Two-pattern strings II - frequency of occurrence and substring complexity

The two previous papers in this series introduced a class of infinite binary strings, called two-pattern strings, that constitute a significant generalization of, and include, the much-studied Sturmian strings. The class of two-pattern strings is a union of a sequence of increasing (with respect to inclusion) subclasses TPλ of two-pattern strings of scope λ, λ = 1, 2, · · · . Prefixes of two-pattern strings are interesting from the algorithmic point of view (their recognition, generation, and computation of repetitions and near-repetitions) and since they include prefixes of the Fibonnaci and the Sturmian strings, they merit investigation of how many finite two-pattern strings of a given size there are among all binary strings of the same length. In this paper we first consider the frequency fλ(n) of occurrence of two-pattern strings of length n and scope λ among all strings of length n on {a, b}: we show that limn→∞ fλ(n) = 0, but that for strings of lengths n ≤ 2λ, two-pattern strings of scope λ constitute more than one-quarter of all strings. Since the class of Sturmian strings is a subset of two-pattern ∗also at School of Computing, Curtin University, Perth WA 6845, Australia, and Department of Computer Science, King’s College London.