113 research outputs found
Binary Patterns in Binary Cube-Free Words: Avoidability and Growth
The avoidability of binary patterns by binary cube-free words is investigated
and the exact bound between unavoidable and avoidable patterns is found. All
avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the
growth rates of the avoiding languages are studied. All such languages, except
for the overlap-free language, are proved to have exponential growth. The exact
growth rates of languages avoiding minimal avoidable patterns are approximated
through computer-assisted upper bounds. Finally, a new example of a
pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons
Days 2012
Polynomial versus Exponential Growth in Repetition-Free Binary Words
It is known that the number of overlap-free binary words of length n grows
polynomially, while the number of cubefree binary words grows exponentially. We
show that the dividing line between polynomial and exponential growth is 7/3.
More precisely, there are only polynomially many binary words of length n that
avoid 7/3-powers, but there are exponentially many binary words of length n
that avoid (7/3+)-powers. This answers an open question of Kobayashi from 1986.Comment: 12 page
Avoiding Abelian powers in binary words with bounded Abelian complexity
The notion of Abelian complexity of infinite words was recently used by the
three last authors to investigate various Abelian properties of words. In
particular, using van der Waerden's theorem, they proved that if a word avoids
Abelian -powers for some integer , then its Abelian complexity is
unbounded. This suggests the following question: How frequently do Abelian
-powers occur in a word having bounded Abelian complexity? In particular,
does every uniformly recurrent word having bounded Abelian complexity begin in
an Abelian -power? While this is true for various classes of uniformly
recurrent words, including for example the class of all Sturmian words, in this
paper we show the existence of uniformly recurrent binary words, having bounded
Abelian complexity, which admit an infinite number of suffixes which do not
begin in an Abelian square. We also show that the shift orbit closure of any
infinite binary overlap-free word contains a word which avoids Abelian cubes in
the beginning. We also consider the effect of morphisms on Abelian complexity
and show that the morphic image of a word having bounded Abelian complexity has
bounded Abelian complexity. Finally, we give an open problem on avoidability of
Abelian squares in infinite binary words and show that it is equivalent to a
well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte
Binary words containing infinitely many overlaps
We characterize the squares occurring in infinite overlap-free binary words
and construct various alpha power-free binary words containing infinitely many
overlaps.Comment: 9 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
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
- …