Related papers: Abelian Primitive Words
In the algebraic theory of codes and formal languages, the set $Q$ of all primitive words over some alphabet $\zi $ has received special interest. With this survey article we give an overview about relevant research to this topic during the…
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…
We show that paper folding words contain arbitrarily large abelian powers.
Let Q be the set of primitive words over a finite alphabet with at least two symbols. We characterize a class of primitive words, Q_I, referred to as ins-robust primitive words, which remain primitive on insertion of any letter from the…
We say that two finite words $u$ and $v$ are abelian equivalent if and only if they have the same number of occurrences of each letter, or equivalently if they define the same Parikh vector. In this paper we investigate various abelian…
An abelian anti-power of order $k$ (or simply an abelian $k$-anti-power) is a concatenation of $k$ consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion…
The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian…
This work considers a natural generalization of primitivity with respect to a language. Given a language $L$, a nonempty word $w$ is said to be $L$-primitive if $w$ is not a proper power of any word in $L$. After ascertaining the number of…
We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.
Answering a recent question of Crochemore, we prove that the language of words that are not abelian squares is not context-free.
In this paper we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection.…
In this paper we study an abelian version of the notion of return word. Our main result is a new characterization of Sturmian words via abelian returns. Namely, we prove that a word is Sturmian if and only if each of its factors has two or…
We study the equality problem for infinite words obtained by iterating morphisms. In particular, we give a practical algorithm to decide whether or not two words generated by primitive morphisms are equal.
Return words constitute a powerful tool for studying symbolic dynamical systems. They may be regarded as a discrete analogue of the first return map in dynamical systems. In this paper we investigate two abelian variants of the notion of…
This article provides a reminder of some properties of primitive words and the morphisms that preserve them. Their proofs, which I have more or less revised, are included. This makes the article almost self-contained. I also contribute by…
In this paper we undertake the general study of the Abelian complexity of an infinite word on a finite alphabet. We investigate both similarities and differences between the Abelian complexity and the usual subword complexity. While the…
We use results on Dyck words and lattice paths to derive a formula for the exact number of binary words of a given length with a given minimal abelian border length, tightening a bound on that number from Christodoulakis et al. (Discrete…
This paper concerns the avoidability of abelian and additive powers in infinite rich words. In particular, we construct an infinite additive $5$-power-free rich word over $\{0,1\}$ and an infinite additive $4$-power-free rich word over…
Abelian complexity of a word $\mathbf{u}$ is a function that counts the number of pairwise non-abelian-equivalent factors of $\mathbf{u}$ of length $n$. We prove that for any $c$-balanced Parry word $\mathbf{u}$, the values of the abelian…
We derive an explicit formula for the Abelian complexity of infinite words associated with quadratic Parry numbers.