On the poset of weighted partitions

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

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^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 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\$.

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