On the (co)homology of the partition lattice and the free Lie algebra Original Research Article

We explore the well-known relationship between the representations of the symmetric group on the homology of the partition lattice and on the ln component of the free Lie algebra. We give two new combinatorial proofs of the sign twisted isomorphism between the two modules. One involves a bijection between bases and the other involves a bijection between generating sets which takes relations to relations.