. N T ] 5 M ay 2 00 4 GRAPHS OF PARTITIONS AND RAMANUJAN ’ S τ - FUNCTION

The partition function p and Ramanujan’s τ -function are related by similar generating functions and consequently by similar recursions involving the sum-ofdivisors function σ(n). These recursions are formally similar to Newton’s symmetric function relations, and one of our goals in this paper is to point out that the GirardWaring solution of Newton’s relations yields a formula for τ(n) as a sum over the partitions of n. The terms of the sum are fractions; the numerators are multiples of various σ(k) and the denominators are zλ, a standard partition invariant. This opens the possibility of an approach to congruences for τ by considering congruences for the terms of this sum. This strategy will not be explored here, but it lies in the background as the reason for our interest in the sum-decomposition we describe. It will become apparent that analyzing the terms of the sum algebraically is an unattractive task; they are too intricate. On the other hand, zλ is a combinatorial object, so it is natural to hope that zλ counts something. Then, perhaps, congruences in zλ might be analyzed by counting methods, much as partition congruences are analyzed in terms of ranks and cranks. Our second aim in this paper is to interpret zλ as a counting function. The objects it counts are automorphisms of graphs representing partitions. The physicist C. M. Bender and his co-authors interpreted Bell numbers in terms of automorphisms of Feynman diagrams that represent partitions [B1,B2]. Our graphs are related to subgraphs of those Feynman diagrams.