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January 18, 2026International Journal of Number Theory0 citations

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

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EDEli R. DeWittWKWilliam J. Keith

Key Points

  • To provide a combinatorial proof of a congruence related to partitions into two sizes of parts and to refine divisibility into subclasses.
  • Developed a combinatorial argument for ν 2 (16n + 14) being congruent to 0 modulo 4.
  • Refined divisibility into finer subclasses of partitions.
  • Related partition counts to the divisor function.
  • Confirmed ν 2 (16n + 14) is congruent to 0 (mod 4).
  • Identified subclasses closely related to d(16n + 14).
  • Proposed a conjecture on a potential rank statistic.

Abstract

Previous work showed that, for ν 2 (n) the number of partitions of n into exactly two part sizes, one has ν 2 (16n + 14) 蠁 0 (mod 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

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Cite This Study

DeWitt et al. (2026) studied this question.

synapsesocial.com/papers/696c77d4eb60fb80d13960b9https://doi.org/10.1142/s1793042126500569
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