145 research outputs found
On the Structure of Bispecial Sturmian Words
A balanced word is one in which any two factors of the same length contain
the same number of each letter of the alphabet up to one. Finite binary
balanced words are called Sturmian words. A Sturmian word is bispecial if it
can be extended to the left and to the right with both letters remaining a
Sturmian word. There is a deep relation between bispecial Sturmian words and
Christoffel words, that are the digital approximations of Euclidean segments in
the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic}
bispecial Sturmian words are precisely the maximal internal factors of
\emph{primitive} Christoffel words. We extend this result by showing that
bispecial Sturmian words are precisely the maximal internal factors of
\emph{all} Christoffel words. Our characterization allows us to give an
enumerative formula for bispecial Sturmian words. We also investigate the
minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167
A Characterization of Bispecial Sturmian Words
A finite Sturmian word w over the alphabet {a,b} is left special (resp. right
special) if aw and bw (resp. wa and wb) are both Sturmian words. A bispecial
Sturmian word is a Sturmian word that is both left and right special. We show
as a main result that bispecial Sturmian words are exactly the maximal internal
factors of Christoffel words, that are words coding the digital approximations
of segments in the Euclidean plane. This result is an extension of the known
relation between central words and primitive Christoffel words. Our
characterization allows us to give an enumerative formula for bispecial
Sturmian words. We also investigate the minimal forbidden words for the set of
Sturmian words.Comment: Accepted to MFCS 201
A Characterization of Infinite LSP Words
G. Fici proved that a finite word has a minimal suffix automaton if and only
if all its left special factors occur as prefixes. He called LSP all finite and
infinite words having this latter property. We characterize here infinite LSP
words in terms of -adicity. More precisely we provide a finite set of
morphisms and an automaton such that an infinite word is LSP if
and only if it is -adic and all its directive words are recognizable by
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
Characterizations of finite and infinite episturmian words via lexicographic orderings
In this paper, we characterize by lexicographic order all finite Sturmian and
episturmian words, i.e., all (finite) factors of such infinite words.
Consequently, we obtain a characterization of infinite episturmian words in a
"wide sense" (episturmian and episkew infinite words). That is, we characterize
the set of all infinite words whose factors are (finite) episturmian.
Similarly, we characterize by lexicographic order all balanced infinite words
over a 2-letter alphabet; in other words, all Sturmian and skew infinite words,
the factors of which are (finite) Sturmian.Comment: 18 pages; to appear in the European Journal of Combinatoric
Representations of Circular Words
In this article we give two different ways of representations of circular
words. Representations with tuples are intended as a compact notation, while
representations with trees give a way to easily process all conjugates of a
word. The latter form can also be used as a graphical representation of
periodic properties of finite (in some cases, infinite) words. We also define
iterative representations which can be seen as an encoding utilizing the
flexible properties of circular words. Every word over the two letter alphabet
can be constructed starting from ab by applying the fractional power and the
cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527
A Coloring Problem for Infinite Words
In this paper we consider the following question in the spirit of Ramsey
theory: Given where is a finite non-empty set, does there
exist a finite coloring of the non-empty factors of with the property that
no factorization of is monochromatic? We prove that this question has a
positive answer using two colors for almost all words relative to the standard
Bernoulli measure on We also show that it has a positive answer for
various classes of uniformly recurrent words, including all aperiodic balanced
words, and all words satisfying
for all sufficiently large, where denotes the number of
distinct factors of of length Comment: arXiv admin note: incorporates 1301.526
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