Counting -letter subwords in compositions

Let be the set of all positive integers and let A be any ordered subset of . Recently, Heubach and Mansour enumerated the number of compositions of n with m parts in A 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.