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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This paper deals with periodic words of the form W0k over an alphabet A of cardinality m, where w0 is fixed and contains p ≤ m different letters (p is also fixed). Let kn be a sequence of positive integers. It is shown that if lim supn→∞ pkn/ln n m then almost all words of length n over A contain the factor w0kn as n → ∞, but if lim supn→ ∞ pkn/ln n 1/ln m then this property is not longer true. Also, if lim infn→∞ pkn/ln n 1/ln m then almost all words of length n over A do not contain the factor w0kn. Moreover, if there exists limn→∞(ln n - pkn ln m)= δ ∈ R, then the proportion of words of length n containing the factor w0kn approaches 1-exp(-(1 - 1/mp) exp(δ)) as n → ∞.