On the Number of Closed Factors in a Word

TL;DR

This paper studies the structure and quantity of closed factors in words, establishing lower bounds, characterizations, and the potential for quadratic growth in the number of such factors.

Contribution

It provides a lower bound of n+1 closed factors in words of length n, characterizes words with minimal closed factors, and shows the potential for quadratic growth in the number of closed factors.

Findings

A word of length n has at least n+1 distinct closed factors.

Words with exactly n+1 closed factors are characterized.

A word can contain on the order of n^2 closed factors.

Abstract

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.