Sets Represented as the Length-n Factors of a Word

Abstract

Let Σ denote a finite alphabet. We say that a subset of Σⁿ is representable if it occurs as the set of all length-n factors of a finite word. In this paper we consider the following problems: how many different subsets of Σⁿ are representable? If a subset is representable, how long a word do we need to represent it? How many such subsets are represented by words of length t? For the first problem, we give upper and lower bounds of the form $\alpha^{2^n}$ in the binary case, where α > 1 is a real number. For the second problem, we give a weak upper bound and some experimental data. For the third problem, we give a closed-form formula in the case where n ≤ t < 2n.