High order congruences for $M$-ary partitions
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.04495
- arXiv:
- arXiv:2403.04495
- Bibcode:
- 2024arXiv240304495Z
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 11P83;
- 11P81;
- 05A15;
- 05A17