Parity of the partition function p(n,k)

Abstract

Let [Formula: see text] denote the number of partitions of [Formula: see text] into parts less than or equal to [Formula: see text]. We show several properties of this function modulo 2. First, we prove that for fixed positive integers [Formula: see text] and [Formula: see text], [Formula: see text] is periodic modulo [Formula: see text]. Using this, we are able to find lower and upper bounds for the number of odd values of the function for a fixed [Formula: see text].