It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7/3. More precisely, there are only polynomially many binary words of length n that avoid 7/3-powers, but there are exponentially many binary words of length n that avoid (7/3+)-powers. This answers an open question of Kobayashi from 1986.
We introduce the notion of general prints of a word, which is substantialized by certain canonical decompositions, to study repetition in words. These associated decompositions, when applied recursively on a word, result in what we term as core prints of the word. The length of the path to attain a core print of a general word is scrutinized. This paper also studies the class of square-free ternary words with respect to the Parikh matrix mapping, which is an extension of the classical Parikh mapping. It is shown that there are only finitely many matrix-equivalence classes of ternary words such…
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.
We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word.
We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of pe…
We consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by Mousavi and Shallit at DLT 2013. Let $\texttt{pexp}_i(w)$ be the supremum of the exponent over the products of $i$ factors of the word $w$. The repetition threshold $\texttt{RT}_i(k)$ is then the infimum of $\texttt{pexp}_i(w)$ over all words $w\inΣ^ω_k$. Mousavi and Shallit obtained that $\texttt{RT}_i(2)=2i$ and $\texttt{RT}_2(3)=\tfrac{13}4$. We show that $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac14$ if $i$ is even and $\texttt{RT}_i(3)=\tfrac{3i}2+\tfrac16$ if $i$ is odd and $i\ge3$.
Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We address lower bounds on the repetition threshold of infinite rich words over 2 and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet $Σ_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.
For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{rep}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{rep}(x) = \mathrm{rep}(y)$ holds for a Sturmian word $x$ and every suffix $y$ of $x$. Let $c$ be the characteristic Sturmian word of slope $θ$. When $θ$ is a quadratic irrational, a formula of $\mathrm{rep}(c)$ is given. This formula shows that $\mathrm{rep}(c) = \mathrm{rep}(c')$ if $c'$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $θ$.
By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x) \leq r_{\max} = \sqrt{10} - 3/2$ if $x$ is a Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulation point of the set of ${\rm rep}(x)$ when $x$ runs over the Sturmian words.
A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a graph coloring is called $long$-$repetition$-$free$ if the word formed by the coloring of each path in the graph is long-square-free. Our results show that every rooted tree of radius less than or equal to seven is long-repetition-free two-colorable. We also prove there exists a class of trees which are not long-repetition-free two-colorable.
We consider two games between two players Ann and Ben who build a word together by adding alternatively a letter at the end of the shared word. In the nonrepetitive game, Ben wins the game if he can create a square of length at least $4$, and Ann wins if she can build an arbitrarily long word before that. In the erase-repetition game, whenever a square occurs the second part of the square is erased and the goal of Ann is still to build an arbitrarily large word (Ben simply wants to limit the size of the word in this game). Grytczuk, Kozik, and Micek showed that Ann has a winning strategy for t…
We consider the following novel variation on a classical avoidance problem from combinatorics on words: instead of avoiding repetitions in all factors of a word, we avoid repetitions in all factors where each individual factor is considered as a "circular word", i.e., the end of the word wraps around to the beginning. We determine the best possible avoidance exponent for alphabet size 2 and 3, and provide a lower bound for larger alphabets.
The repetition threshold is the smallest real number $α$ such that there exists an infinite word over a $k$-letter alphabet that avoids repetition of exponent strictly greater than $α$. This notion can be generalized to graph classes. In this paper, we completely determine the repetition thresholds for caterpillars and caterpillars of maximum degree $3$. Additionally, we present bounds for the repetition thresholds of trees with bounded maximum degrees.
The repetition threshold for words on $n$ letters, denoted $\mbox{RT}(n)$, is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted $\mbox{CRT}_{\mbox{W}}(n)$, $\mbox{CRT}_{\mbox{I}}(n)$, and $\mbox{CRT}_{\mbox{S}}(n)$, respectively. Currie and the present authors conjectured that $\mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n…
In this paper, we analyze the periodic factors of Sturmian words for the findings to lead to a linear-time algorithm for the computation of runs in this class of words which, to our best knowledge, is an open problem in literature.
A word is called $β$-free if it has no factors of exponent greater than or equal to $β$. The repetition threshold $\mathrm{RT}(k)$ is the infimum of the set of all $β$ such that there are arbitrarily long $k$-ary $β$-free words (or equivalently, there are $k$-ary $β$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $β$ such that - there are arbitrarily long $k$-ary $β$-free circular words is called…
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $α$. vAs an analogue of the critical exponent, we introduce the abelian critical …
For a rational number $r$ such that $1<r\leq 2$, an undirected $r$-power is a word of the form $xyx'$, where the word $x$ is nonempty, the word $x'$ is in $\{x,x^R\}$, and we have $|xyx'|/|xy|=r$. The undirected repetition threshold for $k$ letters, denoted $\mbox{URT}(k)$, is the infimum of the set of all $r$ such that undirected $r$-powers are avoidable on $k$ letters. We first demonstrate that $\mbox{URT}(3)=\tfrac{7}{4}$. Then we show that $\mbox{URT}(k)\geq \tfrac{k-1}{k-2}$ for all $k\geq 4$. We conjecture that $\mbox{URT}(k)=\tfrac{k-1}{k-2}$ for all $k\geq 4$, and we confirm this conje…
Ochem, Rampersad, and Shallit gave various examples of infinite words avoiding what they called approximate repetitions. An approximate repetition is a factor of the form xx', where x and x' are close to being identical. In their work, they measured the similarity of x and x' using either the Hamming distance or the edit distance. In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence. Our principal technique is the so-called "entropy compression" method, w…
For an infinite word $\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\text{rep}(\mathbf{x})$ which is called the exponent of repetition of $\mathbf{x}$. They showed $1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$ for any Sturmian word $\mathbf{x}$. Ohnaka and Watanabe found a gap in the set of the exponents of repetition of Sturmian words. For an irrational number $θ\in(0,1)$, let \[ \mathscr{L}(θ):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $θ$}\}.\] In this article, we look into $\mathscr{L}(θ)$. The minimum of $\mathscr{L}(θ)$ is d…
In this paper, we study the critical exponent of infinite words $\ubeta$ coding $β$-integers for $β$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the infinite word in question. We calculate also the ultimate critical exponent that expresses how long repetitions occur in the infinite word $\ubeta$ when the factors of length growing ad infinitum are considered. The basic ingredients of our method are the description of all bispecial factors of $\ubeta$ and the notion of return words. This method can be applied to any…
A word w is rich if it has |w|+1 many distinct palindromic factors, including the empty word. A word is square-free if it does not have a factor uu, where u is a non-empty word. Pelantová and Starosta (Discrete Math. 313 (2013)) proved that every infinite rich word contains a square. We will give another proof for that result. Pelantová and Starosta denoted by r(n) the length of a longest rich square-free word on an alphabet of size n. The exact value of r(n) was left as an open question. We will give an upper and a lower bound for r(n), and make a conjecture that our lower bound is exact. We …
We consider words $w$ over the alphabet $Σ=\{0,1,2\}$. It is shown that there are irreducibly square-free words of all lengths $n$ except 4,5,7 and 12. Such a word is square-free (i.e., it has no repetitions $uu$ as factors), but by removing any one internal letter creates a square in the word.
We prove various results about the largest exponent of a repetition in a factor of the Thue-Morse word, when that factor is considered as a circular word. Our results confirm and generalize previous results of Fitzpatrick and Aberkane & Currie.
We study words that barely avoid repetitions, for several senses of "barely". A squarefree (respectively, overlap-free, cubefree) word is irreducible if removing any one of its interior letters creates a square (respectively, overlap, cube). A squarefree (respectively, overlap-free, cubefree) word is delicate if changing any one of its letters creates a square (respectively, overlap, cube). We classify the lengths of irreducible and delicate squarefree, overlap-free, and cubefree words over binary and ternary alphabets.
Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several …
We introduce new avoidability problems for words by considering equivalence relations, k-abelian equivalences, which lie properly in between equality and commutative equality, i.e. abelian equality. For two k-abelian equivalent words the numbers of occurrences of different factors of length k coincide and the prefixies (resp. suffixies) of length k-1 are equal as well. The size of the smallest alphabet avoiding 2-repetitions of words, i.e. squares, is three and for abelian squares it is four. It follows that for 2-abelian squares this size has to be three or four. Similarly, the size of the sm…
We study repetitions in infinite words coding exchange of three intervals with permutation (3,2,1), called 3iet words. The language of such words is determined by two parameters $\varepsilon,\ell$. We show that finiteness of the index of 3iet words is equivalent to boundedness of the coefficients of the continued fraction of $\varepsilon$. In this case we also give an upper and lower estimate on the index of the corresponding 3iet word.
Lyndon words have been largely investigated and showned to be a useful tool to prove interesting combinatorial properties of words. In this paper we state new properties of both Lyndon and inverse Lyndon factorizations of a word $w$, with the aim of exploring their use in some classical queries on $w$. The main property we prove is related to a classical query on words. We prove that there are relations between the length of the longest common extension (or longest common prefix) $lcp(x,y)$ of two different suffixes $x,y$ of a word $w$ and the maximum length $\mathcal{M}$ of two consecutive fa…
We are interested in the maximal number of distinct squares in a word. This problem was introduced by Fraenkel and Simpson, who presented a bound of 2n for a word of length n, and conjectured that the bound was less than n. Being that the problem is on repetitions, their solution relies on Fine and Wilf's Periodicity lemma. Ilie then refined their result and presented a bound of 2n-O(log n). Lam used an induction to get a bound of 95n/48. Deza, Franek and Thierry achieved a bound of 11n/6 through a combinatorial approach. Using the properties of the core of the interrupt, presented by Thierry,…
In this paper we introduce and study a family of complexity functions of infinite words indexed by $k \in \ints ^+ \cup {+\infty}.$ Let $k \in \ints ^+ \cup {+\infty}$ and $A$ be a finite non-empty set. Two finite words $u$ and $v$ in $A^*$ are said to be $k$-Abelian equivalent if for all $x\in A^*$ of length less than or equal to $k,$ the number of occurrences of $x$ in $u$ is equal to the number of occurrences of $x$ in $v.$ This defines a family of equivalence relations $\thicksim_k$ on $A^*,$ bridging the gap between the usual notion of Abelian equivalence (when $k=1$) and equality (when $…
If $x$ is a non-empty string then the repetition $xx$ is called a tandem repeat. Similarly, a tandem in a two dimensional array $X$ is a configuration consisting of a same primitive block $W$ that touch each other with one side or corner. In \cite{Apostolico:2000}, Apostolico and Brimkov have proved various bounds for the number of tandems in a two dimensional word of size $m \times n$. Of the two types of tandems considered therein, they also proved that, for one type, the number of occurrences in an $m \times n$ Fibonacci array attained the general upper bound, $\mathcal{O}(m^{2}n \hspace{0.…
