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 …

The interplay between words, computability, algebra and arithmetic has now proved its
relevance and fruitfulness. Indeed, the cross-fertilization between formal logic and finite …

We prove that a sequence satisfying a certain symmetry property is $2 $-regular in the sense
of Allouche and Shallit, ie, the $\mathbb {Z} $-module generated by its $2 $-kernel is finitely …

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 …

In combinatorics on words, a classical topic of study is the number of specific patterns
appearing in infinite sequences. For instance, many works have been dedicated to studying …

In a recent paper, one of us posed three open problems concerning squarefree arithmetic
progressions in infinite words. In this paper we solve these problems and prove some …

The Thue–Morse sequence is a prototypical automatic sequence found in diverse areas of
mathematics, and in computer science. We study occurrences of factors w within this …

Although a lot of research has been done on the factor complexity (also called subword
complexity) of morphic words obtained as fixed points of iterated morphisms, there has been …

In this paper we investigate local-to-global phenomena for a new family of complexity
functions of infinite words indexed by k≥ 0. Two finite words u and v are said to be k-abelian …

In this paper we investigate local-to-global phenomena for a new family of complexity
functions of infinite words indexed by k∈ ℕ 1∪+∞ where ℕ 1 denotes the set of positive …