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 $\{A⋖B,A⋖C\}$. 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 conjectured by Bernstein, Striker, and Vorland. This resolves the equivalent conjecture of Hopkins that the order of piecewise-linear rowmotion on the order polytope of $V\times [k]$ has order $2(k+2)$ for all $k\ge 1$.