Sturmian numeration systems and decompositions to palindromes

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 $n$ 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 $Q>0$ it has a prefix which cannot be decomposed to a concatenation of at most $Q$ palindromes.Comment: Submitted to European Journal of Combinatoric

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 $\{0,1\}$. 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

A Classification of Trapezoidal Words

Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us to characterize the special factors of Sturmian palindromes. We end with several open problems.Comment: In Proceedings WORDS 2011, arXiv:1108.341

A geometrical characterization of factors of multidimensional Billiard words and some applications

AbstractWe consider Billiard words in alphabets with k>2 letters. Such words are associated with some k-dimensional positive vector α→=(α1,α2,…,αk). The language of these words is already known in the usual case, i.e. when the αj are linearly independent over Q and so for their inverses. Here we study the language of these words when there exist some linear relationships. We give a new geometrical characterization of the factors of Billiard words. As a consequence, we get some results on the associated language, and on the complexity and palindromic complexity of these words. The situation is quite different from the usual case. The languages of two distinct Billiard words with the same direction generally have a finite intersection. As examples, we get some Standard Billiard words of three letters without any palindromic factor of even length, or Billiard words of three letters whose palindromic factors have a bounded length. These results are obtained by geometrical methods

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 $\{0,1\}$. 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

Palindromes in Sturmian words

Lecture Notes in Computer Science Springer (Berlin 2005