Extended abstract

In this extended abstract we consider the poset of weighted partitions \(\Pi_n^w\), introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of \(\Pi_n\) provide a generalization of the lattice \(\Pi_n\) of partitions, which we show possesses many of the well-known properties of \(\Pi_n\). In particular, we prove these intervals are EL-shellable, we compute the M"obius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted \(\mathfrak{S}_n\)-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 \(\Pi_n^w\) has a nice factorization analogous to that of \(\Pi_n\).