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Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words

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School of Statistics and Data Science, Nankai University, Tianjin 300071, China
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Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
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College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, China
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(5), 778; https://doi.org/10.3390/math8050778
Received: 6 April 2020 / Revised: 8 May 2020 / Accepted: 9 May 2020 / Published: 12 May 2020
(This article belongs to the Section Mathematics and Computer Science)
A universal cycle, or u-cycle, for a given set of words is a circular word that contains each word from the set exactly once as a contiguous subword. The celebrated de Bruijn sequences are a particular case of such a u-cycle, where a set in question is the set A n of all words of length n over a k-letter alphabet A. A universal word, or u-word, is a linear, i.e., non-circular, version of the notion of a u-cycle, and it is defined similarly. Removing some words in A n may, or may not, result in a set of words for which u-cycle, or u-word, exists. The goal of this paper is to study the probability of existence of the universal objects in such a situation. We give lower bounds for the probability in general cases, and also derive explicit answers for the case of removing up to two words in A n , or the case when k = 2 and n 4 . View Full-Text
Keywords: universal cycle; u-cycle; universal word; u-word; de Bruijn sequence universal cycle; u-cycle; universal word; u-word; de Bruijn sequence
MDPI and ACS Style

Chen, H.Z.Q.; Kitaev, S.; Sun, B.Y. Lower Bounds, and Exact Enumeration in Particular Cases, for the Probability of Existence of a Universal Cycle or a Universal Word for a Set of Words. Mathematics 2020, 8, 778.

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