On the number of episturmian palindromes

  • Authors:

  • Michelangelo Bucci;Aldo de Luca;Alessandro De Luca

  • Affiliations:

  • Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy;Dipartimento di Matematica e Applicazioni R. Caccioppoli, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy;LaCIM, Université du Québec à Montréal, Case postale 8888, succursale Centre-ville, Montréal (Québec) H3C 3P8, Canada

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

Episturmian words are a suitable generalization to arbitrary alphabets of Sturmian words. In this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k-letter alphabet A"k. We give a formula for the map g"k giving for any n the number of all palindromes of length n in all episturmian words over A"k. This formula extends to k2 a similar result obtained for k=2 by the second and third authors in 2006. The map g"k is expressed in terms of the map P"k counting for each n the palindromic prefixes of all standard episturmian words (epicentral words). For any n=0, P"2(n)=@f(n+2), where @f is the totient Euler function. The map P"k plays an essential role also in the enumeration formula for the map @l"k counting for each n the finite episturmian words over A"k. Similarly to Euler's function, the behavior of P"k is quite irregular. The first values of P"k and of the related maps g"k, and @l"k for 3@?k@?6 have been calculated and reported in the paper. Some properties of P"k are shown. In particular, broad upper and lower bounds for P"k, as well as for @?"m"="0^nP"k(m) and g"k, are determined. Finally, some conjectures concerning the map P"k are formulated.