Rowmotion on the chain of V's poset and whirling dynamics

TL;DR

This paper builds a general equivariant bijection between $k$-bounded $P$-partitions and order ideals of $P\times[k]$, translating the whirling action into rowmotion and enabling unified dynamical results. Applying this framework to the chain of V's ${\sf V}_k$ yields a precise order for rowmotion, $2(k+2)$, plus center-seeking whorm-based homomesies and a flux-capacitor statistic with a fixed average; the authors extend these phenomena to the chain of claws ${\sf C}_n\times[k]$ through edge-seeking whorms. The periodicity in the claw case is $(k+2)\,\mathrm{LCM}(1,\dots,m)$ with $m=\min(k+1,n)$, and while many homomesies persist (e.g., 0-mesic differences among certain indicators), some flux-related identities fail to generalize in this broader setting. Overall, the work deepens the connection between whirling and rowmotion, expands the class of posets with tractable dynamics, and provides new tools for analyzing dynamical symmetry and periodicity in combinatorial dynamics.

Abstract

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 $V \times [k]$. We are able to generalize some of these results to the more complicated dynamics of rowmotion on $C_{n}\times [k]$, where $C_{n}$ is the claw poset with $n$ unrelated elements each covering $\widehat{0}$.

Figures (16)

• Figure 1: The orbit board of $\mathbf{w}$ on $\mathrm{Inj}_1(3,6)$ containing $f=415$, partitioned into "chunks".
• Figure 2: A linear extension of ${\sf V}$ on the left and the linear extension for ${\sf V}\times[4]$ constructed from it on the right, as in the proof of Theorem \ref{['thm:eqwhirl']}.
• Figure 3: Hasse diagram of ${\sf V}_k$ with our vertex-labeling convention.
• Figure 4: The two orbits of rowmotion on order ideals of ${\sf V} \times [3]$
• Figure 5: An orbit of rowmotion on $P={\sf V} \times [4]$, with corresponding $P$-partitions (which are being whirled), as in Theorem \ref{['thm:eqwhirl']}.

Theorems & Definitions (73)

• Definition 1.1
• Example 1.2
• Definition 1.3
• Example 1.4
• Proposition 1.5: CF95
• Definition 2.1: JPR18
• Example 2.2
• Theorem 2.3: JPR18
• Example 2.4
• Definition 2.5