Rowmotion on hook and two-row alt $ν$-Tamari lattices

Abstract

In 2024, Ceballos and Chenevi{è}re introduced alt $ν$-Tamari lattices, parameterized by a lattice path $ν$ and an increment vector $δ$, as a common generalization of $ν$-Tamari and $ν$-Dyck lattices. We study rowmotion on two families: the alt hook-Tamari lattice $\mathsf{H}_δ(a,b)$ (where $ν=EN^{a-1}E^{b-1}N$) and the alt $2$-row-Tamari lattice $\mathsf{T}_δ(a,b)$ (where $ν=E^aNE^bN$). We explicitly determine the orbit structures of $\mathsf{H}_δ(a,b)$ and $\mathsf{T}_δ(a,b)$ under rowmotion, and prove that their orbit structures are independent of the increment vector $δ$. As a consequence, we show that rowmotion on $\mathsf{H}_δ(a,b)$ exhibits the cyclic sieving phenomenon. We also compute orbit sums for several natural statistics. In the hook case, we evaluate the down-degree, peak, valley, and area statistics; in the $2$-row case, we focus on the down-degree statistic. All of these -- except for the area statistic -- are homometric under rowmotion. Regarding the methodology of this paper, our results in the hook case are obtained by applying a simple local modification to their Hasse diagrams. In the $2$-row case, we introduce a switching property for semidistributive lattices, which allows us to compare the orbit structures arising from different increment vectors.