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

R. S. González D'León, M. L. Wachs

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

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 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 thatthe characteristic polynomial of \(\Pi_n^w\) has a nice factorization analogous to that of \(\Pi_n\).