscule poset and when $P$ is the three element $V$ poset. Finally, we give resonance results for promotion on $P$-strict labelings and rowmotion on $Q$-partitions.