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On the (co)homology of the poset of weighted partitions
We consider the poset of weighted partitions , introduced by
Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The
maximal intervals of provide a generalization of the lattice
of partitions, which we show possesses many of the well-known properties of
. 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 having a fixed number
of descents, we find combinatorial bases for homology and cohomology, and we
give an explicit sign twisted -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
has a nice factorization analogous to that of .Comment: 50 pages, final version, to appear in Trans. AM
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