On universal cycles for multisets

A Universal Cycle for t-multisets of [n] = {1, . . . , n} is a cyclic sequence of ( n+t−1 t ) integers from [n] with the property that each tmultiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides ( n+t−1 t ) , and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t. We prove the conjecture completely for t ∈ {2, 3} and partially for t ∈ {4, 6}. Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287-1804, [email protected] Department of Mathematics, Yale University, New Haven, CT 06520, [email protected]. Research supported in part by NSF grant 0552730. Department of Mathematics, Caltech, Pasadena, CA 91125, [email protected]. Research supported in part by NSF grant 0552730.