High order congruences for M-ary partitions

For a sequence M=(mᵢ)ᵢ₌₀∞ of integers such that m₀=1, mᵢ≥ 2 for i≥ 1, let pM(n) denote the number of partitions of n into parts of the form m₀m₁⋯ mᵣ. In this paper we show that for every positive integer n the following congruence is true: {align*} pM(m₁m₂⋯ mᵣn-1)≡ 0\ \ ({ mod}\ ∏ₜ₌₂ʳM(mₜ,t-1)), {align*} where M(m,r):=m(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 R{}dseth and Sellers.