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

Previous work showed that, for [Formula: see text] the number of partitions of [Formula: see text] into exactly two part sizes, one has [Formula: see text]. The earlier proof required the technology of modular forms, and a combinatorial proof was desired. This paper provides the requested proof, in the process refining divisibility to finer subclasses. Some of these subclasses have counts closely related to the divisor function [Formula: see text], and we offer a conjecture on a potential rank statistic.