Combinatorial proof of a congruence for partitions into two sizes of part

Abstract

Previous work showed that, for ${\nu }_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has ${\nu }_2(16n+14)\equiv 0(\mathrm{mod}4)$. The earlier proof required the technology of modular forms, and a combinatorial proof was desired. This article provides the requested proof, in the process refining divisibility to finer subclasses. Some of these subclasses have counts closely related to the divisor function $d(16n+14)$, and we offer a conjecture on a potential rank statistic.