Nested sets, set partitions and Kirkman-Cayley dissection numbers

In this paper we find exponential formulas for the Betti numbers of the De Concini–Procesi minimal wonderful models YG(r,p,n) associated to the complex reflection groups G(r,p,n). Our formulas are different from the ones already known in the literature: they are obtained by a new combinatorial encoding of the elements of a basis of the cohomology by means of set partitions with weights and exponents. We also point out that a similar combinatorial encoding can be used to describe the faces of the real spherical wonderful models of type An−1(=G(1,1,n)), Bn (=G(2,1,n)) and Dn(=G(2,2,n)). This provides exponential formulas for the f-vectors of the associated nestohedra: the Stasheff’s associahedra (in this case closed formulas are well known) and the graph associahedra of type Dn.