Universal Cycles on 3-Multisets

Tobias L. Johnson, Joshua Zahl

Published 2006-08-30Version 1

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.