The Pop-stack-sorting Operator on Tamari Lattices

TL;DR

This paper studies the dynamics of a pop-stack-sorting operator on Tamari lattices, providing explicit formulas, confirming conjectures about its generating function, and linking its image size to Motzkin numbers.

Contribution

It introduces and analyzes the pop-stack-sorting operator on Tamari lattices, deriving explicit formulas, proving rationality of generating functions, and establishing the image size as Motzkin numbers.

Findings

Derived explicit generating function for t-pop-sortable elements.

Confirmed the rationality of the generating function as conjectured.

Proved the image size of the operator equals the Motzkin number.

Abstract

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.