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On shortening \(u\)-cycles and \(u\)-words for permutations. (English) Zbl 1409.05010
Summary: This paper initiates the study of shortening universal cycles (\(u\)-cycles) and universal words (\(u\)-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for \(n\)-permutations exist of lengths \(n! +(1 - k)(n - 1)\) for \(k = 0, 1, \ldots,(n - 2)!\).

MSC:
05A05 Permutations, words, matrices
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[1] Chen, H. Z.Q.; Kitaev, S.; Mütze, T.; Sun, B. Y., On universal partial words, Discrete Math. Theor. Comput. Sci., 19, 1, 1-19, (2017) · Zbl 1408.68122
[2] Chung, F.; Diaconis, P.; Graham, R., Universal cycles for combinatorial structures, Discrete Math., 110, 43-59, (1992) · Zbl 0776.05001
[3] Compeau, P. E.C.; Pevzner, P. A.; Tesler, G., How to apply de Bruijn graphs to genome assembly, Nature Biotechnol., 29, 11, 987-991, (2011)
[4] Fine, N. J.; Wilf, H. S., Uniqueness theorem for periodic functions, Proc. Amer. Math. Soc., 16, 109-114, (1965) · Zbl 0131.30203
[5] Goeckner, B.; Groothuis, C.; Hettle, C.; Kell, B.; Kirkpatrick, P.; Kirsch, R.; Solava, R., Universal partial words over non-binary alphabets, Theoret. Comput. Sci., 713, 56-65, (2018) · Zbl 1394.68271
[6] Nyu, V.; Fon-Der-Flaass, D., Estimates for the length of a universal sequence for permutations, (Russian), Diskretn. Anal. Issled. Oper. Ser., 17, 2, 65-70, (2000) · Zbl 0995.05003
[7] Pagès, J.; Salvi, J.; Collewet, C.; Forest, J., Optimised De Bruijn patterns for one-shot shape acquisition, Image Vis. Comput., 23, 8, 707-720, (2005)
[8] Ralston, A., De Bruijn sequences—a model example of the interaction of discrete mathematics and computer science, Math. Mag., 55, 3, 131-143, (1982) · Zbl 0492.05014
[9] Scheinerman, E. R., Determining planar location via complement-free De Brujin sequences using discrete optical sensors, IEEE Trans. Robot. Autom., 17, 6, 883-889, (2001)
[10] Sohn, H.; Bricker, D. L.; Simon, J. R.; Hsieh, Y., Optimal sequences of trials for balancing practice and repetition effects, Behav. Res. Methods Instrum. Comput., 29, 4, 574-581, (1997)
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