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.

我们学习 $\nu$-Schröder 路径，这是 Schröder 路径，在给定的晶格路径上方微弱地停留 $\nu$. 一些经典的双射和枚举结果被扩展到$\nu$-设置，包括大小 Schröder 路径之间的关系。我们介绍两个poset$\nu$-施罗德对象，即 $\nu$-Schröder 路径和树木，并证明它们与 $\nu$-副面体 ${\mathrm{一种}}_{\nu }$由 Ceballos、Padrol 和 Sarmiento 介绍。我们的结果的一个结果是$\mathrm{一世}$维面 ${\mathrm{一种}}_{\nu }$ 被索引 $\nu$-施罗德路径与 $\mathrm{一世}$ 对角线步骤，并且我们在特殊情况下获得这些 Schröder 数的封闭形式表达式，当 $\nu$是“有理”格子路径。使用我们对人脸偏位的新描述${\mathrm{一种}}_{\nu }$，我们应用离散莫尔斯理论来证明 ${\mathrm{一种}}_{\nu }$是可收缩的。这产生了为以下事实提供的两个证明之一：${\mathrm{一种}}_{\nu }$是一个。这个的第二个证明是通过一个公式获得的$\nu$-Narayana多项式在 $\nu$-施罗德数。