7 research outputs found
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
On Generating Binary Words Palindromically
We regard a finite word up to word isomorphism as an
equivalence relation on where is equivalent to if
and only if Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form for some
choice of and That is to say,
some finite words are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function defined as the least number of palindromic relations
required to generate We show that every aperiodic infinite word must
contain a factor with and that some infinite words have
the property that for each factor of We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function is unbounded
Existence of an infinite ternary 64-abelian square-free word
We consider a recently defined notion of k-abelian equivalence of words by
concentrating on avoidance problems. The equivalence class of a word depends on the
numbers of occurrences of different factors of length k for a fixed natural
number k and
the prefix of the word. We have shown earlier that over a ternary alphabet k-abelian squares cannot be
avoided in pure morphic words for any natural number k. Nevertheless,
computational experiments support the conjecture that even 3-abelian squares can be
avoided over ternary alphabets. In this paper we establish the first avoidance result
showing that by choosing k to be large enough we have an infinite
k-abelian
square-free word over three letter alphabet. In addition, this word can be obtained as a
morphic image of a pure morphic word
Problems in between words and abelian words: k-abelian avoidability
AbstractWe consider a recently defined notion of k-abelian equivalence of words in connection with avoidability problems. This equivalence relation, for a fixed natural number k, takes into account the numbers of occurrences of the different factors of length k and the prefix and the suffix of length k−1. We search for the smallest alphabet in which k-abelian squares and cubes can be avoided, respectively. For 2-abelian squares this is four–as in the case of abelian words, while for 2-abelian cubes we have only strong evidence that the size is two–as it is in the case of words. However, we are able to prove this optimal value only for 8-abelian cubes