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

Document Type

Article

Publication Date

1-1-2026

Abstract

Previous work showed that, for ν2(n) the number of partitions of n into exactly two part sizes, one has ν2(16n + 14) ≡ 0(mod 4). 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 d(16n + 14), and we offer a conjecture on a potential rank statistic.

Publication Title

International Journal of Number Theory

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