Counting ℓ-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 n with m parts in A that contain the subword τ exactly τ 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 ℓ-letter subword, and (2) to analyze the number of compositions of n with m parts that avoid an ℓ-letter pattern, for given ℓ. We use tools such as asymptotic analysis of generating functions leading to Gaussian asymptotic.