679 research outputs found
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
On a generalization of Abelian equivalence and complexity of infinite words
In this paper we introduce and study a family of complexity functions of
infinite words indexed by k \in \ints ^+ \cup {+\infty}. Let k \in \ints ^+
\cup {+\infty} and be a finite non-empty set. Two finite words and
in are said to be -Abelian equivalent if for all of length
less than or equal to the number of occurrences of in is equal to
the number of occurrences of in This defines a family of equivalence
relations on bridging the gap between the usual notion of
Abelian equivalence (when ) and equality (when We show that
the number of -Abelian equivalence classes of words of length grows
polynomially, although the degree is exponential in Given an infinite word
\omega \in A^\nats, we consider the associated complexity function \mathcal
{P}^{(k)}_\omega :\nats \rightarrow \nats which counts the number of
-Abelian equivalence classes of factors of of length We show
that the complexity function is intimately linked with
periodicity. More precisely we define an auxiliary function q^k: \nats
\rightarrow \nats and show that if for
some k \in \ints ^+ \cup {+\infty} and the is ultimately
periodic. Moreover if is aperiodic, then if and only if is Sturmian. We also
study -Abelian complexity in connection with repetitions in words. Using
Szemer\'edi's theorem, we show that if has bounded -Abelian
complexity, then for every D\subset \nats with positive upper density and for
every positive integer there exists a -Abelian power occurring in
at some position $j\in D.
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
A Note on Efficient Computation of All Abelian Periods in a String
We derive a simple efficient algorithm for Abelian periods knowing all
Abelian squares in a string. An efficient algorithm for the latter problem was
given by Cummings and Smyth in 1997. By the way we show an alternative
algorithm for Abelian squares. We also obtain a linear time algorithm finding
all `long' Abelian periods. The aim of the paper is a (new) reduction of the
problem of all Abelian periods to that of (already solved) all Abelian squares
which provides new insight into both connected problems
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
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
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