Geodesic
Counting l-letter subwords in compositions
Discrete mathematics & theoretical computer science, Tome 8 (2006).

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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. 
@article{DMTCS_2006_8_a17,
     author = {Mansour, Toufik and Sirhan, Basel O.},
     title = {Counting l-letter subwords in compositions},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {8},
     year = {2006},
     doi = {10.46298/dmtcs.376},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.376/}
}
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Mansour, Toufik; Sirhan, Basel O. Counting l-letter subwords in compositions. Discrete mathematics & theoretical computer science, Tome 8 (2006). doi : 10.46298/dmtcs.376. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.376/

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