Generalized Catalan Numbers, Weyl Groups and Arrangements of Hyperplanes

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x)= k, for α∈Φ and k=0, 1, . . . ,m, is denoted here by AΦ . The characteristic polynomial of AΦ is related in the paper to that of the Coxeter arrangement AΦ (corresponding to m=0), and the number of regions into which the fundamental chamber of AΦ is dissected by the hyperplanes of AΦ is deduced to be equal to the product ∏ i=1(ei +mh+1)/(ei +1), where e1, e2, . . . , e are the exponents of Φ and h is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of Φ are included.