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) D k , 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 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) D k , 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.