On some enumerative aspects of generalized associahedra

We prove a conjecture of F. Chapoton relating certain enumerative invariants of (a) the cluster complex associated by S. Fomin and A. Zelevinsky to a finite root system and (b) the lattice of noncrossing partitions associated to the corresponding finite real reflection group. 1. The result Let Φ be a finite root system spanning an n-dimensional Euclidean space V with corresponding finite reflection group W . Let Φ be a positive system for Φ with corresponding simple system Π. The cluster complex ∆(Φ) was introduced by S. Fomin and A. Zelevinsky within the context of their theory of cluster algebras [10, 11, 12]. It is a pure (n − 1)-dimensional simplicial complex on the vertex set Φ ∪ (−Π) which is homeomorphic to a sphere [11]. Although ∆(Φ) was initially defined under the assumption that Φ is crystallographic [11], its definition and main combinatorial properties are valid without this restriction [8, Section 5.3] [9]. In the crystallographic case, ∆(Φ) was realized explicitly in [7] as the boundary complex of an n-dimensional simplicial convex polytope P (Φ), known as the simplicial generalized associahedron associated to Φ; see [8] for an expository treatment of cluster complexes and generalized associahedra. The combinatorics of ∆(Φ) is closely related to that of a finite poset LW , known as the lattice of noncrossing partitions associated to W [3, 4] (see Section 2 for definitions). It is known, for instance (see [8, Theorem 5.9]), that the h-polynomial of ∆(Φ) is equal to the rank generating polynomial of LW . In particular the number of facets of ∆(Φ) is equal to the cardinality of LW . This number is a Catalan number if Φ has type An in the Cartan-Killing classification; in that case P (Φ) is the polar polytope to the classical n-dimensional associahedron [8, Section 3.1] and LW is isomorphic to the lattice of noncrossing partitions of the set {1, 2, . . . , n+1} [8, Section 5.1]. The poset LW is a self-dual graded lattice of rank n which plays an important role in the geometric group theory and topology of finite-type Artin groups; see [16] for a related survey article. The F -triangle for Φ, introduced by F. Chapoton [6, Section 2], is a refinement of the f -vector of ∆(Φ) defined by the generating function (1) F (Φ) = F (x, y) = n ∑