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On a Class of Locally Symmetric Sequences: The Right Infinite Word Λ θ [chapter]

Norman Carey
2011 Lecture Notes in Computer Science  
The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starting point for the development of an infinite class of right infinite Lambda words, Λ θ .  ...  The study sheds additional light on the role of palindromes in musical tone structures. Regions are palindromes on two letters, and form the largest palindromes in the Lambda word, as it develops.  ...  This is, of course, a two dimensional representation of powers of 2 and 3 forming a multiplication table.  ... 
doi:10.1007/978-3-642-21590-2_4 fatcat:qqe5hslfsvhrpcmld3jayqabyi

Various Properties of Sturmian Words

P. Baláži
2005 Acta Polytechnica  
Besides the well known theorem of Hedlund and Morse it also includes recent results on the characterization of Sturmian words using return words or palindromes.  ...  We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.  ...  The author is also very obliged to Edita Pelantová and Zuzana Masáková for their help and discussions on the paper.  ... 
doaj:5d7080b38c4b4cc6a403da2bd191c8e1 fatcat:uszccp35svdjllqp3axd7krciu

Episturmian words: a survey

Amy Glen, Jacques Justin
2009 RAIRO - Theoretical Informatics and Applications  
In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters.  ...  Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words.  ...  Many thanks also to the two anonymous referees whose thoughtful suggestions helped to improve the paper.  ... 
doi:10.1051/ita/2009003 fatcat:b2fwwhlldjftjmgruabyj45xr4

A geometrical approach of palindromic factors of standard billiard words

Jean-Pierre Borel
2007 Discrete Mathematics & Theoretical Computer Science  
International audience Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension  ...  We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.  ...  We made the hypothesis that u is a palindromic word, and so we have v = u = u, and the two segments OH and GM have the same intersection coding sequence with the grid.  ... 
doi:10.46298/dmtcs.410 fatcat:leakulpicjeqnmcyvax2ir47iq

On linear ternary Intersection sequences and their properties [article]

Mahdi Saleh, Majid Jahangiri
2017 arXiv   pre-print
The factor complexity and the palindromic complexity of these linear ternary sequences are also studied.  ...  of Sturmian words.  ...  Three dimensional case In agreement with the two dimensional case, we again use D + to present the first octant of the Euclidean space and use G for the integral cube grid in D + .  ... 
arXiv:1709.03829v1 fatcat:ik5fzujlenexla45honw2swwha

Reversals and palindromes in continued fractions

Boris Adamczewski, Jean-Paul Allouche
2007 Theoretical Computer Science  
We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation and for formal power series.  ...  Acknowledgements The authors want to thank the referees for their remarks and suggestions. The second author was partially supported by ACI NIM 154 Numération (MENESR).  ...  Sturmian sequences and continued fractions Generalities Sturmian sequences can be defined in several ways. We choose the arithmetic definition.  ... 
doi:10.1016/j.tcs.2007.03.017 fatcat:o7uk6op3rvfnfhkgiuqi5wynwy

RECENT RESULTS ON EXTENSIONS OF STURMIAN WORDS

JEAN BERSTEL
2002 International journal of algebra and computation  
In this survey, combinatorial properties of these two families are considered and compared.  ...  Sturmian words are infinite words over a two-letter alphabet that admit a great number of equivalent definitions. Most of them have been given in the past ten years.  ...  Acknowledgments I thank Jacques Justin and Jean Mairesse for helpful comments of an earlier version of this paper.  ... 
doi:10.1142/s021819670200095x fatcat:skg3etiodncdxp2lne7ruex6cq

Counting Berg partitions via Sturmian words and substitution tilings [article]

Artur Siemaszko, Maciej P. Wojtkowski
2012 arXiv   pre-print
Using the symmetries of bi-partitions we obtain geometrically the palindromic properties of Sturmian sequences (Theorem 3) established combinatorially by de Luca and Mignosi, L-M.  ...  We develop the connection of Berg partitions with special substitution tilings of two tiles.  ...  The 1-dimensional tilings defined by the intersection of vertical lines with a bi-partition are associated with infinite Sturmian sequences.  ... 
arXiv:1212.1395v1 fatcat:qoiebx6l5vaxdaizpw67z727ae

Palindromic factors of billiard words

J.-P. Borel, C. Reutenauer
2005 Theoretical Computer Science  
We study palindromic factors of billiard words, in any dimension. There are differences between the two-dimensional case, and higher dimension.  ...  Arbitrary long palindrome factors exist in any dimension, but arbitrary long palindromic prefixes exist in general only in dimension 2.  ...  Acknowledgements The authors thank Srecko Brlek for a preliminary computing in dimension 3, and Aldo de Luca for an important mail exchange on this subject.  ... 
doi:10.1016/j.tcs.2005.03.036 fatcat:vgg4grokdnc7xestn6pfd7n7ly

Palindromic prefixes and episturmian words

Stéphane Fischler
2006 Journal of combinatorial theory. Series A  
We denote by (n i ) i 1 the increasing sequence (assumed to be infinite) of all lengths of palindromic prefixes of w.  ...  Special examples include characteristic Sturmian words, and more generally standard episturmian words.  ...  Acknowledgments I am very thankful to Jean-Paul Allouche and Boris Adamczewski for their help and for pointing out to me crucial references in combinatorics.  ... 
doi:10.1016/j.jcta.2005.12.001 fatcat:2t4dyrobx5axpoimyrdqt7x62q

Palindromic Prefixes and Episturmian Words [article]

Stéphane Fischler
2005 arXiv   pre-print
We denote by (n_i)_i ≥ 1 the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of w.  ...  Special examples include characteristic Sturmian words, and more generally standard episturmian words.  ...  Then A contains exactly two letters, and w is characteristic Sturmian.  ... 
arXiv:math/0501420v3 fatcat:vppm6k6lbbhvjpizt7lmseiwgu

Palindrome complexity [article]

Jean-Paul Allouche, Michael Baake, Julien Cassaigne, David Damanik
2001 arXiv   pre-print
We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for fixed points of primitive morphisms of constant length belonging to the class P substitutions  ...  We study the palindrome complexity of infinite sequences on finite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence.  ...  JPA, MB, and DD began to discuss together on palindromes during the summer school and workshop "The Mathematics of Aperiodic Order" at the University of Alberta at Edmonton.  ... 
arXiv:math/0106121v1 fatcat:rgnjqjkm6rhq5hbf4xabmjjc5y

Extensions of rich words

Jetro Vesti
2014 Theoretical Computer Science  
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.  ...  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.  ...  Acknowledgements I would like to thank my supervisors Luca Zamboni and Tero Harju for several fruitful discussions and for introducing me to rich words.  ... 
doi:10.1016/j.tcs.2014.06.033 fatcat:vpzxhex56fbirafvpc4rle4m74

Palindrome complexity

Jean-Paul Allouche, Michael Baake, Julien Cassaigne, David Damanik
2003 Theoretical Computer Science  
We study the palindrome complexity of inÿnite sequences on ÿnite alphabets, i.e., the number of palindromic factors (blocks) of given length occurring in a given sequence.  ...  We survey the known results and obtain new results for some sequences, in particular for Rote sequences and for ÿxed points of primitive morphisms of constant length belonging to "class P" of Hof-Knill-Simon  ...  Acknowledgements JPA, MB, and DD began to discuss together on palindromes during the summer school and workshop "The Mathematics of Aperiodic Order" at the University of Alberta at Edmonton.  ... 
doi:10.1016/s0304-3975(01)00212-2 fatcat:2olqirboo5fphnyn2afgfd5unm

Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes [article]

David Damanik
2002 arXiv   pre-print
We present applications of our combinatorial results to the spectral theory of discrete one-dimensional Schrödinger operators with potentials given by Arnoux-Rauzy sequences.  ...  What is the number of palindromes of a given length occurring in X?  ...  D. would like to express his gratitude to the Department of Mathematics at the University of North Texas at Denton for its warm hospitality and financial support through the Texas Advanced Research Program  ... 
arXiv:math/0208137v1 fatcat:gq3peetcijhgrch6hagtu7zdyi
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