## On the cohomology of the poset of weighted partitions

**
Rafael Gonz\'alez D'Le\'on and Michelle L. Wachs**
**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 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 $\Pi_n$.

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