Schröder combinatorics and ν-associahedra

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 A_ν introduced by Ceballos, Padrol and Sarmiento. A consequence of our results is that the i-dimensional faces of A_ν 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 A_ν, we apply discrete Morse theory to show that A_ν is contractible. This yields one of two proofs presented for the fact that the Euler characteristic of A_ν is one. A second proof of this is obtained via a formula for the ν-Narayana polynomial in terms of ν-Schröder numbers.