Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows

For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic polytope ${\mathrm{△}}_{\mathbb{H}}$ is the Minkowski sum of the standard simplices ${\mathrm{△}}_H$ for all $H\in \mathbb{H}$. We focus here on interval hypergraphs, where all hyperedges are intervals of $[n]$. They are precisely the deformations of Loday’s associahedron. Their vertex posets are Tamari interval posets, and we describe which Tamari interval poset appears as a vertex poset in which interval hypergraphic polytope. We also characterize the interval hypergraphs $\mathbb{I}$ for which the hypergraphic polytope ${\mathrm{△}}_{\mathbb{I}}$ is simple, and we study their vertex posets, which we call weeping willows.