The Pop-Stack Operator on Ornamentation Lattices

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 $\mathsf{Pop}:\mathcal{O}(\mathsf{T})\to \mathcal{O}(\mathsf{T})$, which sends each element $\hat{I}´$ to the meet of the elements covered by or equal to $\hat{I}´$. 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 $k\ge 0$, 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.

16 pages, 8 figures