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

TL;DR

This paper provides a combinatorial proof for a divisibility congruence related to partitions into two sizes of parts, refining previous modular form-based results and exploring related divisor function properties.

Contribution

It offers a combinatorial proof of a known congruence for partitions into two part sizes, improving understanding and suggesting a new conjecture on rank statistics.

Findings

Proved a divisibility congruence combinatorially

Refined divisibility to subclasses related to divisor function

Proposed a conjecture on a potential rank statistic

Abstract

Previous work showed that, for $\nu_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $\nu_2(16n + 14) \equiv 0 \pmod{4}$. 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.