The Pop-Stack Operator on Ornamentation Lattices

TL;DR

The paper introduces ornamentation lattices $\mathcal{O}(\mathsf{T})$ for rooted plane trees and studies the pop-stack operator $\mathsf{Pop}$ on them, generalizing known results from Tamari lattices. It derives a tight formula for the maximal forward orbit size of $\mathsf{Pop}$ in terms of maximal chains and node depths, and provides a local, hugging-based description of the $\mathsf{Pop}$-image, together with a semidistributivity proof for all ornamentation lattices. The work further analyzes $\mathsf{Pop}^k$, giving necessary conditions and a complete Tamari-chain result, including a closed generating function for the sizes of $\mathsf{Pop}^k$-images on chains. These results unify dynamical and lattice-theoretic aspects of Pop on a broad class of lattices and connect to canonical join complexes. They also yield explicit combinatorial characterizations in the Tamari case and establish fundamental structural properties of 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 $\mathsf{Pop}\colon\mathcal{O}(\mathsf{T})\to\mathcal{O}(\mathsf{T})$, which sends each element $δ$ to the meet of the elements covered by or equal to $δ$. 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.

Figures (8)

• Figure 1: The ornamentation lattice of a rooted tree with $4$ nodes.
• Figure 2: On the left is a plane tree $\mathsf{T}$. Three of the maximal chains, named $C$, $C'$, and $C"$, are labeled under their leaves. For each $\widetilde{C}\in\{C,C',C"\}$, each node $u\in\widetilde{C}$ is labeled with the value of $f_{\widetilde{C}}(u)$. On the right is the tree $\mathsf{T}_v^*$, where $v$ is as indicated. This tree is obtained by adding $f_{\widetilde{C}}(v)$ new nodes to the bottom of each chain $\widetilde{C}\in\{C,C',C"\}$. On the right, each node in $\mathsf{T}$ is labeled with its $v$-rank.
• Figure 3: On the left is an ornamentation $\delta$. On the right is the ornamentation $\delta_u^v$ obtained by reducing the ornament hung at $v$ by the node $u$.
• Figure 4: Applying $\mathsf{Pop}$ to the ornamentation on the left yields the ornamentation on the right.
• Figure 5: On the left is a tree $\mathsf{T}$ with a maximal chain $C^*$ consisting of nodes $v_0,v_1,\ldots,v_8$. We have $k=f_{C^*}(\mathfrak{r})=5$. On the right is the ornamentation $\delta^\dagger$.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof