On the properties of a new classification of partitions into at most three parts

Let n be a positive integer, and let n = x + y + z, x greater than or equal to y greater than or equal to z greater than or equal to 0 be a partition of n into at most three positive parts; throughout the present paper the phase ''partition on n'' will tacitly embody this restriction. The number of such partitions is well known to be p(n, 3) = )(n + 2) (n + 4)/12), where the curly brackets in this formula (but not elsewhere in the paper) indicate nearest integer. Further, let m = x/sup 2/ + y/sup 2/ + z/sup 2/. Fixing n, we run through all p(n, 3) partitions of n, grouping together those which have the same m value. In other words, we impose a structure on the set of partitions by defining equivalence classes C/sub m/ labeled by the value of the sum of squares of the parts. Any given class will consist of some k-tuple of partitions, k = 1, 2,... Let ..mu../sub i/ be the number of distinct i-tuples for given n, so that ..sigma.. r/sub i=2/ i..mu../sub i/ = p(n, 3); here r = r(n) is the maximum number of partitions in any of the classes C/sub m/; we shall call r(n) the rank of the integer n. In the sequel we investigate the properties of this new structure as n increases without limit. A complete characterization of the classification is given in terms of a suitably restricted set of solutions of a well-known binary diophantine equation.