27 research outputs found
Transition Property For Cube-Free Words
We study cube-free words over arbitrary non-unary finite alphabets and prove
the following structural property: for every pair of -ary cube-free
words, if can be infinitely extended to the right and can be infinitely
extended to the left respecting the cube-freeness property, then there exists a
"transition" word over the same alphabet such that is cube free. The
crucial case is the case of the binary alphabet, analyzed in the central part
of the paper.
The obtained "transition property", together with the developed technique,
allowed us to solve cube-free versions of three old open problems by Restivo
and Salemi. Besides, it has some further implications for combinatorics on
words; e.g., it implies the existence of infinite cube-free words of very big
subword (factor) complexity.Comment: 14 pages, 5 figure
Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x^2=x^3
In this paper we investigate the word problem of the free Burnside semigroup
satisfying x^2=x^3 and having two generators. Elements of this semigroup are
classes of equivalent words. A natural way to solve the word problem is to
select a unique "canonical" representative for each equivalence class. We prove
that overlap-free words and so-called almost overlap-free words (this notion is
some generalization of the notion of overlap-free words) can serve as canonical
representatives for corresponding equivalence classes. We show that such a word
in a given class, if any, can be efficiently found. As a result, we construct a
linear-time algorithm that partially solves the word problem for the semigroup
under consideration.Comment: 33 pages, submitted to Internat. J. of Algebra and Compu
Crucial words for abelian powers
A word is "crucial" with respect to a given set of "prohibited words" (or
simply "prohibitions") if it avoids the prohibitions but it cannot be extended
to the right by any letter of its alphabet without creating a prohibition. A
"minimal crucial word" is a crucial word of the shortest length. A word W
contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k
where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with
"abelian squares" and "abelian cubes", respectively.
In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal
crucial word over an n-letter alphabet A_n = {1,2,..., n} avoiding abelian
squares has length 4n-7 for n >= 3. In this paper we show that a minimal
crucial word over A_n avoiding abelian cubes has length 9n-13 for n >= 5, and
it has length 2, 5, 11, and 20 for n=1, 2, 3, and 4, respectively. Moreover,
for n >= 4 and k >= 2, we give a construction of length k^2(n-1)-k-1 of a
crucial word over A_n avoiding abelian k-th powers. This construction gives the
minimal length for k=2 and k=3. For k >= 4 and n >= 5, we provide a lower bound
for the length of crucial words over A_n avoiding abelian k-th powers.Comment: 14 page
Avoiding 2-binomial squares and cubes
Two finite words are 2-binomially equivalent if, for all words of
length at most 2, the number of occurrences of as a (scattered) subword of
is equal to the number of occurrences of in . This notion is a
refinement of the usual abelian equivalence. A 2-binomial square is a word
where and are 2-binomially equivalent.
In this paper, considering pure morphic words, we prove that 2-binomial
squares (resp. cubes) are avoidable over a 3-letter (resp. 2-letter) alphabet.
The sizes of the alphabets are optimal
The critical exponent of the Arshon words
Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov
(for n = 3), we prove that for all n greater than or equal to 2, the critical
exponent of the Arshon word of order is given by (3n-2)/(2n-2), and this
exponent is attained at position 1.Comment: 11 page
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