Abstract
A bordered box repetition-free word is a finite word w where any given factor of the form axa, with \(a\in \varSigma \) and \(x\in \varSigma ^*\), occurs at most once. It is known that the length of a bordered box repetition-free word is at most \(B(n) = n(2+B(n-1))\), with \(B(1)=2\), where n denotes the size of the alphabet. An alternative approach is given to prove that B(n) is an upper bound, which enables a proof that this upper bound is tight.
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Notes
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In [9], the terminology LMRF is used.
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References
Baena-Garcia, M., Morales-Bueno, R., Carmona-Cejudo, J., Castillo, G.: Counting word avoiding factors. Electronic J. Math. Technol. 4(3), 251–267 (2010)
Avoidable patterns in strings of symbols: Bean, D., Ehrenfeucht, A., G., M. Pac. J. Math. 85, 261–294 (1979)
Berstel, J., Karhumäki, J.: Combinatorics on words: a tutorial. Bull. EATCS 79, 178–228 (2003)
Berstel, J., Perrin, D.: The origins of combinatorics on words. Eur. J. Comb. 28, 996–1022 (2007)
Carpi, A., De Luca, A.: Words and special factors. Theoret. Comput. Sci. 259, 145–182 (2001)
Charalambides, C.A.: Enumerative Combinatorics. CRC Press, Boca Raton, USA (2002)
Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discret. Math. 110(1), 43–59 (1992)
Fredricksen, H.: A survey of full length nonlinear shift register cycle algorithms. SIAM Rev. 24(2), 195–221 (1982)
Grobler, T., Habeck, M., Van Zijl, L., Geldenhuys, J.: Search algorithms for the combinatorial generation of bordered box repetition-free words. J. Univ. Comput. Sci. 29(2), 100–117 (2023)
Habeck, M.: The generation of maximum length box repetition-free words. Master’s thesis, Stellenbosch University, South Africa (2021)
Hierholzer, C.: Über die Möglichkeit, einen Linenzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Math. Ann. 6(1), 30–32 (1873)
Johnson, S.: Generation of permutations by adjacent transposition. Math. Comput. 17(83), 282–285 (1963)
Lothaire, M.: Combinatorics on Words. Cambridge University Press, Cambridge Mathematical Library (1997)
Martin, M.: A problem in arrangements. Bull. Am. Math. Soc. 40(12), 859–864 (1934)
Ruskey, F.: Combinatorial generation. Preliminary working draft. University of Victoria, Victoria, BC, Canada 11, 20 (2003)
Stanley, R.: What is enumerative combinatorics? Springer (1986)
Thue, A.: Über unendliche Zeichenreihen. Norske Vid Selsk. Skr. I Mat-Nat Kl. (Christiana) 7, 1–22 (1906)
Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Jacob Dybwad (1912)
Trotter, H.: Algorithm 115: Perm. Commun. ACM 5(8), 434–435 (1962)
Zimin, A.: Blocking sets of terms. Sbornik: Math. 47(2), 353–364 (1984)
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Weight, J., Grobler, T., van Zijl, L., Stewart, C. (2023). A Tight Upper Bound on the Length of Maximal Bordered Box Repetition-Free Words. In: Bordihn, H., Tran, N., Vaszil, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2023. Lecture Notes in Computer Science, vol 13918. Springer, Cham. https://doi.org/10.1007/978-3-031-34326-1_14
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