262 research outputs found
Factorizations of the Fibonacci Infinite Word
The aim of this note is to survey the factorizations of the Fibonacci
infinite word that make use of the Fibonacci words and other related words, and
to show that all these factorizations can be easily derived in sequence
starting from elementary properties of the Fibonacci numbers
23 11 Article 15
Abstract The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers
23 11 Article 15
Abstract The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers
Repetition factorization of automatic sequences
Following Inoue et al., we define a word to be a repetition if it is a
(fractional) power of exponent at least 2. A word has a repetition
factorization if it is the product of repetitions. We study repetition
factorizations in several (generalized) automatic sequences, including the
infinite Fibonacci word, the Thue-Morse word, paperfolding words, and the
Rudin-Shapiro sequence
The number of valid factorizations of Fibonacci prefixes
We establish several recurrence relations and an explicit formula for V(n),
the number of factorizations of the length-n prefix of the Fibonacci word into
a (not necessarily strictly) decreasing sequence of standard Fibonacci words.
In particular, we show that the sequence V(n) is the shuffle of the ceilings of
two linear functions of n.Comment: Version accepted to Theoretical Computer Scienc
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
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
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
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