Meeting Covered Elements in $ν$-Tamari Lattices

Abstract: For each complete meet-semilattice $M$, we define an operator $\mathsf{Pop}M:M\to M$ by [\mathsf{Pop}_M(x)=\bigwedge({y\in M:y\lessdot x}\cup{x}).] When $M$ is the right weak order on a symmetric group, $\mathsf{Pop}_M$ is the pop-stack-sorting map. We prove some general properties of these operators, including a theorem that describes how they interact with certain lattice congruences. We then specialize our attention to the dynamics of $\mathsf{Pop}{\text{Tam}(\nu)}$, where $\text{Tam}(\nu)$ is the $\nu$-Tamari lattice. We determine the maximum size of a forward orbit of $\mathsf{Pop}{\text{Tam}(\nu)}$. When $\text{Tam}(\nu)$ is the $n^\text{th}$ $m$-Tamari lattice, this maximum forward orbit size is $m+n-1$; in this case, we prove that the number of forward orbits of size $m+n-1$ is [\frac{1}{n-1}\binom{(m+1)(n-2)+m-1}{n-2}.] Motivated by the recent investigation of the pop-stack-sorting map, we define a lattice path $\mu\in\text{Tam}(\nu)$ to be $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}{\text{Tam}(\nu)}^t(\mu)=\nu$. We enumerate $1$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(\nu)$ for arbitrary $\nu$. We also give a recursive method to generate $2$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(\nu)$ for arbitrary $\nu$; this allows us to enumerate $2$-$\mathsf{Pop}$-sortable lattice paths in a large variety of $\nu$-Tamari lattices that includes the $m$-Tamari lattices.