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

Abstract In two papers published in Quaestiones Mathematicae , Munagi and Sellers considered the family of functions $$D_{k,t}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> that count the number of integer compositions of weight n in which parts not divisible by k can be of t kinds (subscripted $$0, 1, \dots ,t-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ). 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 $$D_{k,t}(n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> 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.