On Universal Cycles for Multisets

Details On Universal Cycles for Multisets On Universal Cycles for Multisets

2007-01-17

arxiv.org/abs/math/0701488v2

Mathematics

Combinatorics

14 pages, two figures, will appear in Discrete Mathematics' special issue on de Bruijn Cycles, Gray Codes and their generaliza

Scientific paper

A Universal Cycle for t-multisets of [n]={1,...,n} is a cyclic sequence of $\binom{n+t-1}{t}$ 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 $\binom{n+t-1}{t}$, 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 in {2,3} and partially for t in {4,6}. These results also support a positive answer to a question of Knuth.

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