Polynomial versus exponential growth in repetition-free binary words. (English) Zbl 1065.68080

Summary: It is known that the number of overlap-free binary words of length \(n\) grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is \(\frac{7}{3}\). More precisely, there are only polynomially many binary words of length \(n\) that avoid \(\frac{7}{3}\)-powers, but there are exponentially many binary words of length \(n\) that avoid \(\frac{7}{3}^{+}\)-powers. This answers an open question of Kobayashi from 1986.


68R15 Combinatorics on words
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Online Encyclopedia of Integer Sequences:

Number of 7/3+-power-free words over the alphabet {0,1}.


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