  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.
