On the parity of partition functions

Let S denote a subset of the positive integers, and let p_S(n) be the associated partition function, that is, p_S(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then p_S(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by p_S(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.