triangle of the generalised cluster complex

The F-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the F-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the " F = M Conjecture " of Armstrong which predicts a surprising relation between the F-triangle and the Möbius function of his m-divisible partition poset associated to a finite root system. 1. Introduction. Fomin and Zelevinsky created a new exciting research field when they invented cluster algebras in [11]. The classification of cluster algebras of finite type from [12] says that there is a one-to-one correspondence between finite-type cluster algebras and finite root systems. Furthermore, for each finite root system Φ, Fomin and Zelevinsky [13] defined a simplicial complex corresponding to the associated cluster algebra , the cluster complex ∆(Φ). This is a simplicial complex on a subset of the set of roots Φ. As they showed, this complex has many remarkable properties. In particular, the number of facets is given by the Catalan number for the root system Φ, and, moreover, all the face numbers are given by elegant product formulae. Further remarkable (originally, conjectural) properties have been discovered by Chapoton in [8]. In this paper, he refines the face enumeration to, what he calls, the " F-triangle. " He computed the F-triangle for all types (and revealed his findings partially in [8]) and observed a surprising relationship (see [8, Conjecture 1]) between the F-triangle and the Möbius function of the non-crossing partition lattice N C(Φ) associated to Φ, the latter being due to Bessis [4] and Brady and Watt [5]. This relationship, to which we shall refer in the sequel as the " F = M Conjecture , " has been recently proved by Athanasiadis [2]. For further fascinating properties of the F-triangle see [8].