Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.