1,689,233 research outputs found
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
Enumeration and Structure of Trapezoidal Words
Trapezoidal words are words having at most distinct factors of length
for every . They therefore encompass finite Sturmian words. We give
combinatorial characterizations of trapezoidal words and exhibit a formula for
their enumeration. We then separate trapezoidal words into two disjoint
classes: open and closed. A trapezoidal word is closed if it has a factor that
occurs only as a prefix and as a suffix; otherwise it is open. We investigate
open and closed trapezoidal words, in relation with their special factors. We
prove that Sturmian palindromes are closed trapezoidal words and that a closed
trapezoidal word is a Sturmian palindrome if and only if its longest repeated
prefix is a palindrome. We also define a new class of words, \emph{semicentral
words}, and show that they are characterized by the property that they can be
written as , for a central word and two different letters .
Finally, we investigate the prefixes of the Fibonacci word with respect to the
property of being open or closed trapezoidal words, and show that the sequence
of open and closed prefixes of the Fibonacci word follows the Fibonacci
sequence.Comment: Accepted for publication in Theoretical Computer Scienc
Characterization of infinite LSP words and endomorphisms preserving the LSP property
Answering a question of G. Fici, we give an -adic characterization of
thefamily of infinite LSP words, that is, the family of infinite words having
all their left special factors as prefixes.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 one of its directive words is
recognizable by .Then we characterize the endomorphisms that preserve
the property of being LSP for infinite words.This allows us to prove that there
exists no set of endomorphisms for which the set of infinite LSP words
corresponds to the set of -adic words. This implies that an automaton is
required no matter which set of morphisms is used.Comment: arXiv admin note: text overlap with arXiv:1705.0578
Generalized trapezoidal words
The factor complexity function of a finite or infinite word
counts the number of distinct factors of of length for each .
A finite word of length is said to be trapezoidal if the graph of its
factor complexity as a function of (for ) is
that of a regular trapezoid (or possibly an isosceles triangle); that is,
increases by 1 with each on some interval of length , then
is constant on some interval of length , and finally
decreases by 1 with each on an interval of the same length . Necessarily
(since there is one factor of length , namely the empty word), so
any trapezoidal word is on a binary alphabet. Trapezoidal words were first
introduced by de Luca (1999) when studying the behaviour of the factor
complexity of finite Sturmian words, i.e., factors of infinite "cutting
sequences", obtained by coding the sequence of cuts in an integer lattice over
the positive quadrant of made by a line of irrational slope.
Every finite Sturmian word is trapezoidal, but not conversely. However, both
families of words (trapezoidal and Sturmian) are special classes of so-called
"rich words" (also known as "full words") - a wider family of finite and
infinite words characterized by containing the maximal number of palindromes -
studied in depth by the first author and others in 2009.
In this paper, we introduce a natural generalization of trapezoidal words
over an arbitrary finite alphabet , called generalized trapezoidal
words (or GT-words for short). In particular, we study combinatorial and
structural properties of this new class of words, and we show that, unlike the
binary case, not all GT-words are rich in palindromes when , but we can describe all those that are rich.Comment: Major revisio
A characterization of fine words over a finite alphabet
To any infinite word w over a finite alphabet A we can associate two infinite
words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the
lexicographically smallest (resp. greatest) amongst the factors of w of the
same length. We say that an infinite word w over A is "fine" if there exists an
infinite word u such that, for any lexicographic order, min(w) = au where a =
min(A). In this paper, we characterize fine words; specifically, we prove that
an infinite word w is fine if and only if w is either a "strict episturmian
word" or a strict "skew episturmian word''. This characterization generalizes a
recent result of G. Pirillo, who proved that a fine word over a 2-letter
alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but
not periodic) infinite word, all of whose factors are (finite) Sturmian.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and
Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue
of Theoretical Computer Science
On the combinatorics of finite words
AbstractIn this paper we consider a combinatorial method for the analysis of finite words recently introduced in Colosimo and de Luca (Special factors in biological strings, preprint 97/42, Dipt. Matematica, Univ. di Roma) for the study of biological macromolecules. The method is based on the analysis of (right) special factors of a given word. A factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters. We show that in the combinatorics of finite words two parameters play an essential role. The first, denoted by R, represents the minimal integer such that there do not exist special factors of w of length R. The second, that we denote by K, is the minimal length of a factor of w which cannot be extended on the right in a factor of w. Some new results are proved. In particular, a new characterization in terms of special factors and of R and K is given for the set PER of all words w having two periods p and q which are coprimes and such that ¦w¦ = p + q − 2
A Classification of Trapezoidal Words
Trapezoidal words are finite words having at most n+1 distinct factors of
length n, for every n>=0. They encompass finite Sturmian words. We distinguish
trapezoidal words into two disjoint subsets: open and closed trapezoidal words.
A trapezoidal word is closed if its longest repeated prefix has exactly two
occurrences in the word, the second one being a suffix of the word. Otherwise
it is open. We show that open trapezoidal words are all primitive and that
closed trapezoidal words are all Sturmian. We then show that trapezoidal
palindromes are closed (and therefore Sturmian). This allows us to characterize
the special factors of Sturmian palindromes. We end with several open problems.Comment: In Proceedings WORDS 2011, arXiv:1108.341
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