Schroder combinatorics and $\nu$-associahedra

TL;DR

This paper extends classical Schr"oder path results to a new $ u$-setting, introduces $ u$-associahedra, and proves their topological properties using combinatorial and Morse-theoretic methods.

Contribution

It introduces $ u$-Schr"oder paths and $ u$-associahedra, establishing their combinatorial structure and topological properties, including face enumeration and contractibility proofs.

Findings

$ u$-Schr"oder paths are in bijection with faces of $ u$-associahedra.

The face poset of $A_ u$ is isomorphic to $ u$-Schr"oder objects.

$A_ u$ is contractible with Euler characteristic one.

Abstract

We study $\nu$-Schr\"oder paths, which are Schr\"oder 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 Schr\"oder paths. We introduce two posets of $\nu$-Schr\"oder objects, namely $\nu$-Schr\"oder 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 $\nu$-Schr\"oder paths with $i$ diagonal steps, and we obtain a closed-form expression for these Schr\"oder 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 $\nu$-Schr\"oder numbers.