2014/01/12 by Daniel C. McDonald, McDonald, Daniel C.
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.1401.2659
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.