On the Parity of Partition Functions

Abstract

Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is, pS(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 pS(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 pS(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

New theorems on the parity of partition functions

“…1 Research partially supported by grant MDA904-02-1-0065 from the National Security Agency. 2 Research partially supported by a grant from the Number Theory Foundation. 3 Mathematics Subject Classification 2000: Primary, 11P76; Secondary, 11P83.…”

“…The most significant quantitative work on the parity of p(n) in arithmetic progressions has been accomplished by K. Ono [11], [12] and Ahlgren [1] via the theory of modular forms. More details can be found in our first paper [2] and, of course, in the aforementioned authors' papers.…”

“…In [2], the authors developed new methods, which are simple and apply to more general partition functions. In particular, explicit results were proved in [2] on the parity of several classes of partition functions, such as: the number p(r, s; n) of partitions of n into parts congruent to r, s, or r + s modulo r + s; the number c 3 (n) of partitions of n into three colors; the number of partitions of n into parts which are coprime to a given fixed integer b; and the number of partitions of n into parts which are square-free and coprime to a fixed integer b.…”

“…In the present paper we use new ideas, which allow us to develop the theory in a more systematic way, and this leads in turn to vast generalizations of many of the results from [2].…”

“…As in [2], all our work is effected in the ring A of formal power series in one variable over the field of two elements Z/2Z. A key ingredient in the present work is a certain differential equation in A, which is studied in Section 3 below.…”