Recurrent words with constant Abelian complexity

TL;DR

This paper proves that it is impossible to have recurrent words with constant Abelian complexity when containing four or more distinct letters, resolving a previously open question in combinatorics on words.

Contribution

It establishes the non-existence of such words, providing a definitive answer to an open problem in the field.

Findings

Recurrent words with constant Abelian complexity cannot have four or more distinct letters.

The proof resolves an open question posed by Richomme et al.

The result narrows the understanding of Abelian complexity in combinatorics.

Abstract

We prove the non-existence of recurrent words with constant Abelian complexity containing 4 or more distinct letters. This answers a question of Richomme et al.