Introducing Privileged Words: Privileged Complexity of Sturmian Words

TL;DR

This paper explores privileged words, their properties, and their relation to Sturmian and other special words, providing new characterizations and insights into their complexity.

Contribution

It introduces the concept of privileged words, analyzes their properties, and offers a new characterization of Sturmian words based on privileged complexity.

Findings

Privileged words share properties with palindromes.

A new characterization of Sturmian words using privileged complexity.

Brief analysis of privileged complexity in the Thue-Morse word.

Abstract

In this paper we study the class of so-called privileged words which have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian words is given in terms of privileged complexity. The privileged complexity of the Thue-Morse word is also briefly studied.