A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four
Michael D. Hirschhorn
School of Mathematics and Statistics
James A. Sellers
Department of Mathematics
Penn State University
University Park, PA 16802
In 2001, Andrews and Lewis utilized an identity of F. H. Jackson to
derive some new partition generating functions as well as identities
involving some of the corresponding partition functions. In particular,
for 0 < a < b < k, they defined
W1(a, b; k; n) to be the number of
partitions of n in which the parts are congruent to a or
b mod k and such that, for any j, kj +
a and kj + b are not both parts. Our primary goal
in this note is to prove that W1(1,3;4;27n + 17) ≡
0 (mod 3) for all n ≥ 0. We prove this result using
elementary generating function manipulations and classic results from
the theory of partitions.
Full version: pdf,
(Concerned with sequences
Received June 29 2014; revised versions received September 2 2014; September 4 2014.
Published in Journal of Integer Sequences, September 4 2014.
Journal of Integer Sequences home page