Paper detail

Meeting Covered Elements in $ν$-Tamari Lattices

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 $\mathsf{Pop}_{\text{Tam}(ν)}$, where $\text{Tam}(ν)$ is the $ν$-Tamari lattice. We determine the maximum size of a forward orbit of $\mathsf{Pop}_{\text{Tam}(ν)}$. When $\text{Tam}(ν)$ 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 $μ\in\text{Tam}(ν)$ to be $t$-$\mathsf{Pop}$-sortable if $\mathsf{Pop}_{\text{Tam}(ν)}^t(μ)=ν$. We enumerate $1$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(ν)$ for arbitrary $ν$. We also give a recursive method to generate $2$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(ν)$ for arbitrary $ν$; this allows us to enumerate $2$-$\mathsf{Pop}$-sortable lattice paths in a large variety of $ν$-Tamari lattices that includes the $m$-Tamari lattices.