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

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.