High order congruences for M-ary partitions

For a sequence M= (m₈) ₈=₀^ of integers such that m₀=1, m₈ 2 for i 1, let p₌ (n) denote the number of partitions of n into parts of the form m₀m₁ mₑ. In this paper we show that for every positive integer n the following congruence is true: align* p₌ (m₁m₂ mₑn-1) 0\ \ (mod\ ₓ=₂^rM (mₓ, 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 Rdseth and Sellers.