On the (co)homology of the poset of weighted partitions

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 ${\Pi }_n^w$ 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 show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on on node set $\{1,2,\dots ,n\}$ having a fixed number of descents, 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 ${\Pi }_n$.

arXiv comment50 pages, final version, to appear in Trans. AMS