Operahedron Lattices

Abstract

Laplante-Anfossi associated to each rooted plane tree a polytope called an operahedron. He also defined a partial order on the vertex set of an operahedron and asked if the resulting poset is a lattice. We answer this question in the affirmative, motivating us to name Laplante-Anfossi's posets operahedron lattices. The operahedron lattice of a chain with $n+1$ vertices is isomorphic to the $n$-th Tamari lattice, while the operahedron lattice of a claw with $n+1$ vertices is isomorphic to $\mathrm{Weak}(\mathfrak S_n)$, the weak order on the symmetric group $\mathfrak S_n$. We characterize semidistributive operahedron lattices and trim operahedron lattices. Let $Δ_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ be the principal order ideal of $\mathrm{Weak}(\mathfrak S_n)$ generated by the permutation ${w_\circ(k,n)=k(k-1)\cdots 1(k+1)(k+2)\cdots n}$. Our final result states that the operahedron lattice of a broom with $n+1$ vertices and $k$ leaves is isomorphic to the subposet of $\mathrm{Weak}(\mathfrak S_n)$ consisting of the preimages of $Δ_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ under West's stack-sorting map; as a consequence, we deduce that this subposet is a semidistributive lattice.

Figures (10)

• Figure 1: On the left is the chain in $\mathrm{PT}_4$. In the middle is the claw in $\mathrm{PT}_4$. On the right is the broom $\mathsf{Broom}_{3,7}\in\mathrm{PT}_7$. We have identified the vertex set of each tree in $\mathrm{PT}_n$ with $\{0,1,\ldots,n\}$ in a manner such that $0,1,\ldots,n$ is the preorder traversal of the tree.
• Figure 2: On the left is the operahedron lattice of the chain in $\mathrm{PT}_3$, which is isomorphic to the third Tamari lattice. On the right is the operahedron lattice of the claw in $\mathrm{PT}_3$, which is isomorphic to the weak order on $\mathfrak S_3$. Each tube is circled in blue.
• Figure 3: The operahedron lattice of the tree $$. We have identified the vertex set of the tree with the set $\{0,1,2,3,4\}$ so that $0,1,2,3,4$ is the preorder traversal. Each tube is circled in blue. Edges of the lattice corresponding to permutohedron moves are purple, while edges corresponding to associahedron moves are orange.
• Figure 4: Three maximal nestings of a tree in $\mathrm{PT}_9$. The top two maximal nestings cover the bottom maximal nesting. The cover relation on the left corresponds to an associahedron move, while the cover relation on the right corresponds to a permutohedron move.
• Figure 5: Applying $\Psi$ to the maximal nestings in \ref{['fig:Psi1']} yields these three pairs, each of which consists of an ornamentation (which we represent by circling the ornaments in red) and a linear extension.

Theorems & Definitions (40)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Corollary 1.7
• Proposition 3.1
• proof
• Remark 3.2