99 research outputs found

    Occurrences of palindromes in characteristic Sturmian words

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    This paper is concerned with palindromes occurring in characteristic Sturmian words cαc_\alpha of slope α\alpha, where α∈(0,1)\alpha \in (0,1) is an irrational. As cαc_\alpha is a uniformly recurrent infinite word, any (palindromic) factor of cαc_\alpha occurs infinitely many times in cαc_\alpha with bounded gaps. Our aim is to completely describe where palindromes occur in cαc_\alpha. In particular, given any palindromic factor uu of cαc_\alpha, we shall establish a decomposition of cαc_\alpha with respect to the occurrences of uu. Such a decomposition shows precisely where uu occurs in cαc_\alpha, and this is directly related to the continued fraction expansion of α\alpha.Comment: 17 page

    Sturmian numeration systems and decompositions to palindromes

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    We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number nn better reflect the structure of the associated Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every Q>0Q>0 it has a prefix which cannot be decomposed to a concatenation of at most QQ palindromes.Comment: Submitted to European Journal of Combinatoric

    On Theta-palindromic Richness

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    In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism Θ\Theta. It generalizes the notion of a palindrome into a Θ\Theta-palindrome -- a word invariant under Θ\Theta. For languages closed under Θ\Theta we give the relation between Θ\Theta-palindromic complexity and factor complexity. We generalize the notion of richness to Θ\Theta-richness and we prove analogous characterizations of words that are Θ\Theta-rich, especially in the case of set of factors invariant under Θ\Theta. A criterion for Θ\Theta-richness of Θ\Theta-episturmian words is given together with other examples of Θ\Theta-rich words.Comment: 14 page

    On the least number of palindromes contained in an infinite word

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    We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.Comment: Accepted for publication in Theoretical Computer Scienc

    The sequence of open and closed prefixes of a Sturmian word

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    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 nn-th element is 00 if the prefix of length nn of the word is open, or 11 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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