  There is a well-known bijection between parking functions of a fixed length
and maximal chains of the noncrossing partition lattice which we can use to
associate to each set of parking functions a poset whose Hasse diagram is the
union of the corresponding maximal chains. We introduce a decomposition of
parking functions based on the largest number omitted and prove several
theorems about the corresponding posets. In particular, they share properties
with the noncrossing partition lattice such as local self-duality, a nice
characterization of intervals, a readily computable M\"obius function, and a
symmetric chain decomposition. We also explore connections with order
complexes, labeled Dyck paths, and rooted forests.

Combinatorics