Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.11

On Generalized Pseudostandard Words Over Binary Alphabets

Alexandre Blondin Massé
Laboratoire d'informatique formelle
Université du Québec à Chicoutimi
Chicoutimi, QC, G7H 2B1

Geneviève Paquin
Département de mathématiques
Cégep de Saint-Jérôme
Saint-Jérôme, QC J7Z 4V2

Hugo Tremblay
Laboratoire de combinatoire et d'informatique mathématique
Université du Québec à Montréal
Montréal, QC H3C 3P8

Laurent Vuillon
Laboratoire de mathématiques
Université de Savoie
Le-Bourget-du-Lac 73376


In this paper, we study generalized pseudostandard words over a two-letter alphabet, which extend the classes of standard Sturmian, standard episturmian and pseudostandard words, allowing different involutory antimorphisms instead of the usual palindromic closure or a fixed involutory antimorphism. We first discuss pseudoperiods, a useful tool for describing words obtained by iterated pseudopalindromic closure. Then, we introduce the concept of normalized directive bi-sequence (Θ, w) of a generalized pseudostandard word, that is the one that exactly describes all the pseudopalindromic prefixes of it. We show that a directive bi-sequence is normalized if and only if its set of factors does not intersect a finite set of forbidden ones. Moreover, we provide a construction to normalize any directive bi-sequence. Next, we present an explicit formula, generalizing the one for the standard episturmian words introduced by Justin, that computes recursively the next prefix of a generalized pseudostandard word in term of the previous one. Finally, we focus on generalized pseudostandard words having complexity 2n, also called Rote words. More precisely, we prove that the normalized bi-sequences describing Rote words are completely characterized by their factors of length 2.

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Received July 4 2012; revised version received January 1 2013. Published in Journal of Integer Sequences, March 2 2013.

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