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

2025/07/17 by Eli R. DeWitt, DeWitt, Eli R., William J. Keith +1
Mathematics · #05A17 #05A19 #11P81 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2507.13566

openalex publication_date 2025/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \pmod4. 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.

Citations

• Restricted k-color partitions

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