Palindromic Factors Research Articles

On palindromic factorization of words

Given a finite word u, we define its palindromic length|u|pal to be the least number n such that u=v1v2…vn with each vi a palindrome. We address the following open question: let P be a positive integer and w an infinite word such that |u|pal⩽P for every factor u of w. Must w be ultimately periodic? We give a partial answer to this question by proving that for each positive integer k, the word w must contain a k-power, i.e., a factor of the form uk. In particular, w cannot be a fixed point of a primitive morphism. We also prove more: for each pair of positive integers k and l, the word w must contain a position covered by at least l distinct k-powers. In particular, w cannot be a Sierpinski-like word.

Complexity and palindromic defect of infinite words

In this note, we state a conjecture, and prove it in the periodic case, which is an equality relating the number of factors and palindromic factors of infinite words. This equality establishes a link between two inequalities, one due to Droubay, Justin and Pirillo, and the other to Baláži, Masáková and Pelantová, by means of the palindromic defect.

Burrows–Wheeler transform and palindromic richness

The investigation of the extremal case of the Burrows–Wheeler transform leads to study the words w over an ordered alphabet A = { a 1 , a 2 , … , a k } , with a 1 < a 2 < ⋯ < a k , such that b w t ( w ) is of the form a k n k a k − 1 n k − 1 ⋯ a 2 n 2 a 1 n 1 , for some non-negative integers n 1 , n 2 , … , n k . A characterization of these words in the case | A | = 2 has been given in [Sabrina Mantaci, Antonio Restivo, Marinella Sciortino, Burrows-Wheeler transform and Sturmian words, Information Processing Letters 86 (2003) 241–246], where it is proved that they correspond to the powers of conjugates of standard words. The case | A | = 3 has been settled in [Jamie Simpson, Simon J. Puglisi, Words with simple Burrows-Wheeler transforms, Electronic Journal of Combinatorics 15, (2008) article R83], which also contains some partial results for an arbitrary alphabet. In the present paper we show that such words can be described in terms of the notion of “palindromic richness”, recently introduced in [Amy Glen, Jacques Justin, Steve Widmer, Luca Q. Zamboni, Palindromic richness, European Journal of Combinatorics 30 (2) (2009) 510–531]. Our main result indeed states that a word w such that b w t ( w ) has the form a k n k a k − 1 n k − 1 ⋯ a 2 n 2 a 1 n 1 is strongly rich, i.e. the word w 2 contains the maximum number of different palindromic factors.

A connection between palindromic and factor complexity using return words

In this paper we prove that for any infinite word w whose set of factors is closed under reversal, the following conditions are equivalent: (I) all complete returns to palindromes are palindromes; (II) P ( n ) + P ( n + 1 ) = C ( n + 1 ) − C ( n ) + 2 for all n, where P (resp. C ) denotes the palindromic complexity (resp. factor complexity) function of w, which counts the number of distinct palindromic factors (resp. factors) of each length in w.

Palindromic richness

In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are ‘rich’ in palindromes in the utmost sense. A characteristic property of the so-called rich words is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with Droubay and Pirillo in 2001 [X. Droubay, J. Justin, G. Pirillo, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001) 539–553]. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words. Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely weakly rich words and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel’s conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel’s conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency. Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of P - morphisms that was introduced by Hof, Knill, and Simon in 1995 [A. Hof, O. Knill, B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Comm. Math. Phys. 174 (1995) 149–159].

Rich, Sturmian, and trapezoidal words

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.

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

We consider Billiard words in alphabets with k > 2 letters. Such words are associated with some k -dimensional positive vector α → = ( α 1 , α 2 , … , α k ) . The language of these words is already known in the usual case, i.e. when the α j 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.

A geometrical approach of palindromic factors of standard billiard words

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, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.

Occurrences of palindromes in characteristic Sturmian words

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

Palindromic factors of billiard words

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.

ON THE PALINDROMIC COMPLEXITY OF INFINITE WORDS

We study the problem of constructing infinite words having a prescribed finite set P of palindromes. We first establish that the language of all words with palindromic factors in P is rational. As a consequence we derive that there exists, with some additional mild condition, infinite words having P as palindromic factors. We prove that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition. Asymmetric words, those that are not the product of two palindromes, play a fundamental role and an enumeration is provided.

Palindrome complexity

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. 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 “class P” of Hof–Knill–Simon. We also give an upper bound for the palindrome complexity of a sequence in terms of its (block-)complexity.

A constructive analysis of the aperiodic binary correlation function

The aperiodic binary correlation function or correlogram <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X \ast Y</tex> occurs as the output of the detector in the correlation detection of a binary code word <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> in an unsynchronized code stream <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> . In most applications, such as self-synchronizing data links, the correlation properties are determined by the desired system characteristics, and the problem is to find codes that approximate the desired correlation. As a partial solution to this problem, the general algebraic properties of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X \ast Y</tex> as a function of the code words <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> are developed in this paper. In particular, sufficient conditions for the general quadratic equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X \ast Y = W \ast Z</tex> to possess solutions are demonstrated, as well as for the restricted cases in which <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W = \pm Y</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z = \pm X</tex> , i.e., commutative or anticommutative code word pairs, or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y = X</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z = W</tex> . The principal tool used in this development is a repeated Kronecker product of palindromic factor code words, which we call pseudo-Rademacher-Walsh (PRW) codes since the simplest examples of such products are the normal Rademacher-Walsh codes. The PRW codes are used as a basis for constructing arbitrarily many nontrivial solutions to each of the four possible correlogram identities.