On Universal Cycles for Multisets
arXiv:math/0701488 · doi:10.1016/j.disc.2008.04.050
Abstract
A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides , 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 in {2,3} and partially for t in {4,6}. These results also support a positive answer to a question of Knuth.
14 pages, two figures, will appear in Discrete Mathematics' special issue on de Bruijn Cycles, Gray Codes and their generalizations; paper revised according to journal referees' suggestions