Relations on words

TL;DR

This survey explores classical and modern concepts in combinatorics on words, focusing on relations like abelian equivalence and their refinements, providing a unified framework and new insights.

Contribution

It offers a comprehensive overview of combinatorial relations on words, introduces new refinements of abelian equivalence, and unifies various concepts under a common framework.

Findings

Introduction of new refinements of abelian equivalence

Unified framework for combinatorial relations on words

Analysis of Parikh matrices and M-equivalence

Abstract

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.