1,575 research outputs found
Words with many palindrome pair factors
Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many n, every length-n factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue-Morse word does not have this property. We investigate infinite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair."The first author is supported by an NSERC USRA, the second by an NSERC Discovery Grant."http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p2
On the Combinatorics of Palindromes and Antipalindromes
We prove a number of results on the structure and enumeration of palindromes
and antipalindromes. In particular, we study conjugates of palindromes,
palindromic pairs, rich words, and the counterparts of these notions for
antipalindromes.Comment: 13 pages/ submitted to DLT 201
On Generating Binary Words Palindromically
We regard a finite word up to word isomorphism as an
equivalence relation on where is equivalent to if
and only if Some finite words (in particular all binary words) are
generated by "{\it palindromic}" relations of the form for some
choice of and That is to say,
some finite words are uniquely determined up to word isomorphism by the
position and length of some of its palindromic factors. In this paper we study
the function defined as the least number of palindromic relations
required to generate We show that every aperiodic infinite word must
contain a factor with and that some infinite words have
the property that for each factor of We obtain a
complete classification of such words on a binary alphabet (which includes the
well known class of Sturmian words). In contrast for the Thue-Morse word, we
show that the function is unbounded
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
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
On Theta-palindromic Richness
In this paper we study generalization of the reversal mapping realized by an
arbitrary involutory antimorphism . It generalizes the notion of a
palindrome into a -palindrome -- a word invariant under . For
languages closed under we give the relation between
-palindromic complexity and factor complexity. We generalize the notion
of richness to -richness and we prove analogous characterizations of
words that are -rich, especially in the case of set of factors
invariant under . A criterion for -richness of
-episturmian words is given together with other examples of
-rich words.Comment: 14 page
Proof of Brlek-Reutenauer conjecture
Brlek and Reutenauer conjectured that any infinite word u with language
closed under reversal satisfies the equality 2D(u) = \sum_{n=0}^{\infty}T_u(n)
in which D(u) denotes the defect of u and T_u(n) denotes C_u(n+1)-C_u(n) +2 -
P_U(n+1) - P_u(n), where C_u and P_u are the factor and palindromic complexity
of u, respectively. This conjecture was verified for periodic words by Brlek
and Reutenauer themselves. Using their results for periodic words, we have
recently proved the conjecture for uniformly recurrent words. In the present
article we prove the conjecture in its general version by a new method without
exploiting the result for periodic words.Comment: 9 page
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
- …