Smooth words on 2-letter alphabets having same parity

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). Key words: Smooth words, Kolakoski word, Lyndon factorization, letter frequency

• Type: Journal
• Year: 2008
• Where: TCS
• Authors: Srecko Brlek, Damien Jamet, Geneviève Paquin
• Keywords: Alphabets, Generalized Kolakoski Words, Smooth Words, TCS 2008, Theoretical Computer Science