Abelian Properties of Words (Extended abstract)

TL;DR

This paper explores abelian properties of words, analyzing their complexity and repetitions, with specific results on well-known infinite words and answering longstanding questions in the field.

Contribution

It introduces new findings on abelian complexity, including a class of words with constant complexity and the presence of abelian powers in words with bounded complexity.

Findings

Thue-Morse and Tribonacci words analyzed for abelian complexity

Constructed a class of words with constant abelian complexity of 3

Proved that words with bounded abelian complexity contain all abelian powers

Abstract

We say that two finite words $u$ and $v$ are abelian equivalent if and only if they have the same number of occurrences of each letter, or equivalently if they define the same Parikh vector. In this paper we investigate various abelian properties of words including abelian complexity, and abelian powers. We study the abelian complexity of the Thue-Morse word and the Tribonacci word, and answer an old question of G. Rauzy by exhibiting a class of words whose abelian complexity is everywhere equal to 3. We also investigate abelian repetitions in words and show that any infinite word with bounded abelian complexity contains abelian $k$-powers for every positive integer $k$.