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.
翻译:1980年,Edelman定义了一个被称作“非交叉两部分”的物体的图案。这些图案与非交叉分割和停车功能密切相关。在某种程度上,他的定义是停车空间理论的先锋,在有限的反射组框架内。我们展示了这个图案的一些数字和地貌特性。特别是,我们得到了一个公式来计算某些链条,包括惠特尼数字(两种类型)的公式。我们证明了这些图案的可弹性,并把它同理学算为对称组的表示。我们还将它与两个众所周知的多台式联系起来:方形和普穆托赫德龙。