The Pop-stack-sorting Operator on Tamari Lattices

Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator ${\mathsf{P}\mathsf{o}\mathsf{p}}_M:M\to M$ for each complete meet-semilattice $M$ by $${\mathsf{P}\mathsf{o}\mathsf{p}}_M(x)=\bigwedge (\{y\in M:y⋖x\}\cup \{x\})\mathrm{.}$$ This paper concerns the dynamics of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{{\mathrm{T}\mathrm{a}\mathrm{m}}_n}$, where ${\mathrm{T}\mathrm{a}\mathrm{m}}_n$ is the $n$-th Tamari lattice. We say an element $x\in {\mathrm{T}\mathrm{a}\mathrm{m}}_n$ is $t$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable if ${\mathsf{P}\mathsf{o}\mathsf{p}}_M^t(x)$ is the minimal element and we let $h_t(n)$ denote the number of $t$-$\mathsf{P}\mathsf{o}\mathsf{p}$-sortable elements in ${\mathrm{T}\mathrm{a}\mathrm{m}}_n$. We find an explicit formula for the generating function ${\sum }_{n\ge 1}{h}_t(n)z^n$ and verify Defant’s conjecture that it is rational. We furthermore prove that the size of the image of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{{\mathrm{T}\mathrm{a}\mathrm{m}}_n}$ is the Motzkin number $M_n$, settling a conjecture of Defant and Williams.