Abelian Combinatorics on Words: a Survey

TL;DR

This survey reviews the development of abelian combinatorics on words, focusing on abelian equivalence and its applications, highlighting recent research and open problems in the field.

Contribution

It compiles and discusses known results and open questions in abelian combinatorics on words, emphasizing abelian equivalence and its role in the theory.

Findings

Extensive results on abelian equivalence and Parikh vectors.

Identification of open problems in abelian combinatorics.

Recent advances in abelian analogues of classical concepts.

Abstract

We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.