Smooth Words on a 2-letter alphabets having same parity

Srecko Brlek (1) , Damien Jamet (2) , Geneviève Paquin (1)

1 Département d'informatique [Montréal]
2 ADAGIO - Applying Discrete Algorithms to Genomics and Imagery

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).

Mots clés: Smooth words; Kolakoski word; Lyndon factorization; letter frequency

Domaines: Mathématique discrète [cs.DM]

Srecko Brlek, Damien Jamet, Geneviève Paquin. Smooth Words on a 2-letter alphabets having same parity. Theoretical Computer Science, 2008, 393 (1-3), pp.166-181. ⟨10.1016/j.tcs.2007.11.019⟩. ⟨hal-00579854⟩