218,513 research outputs found
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
Construction Of A Rich Word Containing Given Two Factors
A finite word with contains at most distinct
palindromic factors. If the bound is attained, the word is called
\emph{rich}. Let \Factor(w) be the set of factors of the word . It is
known that there are pairs of rich words that cannot be factors of a common
rich word. However it is an open question how to decide for a given pair of
rich words if there is a rich word such that \{u,v\}\subseteq
\Factor(w). We present a response to this open question:\\ If are
rich words, , and
\{w_1,w_2\}\subseteq \Factor(w) then there exists also a rich word
such that \{w_1,w_2\}\subseteq \Factor(\bar w) and , where and is the size
of the alphabet. Hence it is enough to check all rich words of length equal or
lower to in order to decide if there is a rich word containing
factors
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
- …