Enumeration and structure of trapezoidal words

  • Authors:

  • Michelangelo Bucci;Alessandro De Luca;Gabriele Fici

  • Affiliations:

  • Department of Mathematics, University of Turku, FI-20014 Turku, Finland;Dipartimento di Scienze Fisiche, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy;Laboratoire dInformatique, Signaux et Systèmes de Sophia-Antipolis, CNRS & Universitéé de Nice-Sophia Antipolis, 2000, Route des Lucioles - 06903 Sophia Antipolis cedex, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2013

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Abstract

Trapezoidal words are words having at most n+1 distinct factors of length n for every n=0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix; otherwise it is open. We investigate open and closed trapezoidal words, in relation with their special factors. We prove that Sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a Sturmian palindrome if and only if its longest repeated prefix is a palindrome. We also define a new class of words, semicentral words, and show that they are characterized by the property that they can be written as uxyu, for a central word u and two different letters x,y. Finally, we investigate the prefixes of the Fibonacci word with respect to the property of being open or closed trapezoidal words, and show that the sequence of open and closed prefixes of the Fibonacci word follows the Fibonacci sequence.