Avoidability of long kk-abelian repetitions

We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\"akel\"a'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 M\"akel\"a. 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

All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words

We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f: ? ? ?, we construct an infinite binary word whose abelian exponents have limit superior growth rate f. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word

A powerful abelian square-free substitution over 4 letters

AbstractIn 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ4={a,b,c,d} turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g85 of Σ4∗. The size of this g85, i.e., |g85(abcd)|, is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ4 have been based on the structure of g85 and on the endomorphism g98 of Σ4∗ found in 2002.In this paper, a great many new a-2-free endomorphisms of Σ4∗ are reported. The sizes of these endomorphisms range from 4×102 to 4×115. Importantly, twelve of the new a-2-free endomorphisms, of modulus m=109, can be used to construct an a-2-free (commutatively functional) substitution σ109 of Σ4∗ with 12 image words for each letter.The properties of σ109 lead to a considerable improvement for the lower bound of the exponential growth of cn, i.e., of the number of a-2-free words over 4 letters of length n. It is obtained that cn>β−50βn with β=121/m≃1.02306. Originally, in 1998, Carpi established the exponential growth of cn by showing that cn>β−tβn with β=219/t=219/(853−85)≃1.000021, where t=853−85 is the modulus of the substitution that he constructs starting from g85

Relations on words

In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation. In the second part, we mainly focus on abelian equivalence, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced

45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function $f\colon \N \to \R$, we construct an infinite binary word whose abelian exponents have limit superior growth rate $f$. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word.</p

Avoiding abelian powers cyclically

We study a new notion of cyclic avoidance of abelian powers. A finite word $w$ avoids abelian $N$-powers cyclically if for each abelian $N$-power of period $m$ occurring in the infinite word $w^\omega$, we have $m \geq |w|$. Let $\mathcal{A}(k)$ be the least integer $N$ such that for all $n$ there exists a word of length $n$ over a $k$-letter alphabet that avoids abelian $N$-powers cyclically. Let $\mathcal{A}_\infty(k)$ be the least integer $N$ such that there exist arbitrarily long words over a $k$-letter alphabet that avoid abelian $N$-powers cyclically.We prove that $5 \leq \mathcal{A}(2) \leq 8$, $3 \leq \mathcal{A}(3) \leq 4$, $2 \leq \mathcal{A}(4) \leq 3$, and $\mathcal{A}(k) = 2$ for $k \geq 5$. Moreover, we show that $\mathcal{A}_\infty(2) = 4$, $\mathcal{A}_\infty(3) = 3$, and $\mathcal{A}_\infty(4) = 2$.</p

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast