On universal cycles for multisets

Abstract

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 t-multiset 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 Є{2,3} and partially for t Є{4,6}. These results also support a positive answer to a question of Knuth.

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