Related papers: Cyclic Complexity of Words
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 find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation…
By complexity of a finite graph we mean the number of spanning trees in the graph. The aim of the present paper is to give a new approach for counting complexity $\tau(n)$ of cyclic $n$-fold coverings of a graph. We give an explicit…
A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended…
Given a finite alphabet $\Sigma$ and a right-infinite word $w$ over the alphabet $\Sigma$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work…
We say that a finite factor $f$ of a word $w$ is \emph{imaged} if there exists a non-erasing morphism $m$, distinct from the identity, such that $w$ contains $m(f)$. We show that every infinite word contains an imaged factor of length at…
For a finite alphabet $\mathcal{A}$ and a sequence $x \in \mathcal{A}^{\mathbb{N}}$, Kamae and Zamboni defined the maximal pattern complexity function $p^*_x(n)$ as a natural generalization of usual word complexity. They defined a…
For any infinite word $w$ on a finite alphabet $A$, the complexity function $p_w$ of $w$ is the sequence counting, for each non-negative $n$, the number $p_w(n)$ of words of length $n$ on the alphabet $A$ that are factors of the infinite…
We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic function is non-decreasing, submultiplicative and grows at least linearly. We prove that conversely, every…
We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are…
We define a family of natural decompositions of Sturmian words in Christoffel words, called *reversible Christoffel* (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always)…
We consider a new family of factorial languages whose subword complexity grows as $\Theta(n^{\alpha})$, where $\alpha$ is the root of some transcendent equation. Analytical methods and in particular, a corollary of the Wiener-Pitt theorem,…
Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this paper, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded…
A word~$w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word~$w$ is said to be \emph{closed} if $w$ is of length at most $1$ or if $w$ has a border that occurs exactly twice in $w$. A word~$w$ is said to be…
A celebrated theorem by Coven and Hedlund (1973) states that Sturmian words are characterized by their abelian complexity: they are precisely the infinite words with rationally independent letter frequencies and constant abelian complexity…
In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense"…
In the study of infinite words, various notions of balancedness provide quantitative measures for how regularly letters or factors occur, and they find applications in several areas of mathematics and theoretical computer science. In this…
In this paper we explore a new hierarchy of classes of languages and infinite words and its connection with complexity classes. Namely, we say that a language belongs to the class $L_k$ if it is a subset of the catenation of $k$ languages…
We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to…
A cylinder $C^1_u$ is the set of infinite words with fixed prefix $u$. A double-cylinder $C^2_{[1,u]}$ is "the same" for bi-infinite words. We show that for every word $u$ and any automorphism $\varphi$ of the free group $F$ the image…