The Pop-Stack Operator on Ornamentation Lattices

TL;DR

This paper explores the properties of the pop-stack operator on ornamentation lattices associated with rooted plane trees, generalizing known results from Tamari lattices to broader lattice structures.

Contribution

It generalizes the analysis of the pop-stack operator from Tamari lattices to ornamentation lattices of rooted plane trees, including orbit size bounds and image characterizations.

Findings

Maximum size of forward orbits on ornamentation lattices computed

Characterization of the image of the pop-stack operator generalized

Necessary conditions for elements to be in the image of powers of the operator provided

Abstract

Each rooted plane tree $\mathsf{T}$ has an associated ornamentation lattice $\mathcal{O}(\mathsf{T})$. The ornamentation lattice of an $n$-element chain is the $n$-th Tamari lattice. We study the pop-stack operator $\mathsf{Pop}\colon\mathcal{O}(\mathsf{T})\to\mathcal{O}(\mathsf{T})$, which sends each element $\delta$ to the meet of the elements covered by or equal to $\delta$. We compute the maximum size of a forward orbit of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Defant for Tamari lattices. We also characterize the image of $\mathsf{Pop}$ on $\mathcal{O}(\mathsf{T})$, generalizing a result of Hong for Tamari lattices. For each integer $k\geq 0$, we provide necessary conditions for an element of $\mathcal{O}(\mathsf{T})$ to be in the image of $\mathsf{Pop}^k$. This allows us to completely characterize the image of $\mathsf{Pop}^k$ on a Tamari lattice.