Combinatorial proof of a congruence for partitions into two sizes of part
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915397849055232 |
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| author | DeWitt, Eli R. Keith, William J. |
| author_facet | DeWitt, Eli R. Keith, William J. |
| contents | Previous work showed that, for $ν_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $ν_2(16n + 14) \equiv 0 \pmod{4}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial proof of a congruence for partitions into two sizes of part DeWitt, Eli R. Keith, William J. Combinatorics 05A17, 05A19, 11P81, 11P83 Previous work showed that, for $ν_2(n)$ the number of partitions of $n$ into exactly two part sizes, one has $ν_2(16n + 14) \equiv 0 \pmod{4}$. 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. |
| title | Combinatorial proof of a congruence for partitions into two sizes of part |
| topic | Combinatorics 05A17, 05A19, 11P81, 11P83 |
| url | https://arxiv.org/abs/2507.13566 |