The Enumeration Methods of Redfield

Abstract

In 1927, Redfield [13] wrote a remarkable paper on combinatorial theory in which he anticipated several enumeration results [4, 11, 12, 14]. We first became acquainted with this paper while studying the book on group characters by Littlewood [10], which references paper. This is the only paper Redfield ever wrote. lRedfield's theorem was studied by Foulkes [3] using group characters to formulate the result. In a footnote, Foulkes mentioned some of enumeration results. Our object in this expository paper is to give Redfield proper credit for his combinatorial discoveries, while making his methods more available by recasting them in contemporary notation and terminology [6, 7, 8, 9]. In particular, we display explicitly the basic group-theoretic result, which we call Redfield's Lemma, a generalization of the well-known lemma of Burnside [1]. 1. Permutation groups. Let A be any permutation group of order in I AI acting on the set X of d objects. Let jk(a) be the number of eveles of length ic in the disjoint cycle decomposition of a permutation a. Let us denote d variables by a1,a2, , ad. Then the cycle indeX 2 Z (A) is the following formal sum, a polynomial in these variables: 1 d (I)~~~~~ Z(A ) =n -I a7, 1(a 1)m EA k=1