A combinatorial proof on partition function parity

TL;DR

This paper presents a new, self-contained combinatorial proof that the partition function's parity takes both values infinitely often, using bijections and involutions, and extends to a broader class of partition enumeration.

Contribution

It introduces a novel combinatorial proof technique for parity results of the partition function, avoiding generating function manipulations and generalizing previous theorems.

Findings

Proves parity of p(n) occurs infinitely often for all n

Constructs explicit bijections and involutions for partition enumeration

Extends parity results to a broader class of integer partitions

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.