Counting Ordered Pairs

Abstract

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].