Privileged Factors in the Thue-Morse Word - A Comparison of Privileged Words and Palindromes

TL;DR

This paper investigates the privileged complexity of the Thue-Morse word, providing a recursive formula, showing it is unbounded with large zero gaps, and exploring its relation to palindromes and rich words.

Contribution

It introduces a recursive formula for privileged complexity, demonstrating its unbounded nature and differences from palindromic complexity in the Thue-Morse word.

Findings

Privileged complexity is unbounded in the Thue-Morse word.

The function exhibits arbitrarily large gaps of zeros.

Privileged complexity differs significantly from palindromic complexity.

Abstract

In this paper we study the privileged complexity function of the Thue-Morse word. We prove a recursive formula describing this function, and using the formula we show that the function is unbounded and that the values of the function have arbitrarily large gaps of zeros. This demonstrates that the privileged complexity function of an infinite word can drastically differ from its palindromic complexity function, even though there are relations between these functions. Further we study the behavior of palindromes and privileged words in infinite words and the relation between rich words and privileged words.