Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, and designs. Certain classes of objects do not admit universal cycles without either a modification of either the object representation or a generalization of the listing structure. One such generalization of universal cycles, which require almost complete overlap of consecutive words, is s-overlap cycles, which relax such a constraint. In this paper we study permutations and some closely related classes of strings, namely juggling sequences and functions. We prove the existence of s-overlap cycles for these objects, as they do not always lend themselves to the universal cycle structure.
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