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On Universal Cycles for Multisets

Glenn Hurlbert·Tobias Johnson·Joshua Zahl
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Abstract

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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Glenn Hurlbert, Tobias Johnson, Joshua Zahl. 2008-07-28. On Universal Cycles for Multisets. https://doi.org/10.1016/j.disc.2008.04.050

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