In 1980, Edelman defined a poset on objects called the noncrossing 2-partitions. They are closely related with noncrossing partitions and parking functions. To some extent, his definition is a precursor of the parking space theory, in the framework of finite reflection groups. We present some enumerative and topological properties of this poset. In particular, we get a formula counting certain chains, that encompasses formulas for Whitney numbers (of both kinds). We prove shellability of the poset, and compute its homology as a representation of the symmetric group. We moreover link it with two well-known polytopes : the associahedron and the permutohedron.

中文翻译：

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停车功能坐垫的某些属性

1980年，爱德曼（Edelman）在被称为非交叉2分区的对象上定义了一个小样。它们与非交叉分区和停车功能密切相关。在一定程度上，在有限反射组的框架内，他的定义是停车位理论的先驱。我们介绍了此球状体的一些枚举和拓扑性质。特别是，我们得到一个计算某些链的公式，其中包括惠特尼数（两种）的公式。我们证明了该球型的可填充性，并计算其同源性作为对称基团的表示。此外，我们将其与两个著名的多面体联系在一起：缔合体和四面体。