Pointed and multi-pointed partitions of type A and B

The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we show that some operads are Koszul over Z. Introduction For every finite Weyl group W , there exists a generalized partition poset (cf. [2]) defined through the hyperplane arrangement of type W . In the case An−1, this poset is the usual poset of partitions of {1, . . . , n}. B. Fresse proved in [6] that this poset also arises from the theory of operads. A pointed and a multipointed variation of this poset were defined by the second author in [14], once again in the context of Koszul duality of operads. In this article, we study the main properties of these two types of posets. One motivation for this article was the idea that there should also exist a pointed partition poset and a multipointed partition poset for other Weyl groups. Here we propose a definition for the pointed partition posets of type B and check that it satisfies most of the properties which are expected in general and hold in type A. This definition was guessed by similarity, but we hope that there is a general definition of geometric nature, to be found. Let us summarize briefly what properties the generalized pointed partition poset associated to a Weyl group should have. Let h be the Coxeter number and n be the rank of the Weyl group W . Then its characteristic polynomial should be (x−h)n; the number of maximal elements should be h, with a transitive action of the Weyl group. Also the characteristic polynomial of any maximal interval should be (x−1)(x−h)n−1 and the homology must be concentrated in maximal dimension. We prove that all these properties hold in type A and B. One can remark that the expected characteristic polynomial is the same as the characteristic polynomial of the hyperplane arrangement called the Shi arrangement [1, 7]. One difference is that there is no action of the Weyl group on the Shi arrangement. Going to the limit where parallel hyperplanes come together gives the so-called double Coxeter arrangement [12], which is no longer a hyperplane arrangement in the usual sense. Still the double Coxeter arrangement is free, and all its degrees are the Coxeter number. Maybe the pointed partition poset