Extending Arithmetic Properties for Compositions Wherein Parts That Are Non-Multiples of k Can Be of t Kinds

In two papers published in Quaestiones Mathematicae, Munagi and Sellers considered the family of functions Dk,t(n) that count the number of integer compositions of weight n in which parts not divisible by k can be of t kinds (subscripted 0,1,⋯,t-1). These are also closely related to “inplace” integer compositions of weight n. In their second paper, Munagi and Sellers proved a few divisibility properties satisfied by Dk,t(n) for specific values of k, t, and n. In this work, we significantly extend these arithmetic properties, providing infinitely families of congruences. Our proof techniques are quite elementary, relying on the structure of the generating functions in question.