On the 2-Abelian Complexity of the Thue-Morse Word

TL;DR

This paper investigates the 2-abelian complexity of the Thue-Morse word, revealing its 2-regularity, palindrome structure, and bounds on unique factor extensions, contributing to combinatorics on words.

Contribution

It establishes the 2-regularity of the 2-abelian complexity and describes its palindrome concatenation structure, providing new insights into the combinatorial properties of the Thue-Morse word.

Findings

2-abelian complexity of Thue-Morse is 2-regular

Complexity is a concatenation of palindromes of increasing length

Sharp bounds for unique factor extensions

Abstract

We show that the 2-abelian complexity of the infinite Thue-Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue-Morse word.