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4 results

  • A hierarchy for circular codes
  • Giuseppe Pirillo
  • Journal: RAIRO - Theoretical Informatics and Applications / Volume 42 / Issue 4 / October 2008
  • Published online by Cambridge University Press: 22 January 2008, pp. 717-728
  • Print publication: October 2008
  • View abstract

    We first prove an extremal property of the infinite Fibonacci* word f: the family of the palindromic prefixes {hn | n ≥ 6} of f is not only a circular code but “almost” a comma-free one (see Prop. 12 in Sect. 4). We also extend to a more general situation the notion of a necklace introduced for the study of trinucleotides codes on the genetic alphabet, and we present a hierarchy relating two important classes of codes, the comma-free codes and the circular ones.

  • Chapter 4 - Repetitive Mappings and Morphisms
    • By
  • Edited by M. Lothaire
  • Book: Combinatorics on Words
  • Published online: 04 November 2009
  • Print publication: 29 May 1997, pp 55-62
  • View extract

    Summary

    Introduction

    This chapter is devoted to the study of a special type of unavoidable regularities. We consider a mapping φ:A+ → E from A+ to a set E, and we search in a word w for factors of the type w1w2 … wn with φ(w1) = φ(w2)= … = φ(wn). The mapping is called repetitive when such a factor appears in each sufficiently long word. This is related both to square-free words (Chapter 2), by considering the identity mapping, and to van der Waerden's theorem (Chapter 3), as will be shown later on.

    It will first be shown that any mapping from A+ to a finite set is repetitive (Theorem 4.1.1).

    After a direct proof of this fact, it will be shown how the result can also be deduced from Ramsey's theorem (which is stated without proof).

    Investigated also is the special case where φ is a morphism from A+ to a semigroup S. First it is proved that a morphism to the semigroup of positive integers is repetitive when the alphabet is finite (Theorem 4.2.1). Then it is proved that a morphism to a finite semigroup is uniformly repetitive, in the sense that the words w1, w2,…, wn/i> in the foregoing definition can be chosen of equal length (Theorem 4.2.2). This is, as will be shown, a generalization of van der Waerden's theorem. Finally, the chapter mentions a number of extensions and other results.