Counting `-letter subwords in compositions

Let N be the set of all positive integers and let A be any ordered subset of N. Recently, Heubach and Mansour enumerated the number of compositions of nwithm parts inA that contain the subword τ exactly r times, where τ ∈ {111, 112, 221, 123}. Our aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an `-letter subword, and (2) to analyze the number of compositions of n with m parts that avoid an `-letter pattern, for given `. We use tools such as asymptotic analysis of generating functions leading to Gaussian asymptotic.