Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator $\mathsf{ Pop}_M : M \to M$ for each complete meet-semilattice $M$ by $$\mathsf{ Pop}_M(x)=\bigwedge(\{ y\in M: y\lessdot x\}\cup \{ x\}).$$ This paper concerns the dynamics of $\mathsf{ Pop}_{ \mathrm{ Tam}_n}$, where $\mathrm{ Tam}_n$ is the $n$-th Tamari lattice. We say an element $x\in \mathrm{ Tam}_n$ is $t$-$\mathsf{ Pop}$-sortable if $\mathsf{ Pop}_M^t (x)$ is the minimal element and we let $h_t(n)$ denote the number of $t$-$\mathsf{ Pop}$-sortable elements in $\mathrm{ Tam}_n$. 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 $\mathsf{ Pop}_{ \mathrm{ Tam}_n}$ is the Motzkin number $M_n$, settling a conjecture of Defant and Williams.
Categories: math.CO
