High order congruences for M-ary partitions

Abstract

AbstractFor a sequence $$M=(m_{i})_{i=0}^{\infty }$$ M = ( m i ) i = 0 ∞ of integers such that $$m_{0}=1$$ m 0 = 1 , $$m_{i}\ge 2$$ m i ≥ 2 for $$i\ge 1$$ i ≥ 1 , let $$p_{M}(n)$$ p M ( n ) denote the number of partitions of n into parts of the form $$m_{0}m_{1}\cdots m_{r}$$ m 0 m 1 ⋯ m r . In this paper we show that for every positive integer n the following congruence is true: $$\begin{aligned} p_{M}(m_{1}m_{2}\cdots m_{r}n-1)\equiv 0\ \ \left( \textrm{mod}\ \prod _{t=2}^{r}\mathcal {M}(m_{t},t-1)\right) , \end{aligned}$$ p M ( m 1 m 2 ⋯ m r n - 1 ) ≡ 0 mod ∏ t = 2 r M ( m t , t - 1 ) , where $$\mathcal {M}(m,r):=\frac{m}{\textrm{gcd}\big (m,\textrm{lcm}(1,\ldots ,r)\big )}$$ M ( m , r ) : = m gcd ( m , lcm ( 1 , … , 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.