Cumulants and Partition Lattices

This is the first paper to appear in the statistical literature pointing out the importance of the partition lattice in the theory of statistical moments and their close cousins, the cumulants. The paper was first brought to my attention by Susan Wilson, shortly after I had given a talk at Imperial College on the Leonov-Shiryaev result expressed in graph-theoretic terms. Speed’s paper was hot off the press, arriving a day or two after I had first become acquainted with the partition lattice from conversations with Oliver Pretzel. Naturally, I read the paper with more than usual attention to detail because I was still unfamiliar with Rota [18], and because it was immediately clear that Mobius inversion on the partition lattice \({\mathcal{E}}_{n}\), partially ordered by sub-partition, led to clear proofs and great simplification. It was a short paper packing a big punch, and for me it could not have arrived at a more opportune moment.