Rowmotion on the Chain of V's Poset and Whirling Dynamics

Given a finite poset $P$, we study the whirling action on vertex-labelings of $P$ with the elements $\{0,1,2,\dotsc,k\}$. When such labelings are (weakly) order-reversing, we call them $k$-bounded $P$-partitions. We give a general equivariant bijection between $k$-bounded $P$-partitions and order ideals of the poset $P\times [k]$ which conveys whirling to the well-studied rowmotion operator. As an application, we derive periodicity and homomesy results for rowmotion acting on the chain of V's poset $\sf V \times [k]$. We are able to generalize some of these results to the more complicated dynamics of rowmotion on $\sf C _{n}\times [k]$, where $\sf C _{n}$ is the claw poset with $n$ unrelated elements each covering $\widehat{0}$.