On the arithmetic properties of partitions into parts simultaneously $4$-regular and $9$-distinct
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916785340547072 |
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| author | Nadji, Mohammed L. Ahmia, Moussa |
| author_facet | Nadji, Mohammed L. Ahmia, Moussa |
| contents | In 2017, Keith presented a comprehensive survey on integer partitions into parts that are simultaneously regular, distinct, and/or flat. Recently, the authors initiated a study of partitions into parts that are simultaneously regular and distinct, examining them from both arithmetic and combinatorial perspectives. In particular, several Ramanujan-like congruences were obtained for $\myRD^{(\ell, t)}(n)$, the number of partitions of $n$ into parts that are simultaneously $\ell$-regular and $t$-distinct (parts appearing fewer than $t$ times), for various pairs $(\ell, t)$. In this paper, we focus on the case $(\ell, t)=(4,9)$ and conduct a thorough investigation of the arithmetic properties of $\myRD^{(4, 9)}(n)$. We establish several infinite families of congruences modulo $4$, $6$, and $12$, along with a collection of Ramanujan-like congruences modulo $24$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07704 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the arithmetic properties of partitions into parts simultaneously $4$-regular and $9$-distinct Nadji, Mohammed L. Ahmia, Moussa Number Theory 11P83, 05A15, 05A17 In 2017, Keith presented a comprehensive survey on integer partitions into parts that are simultaneously regular, distinct, and/or flat. Recently, the authors initiated a study of partitions into parts that are simultaneously regular and distinct, examining them from both arithmetic and combinatorial perspectives. In particular, several Ramanujan-like congruences were obtained for $\myRD^{(\ell, t)}(n)$, the number of partitions of $n$ into parts that are simultaneously $\ell$-regular and $t$-distinct (parts appearing fewer than $t$ times), for various pairs $(\ell, t)$. In this paper, we focus on the case $(\ell, t)=(4,9)$ and conduct a thorough investigation of the arithmetic properties of $\myRD^{(4, 9)}(n)$. We establish several infinite families of congruences modulo $4$, $6$, and $12$, along with a collection of Ramanujan-like congruences modulo $24$. |
| title | On the arithmetic properties of partitions into parts simultaneously $4$-regular and $9$-distinct |
| topic | Number Theory 11P83, 05A15, 05A17 |
| url | https://arxiv.org/abs/2506.07704 |