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

R. S. González D'León, M. L. Wachs

Trans. Amer. Math. Soc. 368 (2016) 6779-6818.

### 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 M"obius 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 thatthe characteristic polynomial of $$\Pi_n^w$$ has a nice factorization analogous to that of $$\Pi_n$$.