Abstract
In this note we prove that a binary string of lengthn can have no more than\(2^{k + 1} - 1 + \left( {\mathop {n - k + 1}\limits_2 } \right)\) distinct factors, wherek is the unique integer such that 2k + k - 1 ≤ n < 2k+1 + k. Furthermore, we show that for eachn, this bound is actually achieved. The proof uses properties of the de Bruijn graph.
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References
de Bruijn, N.G. A combinatorial problem. Nederl. Akad. Wetensch. Proc.49 758–764 (1946)
Bondy, J.A., Murty, U.S.R. Graph Theory with Applications. London, New York: Macmillan 1976
Fredricksen, H. A survey of full length nonlinear shift register cycle algorithms. SIAM Rev.24, 195–221 (1982)
Flye-Sainte Marie, C. Solution to problem number 58. L’Intermrdiaire des Mathématiciens1, 107–110 (1894)
Good, I.J. Normally recurring decimals. J. London Math. Soc.21, 167–169 (1946)
van Lint, J.H. Combinatorial Theory Seminar, Eindhoven University of Technology (Lect. Notes Math. vol. 382) Berlin Heidelberg New York: Springer-Verlag 1974.
Yoeli, M. Binary ring sequences. Amer. Math. Monthly69, 852–855 (1962)
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Research supported in part by an NSERC operating grant.
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Shallit, J. On the maximum number of distinct factors of a binary string. Graphs and Combinatorics 9, 197–200 (1993). https://doi.org/10.1007/BF02988306
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DOI: https://doi.org/10.1007/BF02988306