A combinatorial proof on partition function parity (Q2926266)

For a positive integer \(n\), the partition function \(p(n)\) counts the number of partitions of \(n\) into positive integral parts; here a partition of \(n\) is a non-increasing list of positive integers that sum to \(n\). The folklore conjecture that asymptotically the number of \(n\leq x\) such that \(p(n)\) is even is \(\sim \frac{x}{2}\), is an important open question. The result of \textit{O. Kolberg} [Math. Scand. 7, 377--378 (1960; Zbl 0091.04402)] stating that \(p(n)\) assumes even and odd values infinitely often was later strengthened. For instance, a result of \textit{J. L. Nicolas} et al. [J. Number Theory 73, No. 2, 292--317 (1998; Zbl 0921.11050)] says that there is a constant \(C>0\) and a positive integer \(N_0\) such that for \(N>N_0\), there are at least \(CN^{1/2}\) integers \(n\leq N\) such that \(p(n)\) is even. There are similar results for the other parity. Here the author gives a new proof of a result of Kolberg [loc. cit.] and \textit{M. V. Subbarao} [Am. Math. Mon. 73, 851--854 (1966; Zbl 0173.01803)] that both \(p(2n)\) and \(p(2n+1)\) take each value of parity infinitely often. The authors provide a new proof of Subbarao's result [loc. cit.].