High order congruences for $M$-ary partitions

Abstract

For a sequence $M=(m_i{)}_{i=0}^{\infty }$ of integers such that $m_0=1$, $m_i\ge 2$ for $i\ge 1$, 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 $\mathcal{M}(m,r):=\frac{m}{\gcd (m,\mathrm{lcm}(1,\dots ,r))}$. 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.