Remarks on a certain restricted partition function of Lin
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918278043009024 |
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| author | Guadalupe, Russelle |
| author_facet | Guadalupe, Russelle |
| contents | Let $b(n)$ be the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ consists of distinct odd parts, and $π_2$ and $π_3$ consist of parts divisible by $4$. Utilizing modular forms, Lin obtained the generating functions for $b(3n+1)$ and $b(3n+2)$, which yields the congruence $b(3n+2)\equiv 0\pmod{3}$ for all $n\geq 0$. We provide in this note elementary proofs of these generating functions by employing $q$-series manipulations and dissection formulas. We also establish infinite families of internal congruences modulo $3$ for $b(n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on a certain restricted partition function of Lin Guadalupe, Russelle Number Theory Combinatorics 11P83, 05A17, 11P81 Let $b(n)$ be the number of partition triples $π=(π_1,π_2,π_3)$ of $n$ such that $π_1$ consists of distinct odd parts, and $π_2$ and $π_3$ consist of parts divisible by $4$. Utilizing modular forms, Lin obtained the generating functions for $b(3n+1)$ and $b(3n+2)$, which yields the congruence $b(3n+2)\equiv 0\pmod{3}$ for all $n\geq 0$. We provide in this note elementary proofs of these generating functions by employing $q$-series manipulations and dissection formulas. We also establish infinite families of internal congruences modulo $3$ for $b(n)$. |
| title | Remarks on a certain restricted partition function of Lin |
| topic | Number Theory Combinatorics 11P83, 05A17, 11P81 |
| url | https://arxiv.org/abs/2503.23996 |