118 research outputs found
Factor versus palindromic complexity of uniformly recurrent infinite words
We study the relation between the palindromic and factor complexity of
infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1)
\leq \Delta C(n) + 2, for all n \in N. For a large class of words it is a
better estimate of the palindromic complexity in terms of the factor complexity
then the one presented by Allouche et al. We provide several examples of
infinite words for which our estimate reaches its upper bound. In particular,
we derive an explicit prescription for the palindromic complexity of infinite
words coding r-interval exchange transformations. If the permutation \pi
connected with the transformation is given by \pi(k)=r+1-k for all k, then
there is exactly one palindrome of every even length, and exactly r palindromes
of every odd length.Comment: 16 pages, submitted to Theoretical Computer Scienc
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
Palindromic complexity of trees
We consider finite trees with edges labeled by letters on a finite alphabet
. Each pair of nodes defines a unique labeled path whose trace is a
word of the free monoid . The set of all such words defines the
language of the tree. In this paper, we investigate the palindromic complexity
of trees and provide hints for an upper bound on the number of distinct
palindromes in the language of a tree.Comment: Submitted to the conference DLT201
Extensions of rich words
In [X. Droubay et al, Episturmian words and some constructions of de Luca and
Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has
at most |w|+1 many distinct palindromic factors, including the empty word. The
unified study of words which achieve this limit was initiated in [A. Glen et
al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these
words rich (in palindromes).
This article contains several results about rich words and especially
extending them. We say that a rich word w can be extended richly with a word u
if wu is rich. Some notions are also made about the infinite defect of a word,
the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure
On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet
We exhibit a recurrence on the number of discrete line segments joining two
integer points in the plane using an encoding of such segments as balanced
words of given length and height over the two-letter alphabet . We
give generating functions and study the asymptotic behaviour. As a particular
case, we focus on the symmetrical discrete segments which are encoded by
balanced palindromes.Comment: 24 page
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