Combinatorial proof of a congruence for partitions into two sizes of part
Combinatorics
2025-07-21 v1
Abstract
Previous work showed that, for the number of partitions of into exactly two part sizes, one has . 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 , and we offer a conjecture on a potential rank statistic.
Keywords
Cite
@article{arxiv.2507.13566,
title = {Combinatorial proof of a congruence for partitions into two sizes of part},
author = {Eli R. DeWitt and William J. Keith},
journal= {arXiv preprint arXiv:2507.13566},
year = {2025}
}