### Counting ℓ-letter subwords in compositions

*Toufik Mansour, Basel Sirhan*

#### Abstract

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 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 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.Full Text: PDF PostScript