Arithmetic properties of $5$-regular partitions into distinct parts
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913571929063424 |
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| author | Baruah, Nayandeep Deka Sarma, Abhishek |
| author_facet | Baruah, Nayandeep Deka Sarma, Abhishek |
| contents | A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02978 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetic properties of $5$-regular partitions into distinct parts Baruah, Nayandeep Deka Sarma, Abhishek Number Theory 05A17, 11P81, 11P83, 11B37 A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5. |
| title | Arithmetic properties of $5$-regular partitions into distinct parts |
| topic | Number Theory 05A17, 11P81, 11P83, 11B37 |
| url | https://arxiv.org/abs/2411.02978 |