The 2-associahedra are Eulerian

The 2-associahedra W n W_{\mathbf {n}} are a family of abstract polytopes, indexed by sequences of nonnegative integers of arbitrary length. They arise in symplectic geometry, where they govern the functoriality properties of the Fukaya category, but they have purely combinatorial and topological interpretations: for instance, W n W_{\mathbf {n}} indexes the possible degenerations in a configuration space of marked vertical lines in R 2 \mathbb {R}^{2} , up to translations and dilations. In this paper, we exploit the recursive structure of the 2-associahedra in order to further explore their combinatorial properties. First, we define a family of generating functions that count elements of 2-associahedra, and establish a system of equations satisfied by these generating functions. Next, we use this computation to prove that the 2-associahedra are Eulerian.