Plethysm, Partitions with an Even Number of Blocks 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 combinatorialists. 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 conjectures themselves are completely elementary, 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 André 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 representations of posets of partitions are 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 definition for the class of numbers bi(n) (Definition 1.2) was discovered by performing plethystic manipulations on a recurrence (Theorem 4.8) for the Frobenius characteristic of the homology representation. In an effort 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. 1991 Mathematics Subject Classification. Primary 05E25; Secondary 20C30.