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 $\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ 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 $\Delta_{\mathrm{Weak}(\mathfrak S_n)}(w_\circ(k,n))$ under West's stack-sorting map; as a consequence, we deduce that this subposet is a semidistributive lattice.