Pattern Avoidability with Involution

An infinte word w avoids a pattern p with the involution t if there is no substitution for the variables in p and no involution t such that the resulting word is a factor of w. We investigate the avoidance of patterns with respect to the size of the alphabet. For example, it is shown that the pattern a t(a) a can be avoided over three letters but not two letters, whereas it is well known that a a a is avoidable over two letters.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Avoidability index for binary patterns with reversal

For every pattern $p$ over the alphabet $\{x,y,x^R,y^R\}$, we specify the least $k$ such that $p$ is $k$-avoidable.Comment: 15 pages, 1 figur

Growth rate of binary words avoiding xxxRxxx^R

Consider the set of those binary words with no non-empty factors of the form $xxx^R$. Du, Mousavi, Schaeffer, and Shallit asked whether this set of words grows polynomially or exponentially with length. In this paper, we demonstrate the existence of upper and lower bounds on the number of such words of length $n$, where each of these bounds is asymptotically equivalent to a (different) function of the form $Cn^{\lg n+c}$, where $C$, $c$ are constants

Unary patterns under permutations

Thue characterized completely the avoidability of unary patterns. Adding function variables gives a general setting capturing avoidance of powers, avoidance of patterns with palindromes, avoidance of powers under coding, and other questions of recent interest. Unary patterns with permutations have been previously analysed only for lengths up to 3. Consider a pattern $p=\pi_{i_1}(x)\ldots \pi_{i_r}(x)$, with $r\geq 4$, $x$ a word variable over an alphabet $\Sigma$ and $\pi_{i_j}$ function variables, to be replaced by morphic or antimorphic permutations of $\Sigma$. If $|\Sigma|\ge 3$, we show the existence of an infinite word avoiding all pattern instances having $|x|\geq 2$. If $|\Sigma|=3$ and all $\pi_{i_j}$ are powers of a single morphic or antimorphic $\pi$, the length restriction is removed. For the case when $\pi$ is morphic, the length dependency can be removed also for $|\Sigma|=4$, but not for $|\Sigma|=5$, as the pattern $x\pi^2(x)\pi^{56}(x)\pi^{33}(x)$ becomes unavoidable. Thus, in general, the restriction on $x$ cannot be removed, even for powers of morphic permutations. Moreover, we show that for every positive integer $n$ there exists $N$ and a pattern $\pi^{i_1}(x)\ldots \pi^{i_n}(x)$ which is unavoidable over all alphabets $\Sigma$ with at least $N$ letters and $\pi$ morphic or antimorphic permutation

Avoiding and Enforcing Repetitive Structures in Words

The focus of this thesis is on the study of repetitive structures in words, a central topic in the area of combinatorics on words. The results presented in the thesis at hand are meant to extend and enrich the existing theory concerning the appearance and absence of such structures. In the first part we examine whether these structures necessarily appear in infinite words over a finite alphabet. The repetitive structures we are concerned with involve functional dependencies between the parts that are repeated. In particular, we study avoidability questions of patterns whose repetitive structure is disguised by the application of a permutation. This novel setting exhibits the surprising behaviour that avoidable patterns may become unavoidable in larger alphabets. The second and major part of this thesis deals with equations on words that enforce a certain repetitive structure involving involutions in their solution set. Czeizler et al. (2009) introduced a generalised version of the classical equations u` Æ vmwn that were studied by Lyndon and Schützenberger. We solve the last two remaining and most challenging cases and thereby complete the classification of these equations in terms of the repetitive structures appearing in the admitted solutions. In the final part we investigate the influence of the shuffle operation on words avoiding ordinary repetitions. We construct finite and infinite square-free words that can be shuffled with themselves in a way that preserves squarefreeness. We also show that the repetitive structure obtained by shuffling a word with itself is avoidable in infinite words

On the aperiodic avoidability of binary patterns with variables and reversals

In this work we present a characterisation of the avoidability of all unary and binary patterns, that do not only contain variables but also reversals of their instances, with respect to aperiodic infinite words. These types of patterns were studied recently in either more general or particular cases

Theoretical and Practical Aspects Related to the Avoidability of Patterns in Words

This thesis concerns repetitive structures in words. More precisely, it contributes to studying appearance and absence of such repetitions in words. In the first and major part of this thesis, we study avoidability of unary patterns with permutations. The second part of this thesis deals with modeling and solving several avoidability problems as constraint satisfaction problems, using the framework of MiniZinc. Solving avoidability problems like the one mentioned in the past paragraph required, the construction, via a computer program, of a very long word that does not contain any word that matches a given pattern. This gave us the idea of using SAT solvers. Representing the problem-based SAT solvers seemed to be a standardised, and usually very optimised approach to formulate and solve the well-known avoidability problems like avoidability of formulas with reversal and avoidability of patterns in the abelian sense too. The final part is concerned with a variation on a classical avoidance problem from combinatorics on words. Considering the concatenation of i different factors of the word w, pexp_i(w) is the supremum of powers that can be constructed by concatenation of such factors, and RTi(k) is then the infimum of pexp_i(w). Again, by checking infinite ternary words that satisfy some properties, we calculate the value RT_i(3) for even and odd values of i

Testing Generalised Freeness of Words

Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding efficiently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variable x and its image under f, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect efficiently a factor of the form xx^Rxxx^R, or any other pattern of such type. We also address the problem of testing efficiently, in the same setting, whether the word w contains an arbitrary pseudo-repetition of a given exponent