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

González D’León, Rafael; Wachs, Michelle

doi:10.1090/tran/6483

On the (co)homology of the poset of weighted partitions
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by Rafael S. González D’León and Michelle L. Wachs PDF

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

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 Möbius 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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