On the (co)homology of the Poset of Weighted Partitions

We consider the poset of weighted partitions Πn , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πn provide a generalization of the lattice Πn of partitions, which we show possesses many of the well-known properties of Πn. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set {1, 2, . . . , n} having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted Sn-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of Πn has a nice factorization analogous to that of Πn.