Parking functions and triangulation of the associahedron

We show that a minimal triangulation of the associahedron (Stasheff polytope) of dimension n is made of (n+ 1)n−1 simplices. We construct a natural bijection with the set of parking functions from a new interpretation of parking functions in terms of shuffles. Introduction The Stasheff polytope, also known as the associahedron, is a polytope which comes naturally with a poset structure on the set of vertices (Tamari poset), hence a natural orientation on each edge. We decompose this polytope into a union of oriented simplices, the orientation being compatible with the poset structure. This construction defines the associahedron as the geometric realization of a simplicial set. In dimension n the number of (non-degenerate) simplices is (n+ 1). A parking function is a permutation of a sequence of integers i1 ≤ · · · ≤ in such that 1 ≤ ik ≤ k for any k. For fixed n the number of parking functions is (n+ 1). We show that the set PFn of parking functions of length n admits the following inductive description: PFn = ⋃ p+q=n−1 p≥0,q≥0 {1, . . . , p+ 1} × Sh(p, q)× PFp × PFq where Sh(p, q) is the set of (p, q)-shuffles. From this bijection we deduce a natural bijection between the top dimensional simplices of the associahedron and the parking functions.