Three more proofs of two congruences for Merca's partition function
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908716515721216 |
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| author | Zanello, Fabrizio |
| author_facet | Zanello, Fabrizio |
| contents | In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function $a(n)$, which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame $a(n)$ into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that $a(n)$ is odd precisely 25\% of the time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09833 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Three more proofs of two congruences for Merca's partition function Zanello, Fabrizio Combinatorics Commutative Algebra Number Theory Primary: 11P83, Secondary: 05A17, 11P84 In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function $a(n)$, which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame $a(n)$ into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that $a(n)$ is odd precisely 25\% of the time. |
| title | Three more proofs of two congruences for Merca's partition function |
| topic | Combinatorics Commutative Algebra Number Theory Primary: 11P83, Secondary: 05A17, 11P84 |
| url | https://arxiv.org/abs/2509.09833 |