Enumerative combinatorics
Generating functionology
On words containing all short subwords
Theoretical Computer Science
Information and Randomness: An Algorithmic Perspective
Information and Randomness: An Algorithmic Perspective
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Let A be an alphabet of cardinality m, k$_n$ be a sequence of positive integers and ω∈ A* (|ω|=k$_n$). In this paper it is shown that if lim sup$_{n→∞}$k$_n$/ln n1/ln m, then this property is not true. Also, if lim inf$_{n→∞}$k$_n$/ln n1/ln m, then almost all words of length n over A do not contain the factor ω. Moreover, if lim$_{n→∞}$(ln n-k$_n$ln m)=α∈ R, then lim sup$_{n→∞}$|W(n,k$_n$,ω,A)| /m$^m$≤1−exp (−exp(α)) and lim inf$_{n→∞}$|W (n, k$_n$,ω,A)|/m$^n$≥1−exp (−(1−1/m) exp(α)), where W(n, k$_n$, ω, A) denotes the set of words of length n over A containing the factor ω of length k$_n$.