A congruence modulo 3 for partitions into distinct non-multiples of four

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.