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

Abstract

The previous paper 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 T λ 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 Fibonacci 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 f λ ( 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 lim n → ∞ f λ ( n ) = 0 , but that for strings of lengths n ⩽ 2 λ , 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 strings of scope 1, it was natural to focus the study of the substring complexity of two-pattern strings to those of scope 1. Though preserving the aperiodicity of the Sturmian strings, the generalization to two-pattern strings greatly relaxes the constrained substring complexity (the number of distinct substrings of the same length) of the Sturmian strings. We derive upper and lower bounds on C 1 ( k ) (the number of distinct substring of length k) of two-pattern strings of scope 1, and we show that it can be considerably greater than that of a Sturmian string. In fact, we describe circumstances in which lim k → ∞ ( C 1 ( k ) − k ) = ∞ .