Abstract: There is a well-known relationship between the homology of the partition lattice and the free Lie algebra, namely that the representation of the symmetric group on the homology of the partition lattice is isomorphic to the representation of the symmetric group on the $1^n$ component of the free Lie algebra tensored with the sign representation. Barcelo combinatorially explained this result by comparing the representation matrices for a basis for homology due to Bj\"orner with the representation matrices for the Lyndon basis for the free Lie algebra. In this paper we further explain the result by giving an elegant $\Cal S_n$-module isomorphism from the cohomology of the partition lattice to the $1^n$ component of the free Lie algebra tensored with the sign representation. This isomorphism is described as a map from the natural defining generating set for one module to the natural defining generating set for the other module, rather than as a bijection between bases. The natural generating set for the cohomology module contains the basis dual to the Bj\"orner basis and the natural generating set for the free Lie algebra module contains the Lyndon basis. Moreover, our isomorphism is shown to map the dual Bj\"orner basis to the Lyndon basis thereby recovering Barcelo's result. The natural generating sets contain other known bases for cohomology and the free Lie algebra; and we show that our isomorphism maps the splitting basis for cohomology to the comb basis for the free Lie algebra. A new basis for homology is introduced and studied in this context.
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