Partitions of Even Rank and Euler Numbers

This paper is based on an hour address given at the Sixth Conference on Formal Power Series and Algebraic Combinatorics, held at DIMACS in 1994. It is written primarily for an audience of combinato-rialists. Our hope is to publicise some intriguing enumerative conjectures which arise in the study of the homology representations of the poset of (set) partitions with an even number of blocks. The conjecturesthemselvesare completelyelementary, and can be stated without reference to the representation-theoretic context in which they arose. These conjectures seem to have connections to the theory of Andr e permutations, which is currently enjoying a renewed attention in the literature , and questions concerning the cd-index, another topic of interest in recent research. Homology representationsof posets of partitionsare more elegantly computed by exploiting the machinery of symmetric functions, and in particular the role of the plethysm operation in describing wreath product representations of the symmetric group. The somewhat intricate deenition for the class of numbers b i (n) (Deenition 1.2) was discovered by performing plethystic manipulations on a recurrence (Theorem 4.8) for the Frobenius characteristic of the homology representation. In an eeort to dispel some of the mystery surrounding plethysm, and to explain its connection with representations of the symmetric group, we give a careful discussion of wreath product modules, and a complete analysis of the relationship between the plethysm operation and wreath product modules for the symmetric groups.