High order congruences for $M$-ary partitions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917607561494528 |
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| author | Żmija, Błażej |
| author_facet | Żmija, Błażej |
| contents | For a sequence $M=(m_{i})_{i=0}^{\infty}$ of integers such that $m_{0}=1$, $m_{i}\geq 2$ for $i\geq 1$, let $p_{M}(n)$ denote the number of partitions of $n$ into parts of the form $m_{0}m_{1}\cdots m_{r}$. In this paper we show that for every positive integer $n$ the following congruence is true: \begin{align*} p_{M}(m_{1}m_{2}\cdots m_{r}n-1)\equiv 0\ \ \left({\rm mod}\ \prod_{t=2}^{r}\mathcal{M}(m_{t},t-1)\right), \end{align*} where $\mathcal{M}(m,r):=\frac{m}{\gcd\big(m,{\rm lcm} (1,\ldots ,r)\big)}$.
Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for $m$-ary partitions found by Andrews, Gupta, and Rødseth and Sellers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04495 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | High order congruences for $M$-ary partitions Żmija, Błażej Number Theory Combinatorics 11P83, 11P81, 05A15, 05A17 For a sequence $M=(m_{i})_{i=0}^{\infty}$ of integers such that $m_{0}=1$, $m_{i}\geq 2$ for $i\geq 1$, let $p_{M}(n)$ denote the number of partitions of $n$ into parts of the form $m_{0}m_{1}\cdots m_{r}$. In this paper we show that for every positive integer $n$ the following congruence is true: \begin{align*} p_{M}(m_{1}m_{2}\cdots m_{r}n-1)\equiv 0\ \ \left({\rm mod}\ \prod_{t=2}^{r}\mathcal{M}(m_{t},t-1)\right), \end{align*} where $\mathcal{M}(m,r):=\frac{m}{\gcd\big(m,{\rm lcm} (1,\ldots ,r)\big)}$. Our result answers a conjecture posed by Folsom, Homma, Ryu and Tong, and is a generalisation of the congruence relations for $m$-ary partitions found by Andrews, Gupta, and Rødseth and Sellers. |
| title | High order congruences for $M$-ary partitions |
| topic | Number Theory Combinatorics 11P83, 11P81, 05A15, 05A17 |
| url | https://arxiv.org/abs/2403.04495 |