Schröder combinatorics and [formula omitted]-associahedra

Abstract

We study ν-Schröder paths, which are Schröder paths which stay weakly above a given lattice path ν. Some classical bijective and enumerative results are extended to the ν-setting, including the relationship between small and large Schröder paths. We introduce two posets of ν-Schröder objects, namely ν-Schröder paths and trees, and show that they are isomorphic to the face poset of the ν-associahedron Aν introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the i-dimensional faces of Aν are indexed by ν-Schröder paths with i diagonal steps, and we obtain a closed-form expression for these Schröder numbers in the special case when ν is a ‘rational’ lattice path. Using our new description of the face poset of Aν, we apply discrete Morse theory to show that Aν is contractible. This yields one of two proofs presented for the fact that the Euler characteristic of Aν is one. A second proof of this is obtained via a formula for the ν-Narayana polynomial in terms of ν-Schröder numbers.