## Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups.(English)Zbl 0671.10050

The (Prouhet)-Thue-Morse sequence is one of the fixed points of the morphism $$a\to ab$$, $$b\to ba$$. The authors define $$\phi$$ (n) to be the number of factors w of the Thue-Morse sequence such that both wa and wb are factors, (“special factors”). They then prove that $$\phi$$ (n) is equal either to 2 or to 4 for $$n\geq 1$$. They also characterize those n such that $$\phi (n)=2$$ and give an algorithm to construct all special factors of given length.
Denoting by F(n) the total number of factors of length n (in the Thue- Morse sequence), they deduce the exact value of F(n) (using the relation $$F(n+1)=F(n)+\phi (n))$$, and obtain the inequalities $$3n\leq F(n+1)\leq 10n/3.$$ (Note that every automatic sequence satisfies F(n)$$\leq Cn$$, see A. Cobham, Math. Syst. Theory 6, 164-192 (1972; Zbl 0253.02029). As a consequence they prove that the Thue-Morse monoid is finitely generated, periodic, infinite and weakly permutable.
Note, as the authors point out, that an enumeration formula for the factors of the Thue-Morse sequence has been independently obtained by S. Brlek [Enumeration of factors in the Thue-Morse word, in Proc. Coll. Montréalais sur la Combinatoire et l’Informatique (to appear)].
Reviewer: J.-P.Allouche

### MSC:

 11B99 Sequences and sets 68Q45 Formal languages and automata 20M10 General structure theory for semigroups

Zbl 0253.02029
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### References:

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