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

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