On the poset of weighted partitions

In this extended abstract we consider the poset of weighted partitions ∏^w_n , introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of ∏^w_n provide a generalization of the lattice ∏_n of partitions, which we show possesses many of the well-known properties of ∏_n. In particular, we prove these intervals are EL-shellable, we compute the M̈obius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted O_n-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 ∏^w_nhas a nice factorization analogous to that of ∏_n.