The Image of the Pop Operator on Various Lattices

TL;DR

This paper generalizes the pop-stack sorting map to various lattices, analyzing their images and settling four conjectures related to generating functions of these images across different lattice types.

Contribution

It extends the classical pop-stack sorting map to multiple lattice structures and proves four conjectures about their generating functions.

Findings

Established the structure of $ ext{Pop}_M(M)$ for several lattices.

Proved four conjectures on the generating function $ ext{Pop}(M; q)$.

Connected the pop operator's image to combinatorial properties of different lattices.

Abstract

Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice $M$, a map $\mathsf{Pop}_{M}: M \to M$ that sends an element $x\in M$ to the meet of $x$ and the elements covered by $x$. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study $\mathsf{Pop}_{M}(M)$ when $M$ is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $B_n$. In particular, we settle four conjectures proposed by Defant and Williams on the generating function \begin{equation*} \mathsf{Pop}(M; q) = \sum_{b \in \mathsf{Pop}_{M}(M)} q^{|\mathscr{U}_{M}(b)|}, \end{equation*} where $\mathscr{U}_{M}(b)$ is the set of elements of $M$ that cover $b$.