Congruences for a restricted m-ary partition function

We discuss a family of restricted m-ary partition functions bm,j(n), which is the number of m-ary partitions of n with at most i+j copies of the part mi allowed. We then use generating function dissections to prove the following family of congruences for k⩾2 and 1⩽t⩽m−k+1:bm,m−1(mk+tn+mk+t−1+mk+t−2+⋯+mk)≡0 (mod 2t−1k).