On the poset of weighted partitions

R. S. González D'León, M. L. Wachs

DMTCS Proceedings 01 (2013) 1029-1040.
Extended abstract

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 $$\Pi_n$$.