Partitions

Abstract

Abstract The general problem of additive arithmetic. In this and the next two chapters we shall be occupied with the additive theory of numbers. The general problem of the theory may be stated as follows. Suppose that Ao is a given system of integers. Thus Amight contain all the positive integers, or the squares, or the primes. We consider all possible representations of an arbitrary positive integer nin the form where smay be fixed or unrestricted, the amay or may not be necessarily different, and order may or may not be relevant, according to the particular problem considered. We denote by r(n)the number of such representations. Then what can we say about r(n)? For example, is r(n)always positive? Is there always at any rate one representation of every n? Partitions of numbers. We take first the case in which Ais the set 1, 2, 3, ...of all positive integers, sis unrestricted, repetitions are allowed, and order is irrelevant. This is the problem of ‘unrestricted partitions’. A partitionof a number nis a representation of nas the sum of any number of positive integral parts. Thus has 7 partitions.† The order of the parts is irrelevant, so that we may, when we please, suppose the parts to be arranged in descending order of magnitude. We denote by p(n)the number of partitions of n; thus p(5)= 7. We can represent a partition graphically by an array of dots or ‘nodes’ such as † We have, of course, to count the representation by one part only.