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 less than or equal to N. New very general theorems axe obtained, and applications are made to several partition functions.