Meeting Covered Elements in $\hat{I}½$-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 ${\mathsf{Pop}}_{\text{Tam}(\hat{I}½)}$, where $\text{Tam}(\hat{I}½)$ is the $\hat{I}½$-Tamari lattice. We determine the maximum size of a forward orbit of ${\mathsf{Pop}}_{\text{Tam}(\hat{I}½)}$. When $\text{Tam}(\hat{I}½)$ 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 $\hat{I}¼\in \text{Tam}(\hat{I}½)$ to be $t$-$\mathsf{Pop}$-sortable if ${\mathsf{Pop}}_{\text{Tam}(\hat{I}½)}^t(\hat{I}¼)=\hat{I}½$. We enumerate $1$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(\hat{I}½)$ for arbitrary $\hat{I}½$. We also give a recursive method to generate $2$-$\mathsf{Pop}$-sortable lattice paths in $\text{Tam}(\hat{I}½)$ for arbitrary $\hat{I}½$; this allows us to enumerate $2$-$\mathsf{Pop}$-sortable lattice paths in a large variety of $\hat{I}½$-Tamari lattices that includes the $m$-Tamari lattices.

25 pages, 4 figures