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

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    We consider the poset of weighted partitions Πnw\Pi_n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Πnw\Pi_n^w provide a generalization of the lattice Πn\Pi_n of partitions, which we show possesses many of the well-known properties of Πn\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,…,n}\{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 Sn\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 Πnw\Pi_n^w has a nice factorization analogous to that of Πn\Pi_n.Comment: 50 pages, final version, to appear in Trans. AM
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