On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice Q̌, spanning a Euclidean space V . Let m be a positive integer and AmΦ be the arrangement of hyperplanes in V of the form (α, x) = k for α ∈ Φ and k = 0, 1, . . . ,m. It is known that the number N(Φ, m) of bounded dominant regions of AmΦ is equal to the number of facets of the positive part ∆ m + (Φ) of the generalized cluster complex associated to the pair (Φ, m) by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of AmΦ and conjecture that the corresponding refinement of N(Φ, m) coincides with the h-vector of ∆m+ (Φ). We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient Q̌/ (mh − 1) Q̌ and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of AmΦ . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1.