On a restricted m-ary partition function

We discuss a family of restricted m-ary partition functions bm,j(s)(n), which is the number of m-ary partitions of n with only 0,s,2s,…,(i+j)s copies of the part mi allowed, and obtain the following family of congruences for 1⩽l⩽m1/2, l+1⩽k⩽m1−l+1 and 1⩽t⩽m1−k+1:bm,lm−1(s)(mk+tn+lsmk+t−1+⋯+lsmk)≡0mod (l+1)t−1∑0⩽r⩽kr≠k−1 lr,where m1=m/(m,s), (m,s)