Related papers: Circularly squarefree words and unbordered conjuga…

In 2008, Kurosaki gave a new construction of a (bi-)infinite squarefree word over three letters. We show that in fact Kurosaki's word avoids 7/4+-powers, which, as shown by Dejean, is optimal over a 3-letter alphabet.

We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of…

We characterize exactly the lengths of binary circular words containing no squares other than 00, 11, and 0101. Key words: combinatorics on words, circular words, necklaces, square-free words, non-repetitive sequences

Improved upper and lower bounds on the number of square-free ternary words are obtained. The upper bound is based on the enumeration of square-free ternary words up to length 110. The lower bound is derived by constructing generalised…

A square-free word $w$ over a fixed alphabet $\Sigma$ is extremal if every word obtained from $w$ by inserting a single letter from $\Sigma$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal…

In 2007, Grytczuk conjecture that for any sequence $(\ell_i)_{i\ge1}$ of alphabets of size $3$ there exists a square-free infinite word $w$ such that for all $i$, the $i$-th letter of $w$ belongs to $\ell_i$. The result of Thue of 1906…

A word is square-free if it does not contain any square (a word of the form $XX$), and is extremal square-free if it cannot be extended to a new square-free word by inserting a single letter at any position. Grytczuk, Kordulewski, and…

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained…

In this paper we answer two recent questions from Charlier et al. and Harju about self-shuffling words. An infinite word $w$ is called self-shuffling, if $w=\prod_{i=0}^\infty U_iV_i=\prod_{i=0}^\infty U_i=\prod_{i=0}^\infty V_i$ for some…

A non-empty word $w$ is a border of the word $u$ if $\vert w\vert<\vert u\vert$ and $w$ is both a prefix and a suffix of $u$. A word $u$ with the border $w$ is closed if $u$ has exactly two occurrences of $w$. A word $u$ is privileged if…

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo $p$ and $q$ are also square-free for relatively prime integers $p$ and $q$. Among such pairs $(p, q)$ with $p, q…

A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and…

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of…

A word is \emph{square-free} if it does not contain non-empty factors of the form $XX$. In 1906 Thue proved that there exist arbitrarily long square-free words over $3$-letter alphabet. We consider a new type of square-free words. A…

We prove that for any sequence of binary alphabets $\mathcal{A}_1,\mathcal{A}_2,\dots$, there exists a cube-free word $c_1c_2\dots$ so that $c_1\in\mathcal{A}_1,c_2\in\mathcal{A}_2,\dots$. In particular, for every $n$, there are at least…

A \emph{square} is a finite non-empty word consisting of two identical adjacent blocks. A word is \emph{square-free} if it does not contain a square as a factor. In any finite word one may delete the repeated block of a square, obtaining…

This paper begins with a comprehensive overview of combinatorics on words and symbolic dynamics, covering their historical origins, fundamental concepts, and interconnections. Building upon this foundation, we introduce novel mathematical…

We present a new recursive generation algorithm for prefix normal words. These are binary strings with the property that no substring has more 1s than the prefix of the same length. The new algorithm uses two operations on binary strings,…

In this paper we study the enumeration and the construction, according to the number of ones, of particular binary words avoiding a fixed pattern. The growth of such words can be described by particular jumping and marked succession rules.…

Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give…