A combinatorial proof on partition function parity

Abstract

One of the most basic results concerning the number-theoretic properties of the partition function $p(n)$ is that $p(n)$ takes each value of parity infinitely often. This statement was first proved by Kolberg in 1959, and it was strengthened by Subbarao in 1966 to say that both $p(2n)$ and $p(2n+1)$ take each value of parity infinitely often. These results have received several other proofs, each relying to some extent on manipulating generating functions. We give a new, self-contained proof of Subbarao's result by constructing a series of bijections and involutions, along the way getting a more general theorem concerning the enumeration of a special subset of integer partitions.