Abstract
Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.
Highlights
Smooth infinite words over Σ = {1, 2} are connected to the Kolakoski word [Kolakoski (1965)]K = 22112122122112112212112122112112122122112122121121122 · · ·, defined as the fixed point of the run-length encoding function ∆
We know from Carpi [Carpi (1993, 1994)] that K does contain only a finite number of squares, implying by direct inspection that K is cube-free, a result that was extended in [Brlek et al (2006)] to the infinite class K of smooth words
Using a result from [Brlek et al (2006)] about the number of squares in smooth words, we show that the Lyndon factorization of the minimal infinite smooth word is a strictly decreasing sequence of Lyndon words, while the Lyndon factorization of the maximal one is strictly decreasing following the second term, since the first two Lyndon words are equal
Summary
K = 22112122122112112212112122112112122122112122121121122 · · ·, defined as the fixed point of the run-length encoding function ∆ They are characterized by the property that the orbit obtained by iterating ∆ is contained in {1, 2}∗. In the last part of this paper, we establish a link between the Fibonacci word F and the minimal infinite smooth word over the alphabet Σ = {1, 3}. It turns out that the minimal smooth word over Σ = {1, 3} may be computed in linear time and we give a transducer generating it. This minimal word appeared in Berthe et al [Berthe et al (2005)], but the authors did not point out its minimality
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