Relations on words

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.