Srecko Brlek

Enumeration of factors in the Thue-Morse word

Abstract We give a new combinatorial property of square factors in the infinite Thue-Morse word M 2 : every nontrivial factorization M 2 = w 1 w 2 implies that w 1 has a square suffix, or w 2 has a square prefix. The proof is based on a coding which yields also a new demonstration of the characterization of square factors. Finally, we present results on the enumeration of factors: first explicit formulas, then the asymptotic values as well as a recurrence formula are given.

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.

Addition chains using continued fractions

Abstract This paper introduces a new algorithm for the evaluation of monomials in two variables x a y b based upon the continued fraction expansion of a b . A method for fast explicit generation of addition chains of small length for a positive integer n is deduced from this Algorithm. As an illustration of the properties of the method, a Scholz-Brauer-like inequality p ( N ) ≤ nb + k + p ( n + 1), is shown to be true whenever N is an integer of the form 2 k (1 + 2 b + … + 2 nb ). Computer experimentation has shown that the length of the chains constructed are of optimal length for all integers up to 1000, with 29 exceptions for which the length is equal to the optimal length plus one.

Lyndon + Christoffel = digitally convex

Discrete geometry redefines notions borrowed from Euclidean geometry creating a need for new algorithmical tools. The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis. To the many different approaches of digital convexity, we propose the combinatorics on words point of view, unnoticed until recently in the pattern recognition community. In this paper, we provide first a fast optimal algorithm checking digital convexity of polyominoes coded by their contour word. The result is based on linear time algorithms for both computing the Lyndon factorization of the contour word and the recognition of Christoffel factors that are approximations of digital lines. By avoiding arithmetical computations the algorithm is much simpler to implement and much faster in practice. We also consider the convex hull computation and relate previous work in combinatorics on words with the classical Melkman algorithm.

On the tiling by translation problem

On square or hexagonal lattices, tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. We also have a linear algorithm for pseudo-hexagon polyominoes not containing arbitrarily large square factors. The results are extended to more general tiles.

A note on differentiable palindromes

Abstract We give a characterization of the palindromes in a class of infinite words over Σ ={1,2} related to the Kolakoski word K . This characterization, based on the left palindromic closure of all prefixes of K , is obtained by using a bijection between the class of right infinite words over Σ and a class of words over the same alphabet, and reveals the first link between the existence of some palindromes and the recurrence of K . Indeed, the existence of arbitrarily long palindromes implies the recurrence of K , and a stronger assumption implies the closure of the set of its factors by permutation of the letters in Σ .

A relative of the Thue-Morse sequence

Abstract We study a sequence, c, which encodes the lengths of blocks in the Thue-Morse sequence. In particular, we show that the generating function for c is a simple product.

Combinatorial properties of smooth infinite words

We describe some combinatorial properties of an intriguing class of infinite words, called smooth, connected with the Kolakoski one, K, defined as the fixed point of the run-length encoding Δ. It is based on a bijection on the free monoid over Σ = {1, 2}, that shows some surprising mixing properties. All words contain the same finite number of square factors, and consequently they are cube-free. This suggests that they have the same complexity as confirmed by extensive computations. We further investigate the occurrences of palindromic subwords. Finally, we show that there exist smooth words obtained as fixed points of substitutions (realized by transducers) as in the case of K.

The discrete Green Theorem and some applications in discrete geometry

The discrete version of Greens Theorem and bivariate difference calculus provide a general and unifying framework for the description and generation of incremental algorithms. It may be used to compute various statistics about regions bounded by a finite and closed polygonal path. More specifically, we illustrate its use for designing algorithms computing many statistics about polyominoes, regions whose boundary is encoded by four letter words: area, coordinates of the center of gravity, moment of inertia, set characteristic function, the intersection with a given set of pixels, hook-lengths, higher order moments and also q-statistics for projections.

A note on a result of daurat and nivat

We consider paths in the square lattice and use a valuation called the winding number in order to exhibit some combinatorial properties on these paths. As a corollary, we obtain a characteristic property of self-avoiding closed paths, generalizing in this way a recent result of Daurat and Nivat (2003) on the boundary properties of polyominoes concerning salient and reentrant points.