3 0 A ug 2 00 6 Universal Cycles on 3 – Multisets

Consider the collection of all t–multisets of {1, . . . , n}. A universal cycle on multisets is a string of numbers, each of which is between 1 and n, such that if these numbers are considered in t–sized windows, every multiset in the collection is present in the string precisely once. The problem of finding necessary and sufficient conditions on n and t for the existence of universal cycles and similar combinatorial structures was first addressed by DeBruijn in 1946 (who considered t–tuples instead of t–multisets). The past 15 years has seen a resurgence of interest in this area, primarily due to Chung, Diaconis, and Graham’s 1992 paper on the subject. For the case t = 3, we determine necessary and sufficient conditions on n for the existence of universal cycles, and we examine how this technique can be generalized to other values of t.