Schroder combinatorics and $\nu$-associahedra

Abstract

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

20 pages, 11 figures