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

Chen, Herman Z.Q. and Kitaev, Sergey and Sun, Brian Y. (2020) 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, 8 (5). 778. (https://doi.org/10.3390/math8050778)

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Abstract

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 thuniversal cycle; u-cycle; universal word; u-word; de Bruijn sequencee 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.

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