On the Order of P-Strict Promotion on $V\times [\ell]$

Abstract

Denote by $V$ the poset consisting of the elements $\{A,B,C\}$ with cover relations $\{A\lessdot B, A\lessdot C\}$. We show that $P$-strict promotion, as defined by Bernstein, Striker, and Vorland, on $P$-strict labelings of $V\times [\ell]$ with labels in the set $[q]$ has order $2q$ for every $\ell\ge 1$ and $q\ge 3$.  As a consequence of results of Bernstein, Striker, and Vorland, this result proves that piecewise-linear rowmotion on $V\times [k]$ has order $2(k+2)$ for all $k\ge 1$, as conjectured by Hopkins.