论文标题
Schroder Comminatorics和$ν$ -associahedra
Schroder combinatorics and $ν$-associahedra
论文作者
论文摘要
我们研究$ν$-Schröder路径，这是Schröder路径，它们在给定的晶格路径$ν$上弱弱。一些经典的徒和列举结果扩展到$ν$ - 设定，包括小型和大型Schröder路径之间的关系。我们介绍了两个$ν$-schröder对象的posets，即$ν$-schröder路径和树木，并表明它们是$ν$ sassociahedron $a_ν$的脸部poset的同构。结果的结果是，$ i $维$a_ν$的面孔由$ν$-schröder路径用$ i $ i $ i $ i $ i $ i $ i $ i $ $ $ $ n $ n是“合理的” lattice'lattice path。使用我们对$a_ν$面部poset的新描述，我们应用离散的摩尔斯理论表明$a_ν$是合同的。这产生了两个证据之一，因为$a_ν$的欧拉特征是一个。第二个证明是通过$ν$ -Narayana多项式的公式来获得的。
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.
