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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.

MSC:

68R15 Combinatorics on words
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[1] Allouche, J.-P.; Shallit, J., Automatic sequences: theory, applications, generalizations, (2003), Cambridge University Press Cambridge · Zbl 1086.11015
[2] Brandenburg, F.-J., Uniformly growing k-th power-free homomorphisms, Theoret. comput. sci, 23, 69-82, (1983) · Zbl 0508.68051
[3] Carpi, A., Overlap-free words and finite automata, Theoret. comput. sci, 115, 243-260, (1993) · Zbl 0784.68049
[4] Cassaigne, J., Counting overlap-free binary words, (), 216-225 · Zbl 0799.68153
[5] Dejean, F., Sur un théorème de thue, J. combin. theory. ser. A, 13, 90-99, (1972) · Zbl 0245.20052
[6] Dekking, F.M., On repetitions of blocks in binary sequences, J. combin. theory. ser. A, 20, 292-299, (1976) · Zbl 0325.05010
[7] Edlin, A.E., The number of binary cube-free words of length up to 47 and their numerical analysis, J. differential equations appl, 5, 353-354, (1999) · Zbl 0939.05007
[8] U. Grimm, Counting power-free words in two letters, Poster Presentation at Oberwohlfach Meeting on Aperiodic Order, May, 2001.
[9] Harju, T.; Karhumäki, J., Morphisms, (), 439-510
[10] Kfoury, A.-J., A linear-time algorithm to decide whether a binary word contains an overlap, RAIRO inform. théory appl, 22, 135-145, (1988) · Zbl 0645.68087
[11] Kobayashi, Y., Repetition-free words, Theoret. comput. sci, 44, 175-197, (1986) · Zbl 0596.20058
[12] Kobayashi, Y., Enumeration of irreducible binary words, Discrete appl. math, 20, 221-232, (1988) · Zbl 0673.68046
[13] Kolpakov, R.; Kucherov, G., Minimal letter frequency in n-th power-free binary words, (), 347-357 · Zbl 0941.68103
[14] Kolpakov, R.; Kucherov, G.; Tarannikov, Y., On repetition-free binary words of minimal density, Theoret. comput. sci, 218, 161-175, (1999) · Zbl 0916.68118
[15] A. Lepistö, A characterization of 2^{+}-free words over a binary alphabet, Master’s Thesis, University of Turku, Finland, 1995.
[16] Lothaire, M., Combinatorics on words, Encyclopedia of mathematics and its applications, Vol. 17, (1983), Addison-Wesley Reading, MA · Zbl 0514.20045
[17] Lothaire, M., Algebraic combinatorics on words, Encyclopedia of mathematics and its applications, Vol. 90, (2002), Cambridge University Press Cambridge · Zbl 1001.68093
[18] Noonan, J.; Zeilberger, D., The goulden – jackson cluster methodextensions, applications and implementations, J. differential equations appl, 5, 355-377, (1999) · Zbl 0935.05003
[19] Restivo, A.; Salemi, S., On weakly square free words, Bull. European assoc. theoret. comput. sci. no, 21, 49-56, (1983)
[20] Restivo, A.; Salemi, S., Overlap free words on two symbols, (), 198-206
[21] J. Shallit, Simultaneous avoidance of large squares and fractional powers in infinite binary words, Internat. J. Found. Comput. Sci., to appear, preprint, available at , 2003. · Zbl 1067.68119
[22] Shur, A.M., The structure of the set of cube-free Z-words in a two-letter alphabet, Izv. ross. akad. nauk ser. mat, 64, 4, 201-224, (2000), (in Russian) (English trans.: Izv. Math. 64 (2000) 847-871) · Zbl 0972.68131
[23] Y. Tarannikov, The minimal density of a letter in an infinite ternary square-free word is 0.2746⋯ . J. Integer Sequences 5 (2002) 02.2.2 (electronic). · Zbl 1121.11303
[24] A. Thue, Über unendliche Zeichenreihen, Norske vid. Selsk. Skr. Mat. Nat. Kl. 7 (1906) 1-22 (Reprinted in: T. Nagell (Ed.), Selected Mathematical Papers of Axel Thue, Universitetsforlaget, Oslo, 1977, pp. 139-158).
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