Rich square-free words

TL;DR

This paper investigates the properties of rich square-free words, providing bounds for their maximum length on n-letter alphabets, and explores related concepts like repetition thresholds in infinite words.

Contribution

It offers new bounds for the length of the longest rich square-free words and conjectures their exact value, advancing understanding of combinatorial properties of such words.

Findings

Provided upper and lower bounds for r(n)

Conjectured the exact value of r(n)

Extended the concept of repetition threshold

Abstract

A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantov\'a and Starosta (Discrete Math. 313 (2013)) proved that every infinite rich word contains a square. We will give another proof for that result. Pelantov\'a and Starosta denoted by r(n) the length of a longest rich square-free word on an alphabet of size n. The exact value of r(n) was left as an open question. We will give an upper and a lower bound for r(n), and make a conjecture that our lower bound is exact. We will also generalize the notion of repetition threshold for a limited class of infinite words. The repetition thresholds for episturmian and rich words are left as an open question.