## On the factors of the Thue-Morse word on three symbols.(English)Zbl 0746.68067

Summary: Let $$m$$ be the Thue-Morse word in a three-letter alphabet. A finite factor $$v$$ of $$m$$ is called special if there exist two distinct letters $$x$$, $$y$$ of the alphabet such that $$vx$$ and $$vy$$ are factors of $$m$$. Define, for each $$n>0$$, $$\psi(n)$$ (respectively $$g(n)$$) to be the number of special factors (respectively factors) of length $$n$$. By using some of our previous results Aldo De Luca and Stefano Varricchio [Theor. Comput. Sci. 63(3), 333-348 (1989; Zbl 0671.10050)] on the Thue- Morse word on two symbols we prove the following proposition: For each $$n>0$$, $$\psi(n)=4$$ if $$n\leq3\times 2^{[\log_ 2n]-1}-1$$, and $$\psi(n)=2$$ otherwise (where $$[x]$$ is the integer part of $$x$$). As a corollary one has: For each $$n>1$$, $$g(n)=4n-2^{[\log_ 2n]}$$ if $$n\leq 3\times 2^{[\log_ 2n]-1}$$ and $$g(n)=2n+2^{[\log_ 2n]+1}$$ otherwise.

### MSC:

 68R15 Combinatorics on words 68Q45 Formal languages and automata

### Keywords:

square-free word; special factor; structure-function

Zbl 0671.10050
Full Text:

### References:

  Berstel, J., Mots de Fibonacci, (Seminaire d’Informatique Théorique (1980/81), Univ. Paris VI and VII), 57-78, L.I.T.P.  Berstel, J.; Crochemore, M.; Pin, J. E., Thue-Morse Sequence and p-adic Technology of Free Monoid (February 1987), Univ. Paris VI and VII, Preprint, L.I.T.P.  de Luca, A.; Varricchio, S., Some Combinatorial Properties of the Thue-Morse Sequence and a Problem in Semigroups (June 1987), Dept. of Mathematics, Univ. of Rome “La Sapienza”, Preprint  Lothaire, M., Combinatorics on Words (1983), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0514.20045  Morse, M.; Hedlund, G., Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J., 11, 1-7 (1944) · Zbl 0063.04115  Pansiot, J. J., The Morse sequence and iterated morphisms, Inform. Process. Lett., 12, 68-70 (1981) · Zbl 0464.68075  Thue, A., Über unendlichen Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 7, 1-22 (1906) · JFM 39.0283.01  Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 1, 1-67 (1912) · JFM 44.0462.01
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