TY - JOUR

T1 - On the parity of partition functions

AU - Berndt, Bruce C.

AU - Yee, Ae Ja

AU - Zaharescu, Alexandru

N1 - Funding Information:
The authors are grateful to Scott Ahlgren and George Andrews for helpful comments. The research of the first author is partially supported by grant MDA904-02-1-0065 from the National Security Agency. The research of the second author is partially supported by a grant from the Number Theory Foundation.

PY - 2003/6

Y1 - 2003/6

N2 - 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.

AB - 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.

KW - Formal power series

KW - Generating functions

KW - Parity problems

KW - Partition functions

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U2 - 10.1142/S0129167X03001740

DO - 10.1142/S0129167X03001740

M3 - Article

AN - SCOPUS:0038505593

SN - 0129-167X

VL - 14

SP - 437

EP - 459

JO - International Journal of Mathematics

JF - International Journal of Mathematics

IS - 4

ER -