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Smooth Words on a 2-letter alphabets having same parity

Srecko Brlek 1 Damien Jamet 2 Geneviève Paquin 1
2 ADAGIO - Applying Discrete Algorithms to Genomics and Imagery
LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b − 1 + 1).
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Srecko Brlek, Damien Jamet, Geneviève Paquin. Smooth Words on a 2-letter alphabets having same parity. Theoretical Computer Science, Elsevier, 2008, 393 (1-3), pp.166-181. ⟨hal-00579854⟩

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