On Universal Cycles for Multisets

TL;DR

It is proven completely and partially that the conjecture that n divides (n+t-1t), and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t.

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.

Figures (2)

• Figure 1: The transition graph ${\cal T}_{5,3}$
• Figure 2: The transition graph ${\cal G}_{8,3}$

Theorems & Definitions (8)

• Conjecture 1
• Theorem 2
• Lemma 4
• Lemma 5
• Lemma 7
• Theorem 8
• Theorem 9
• Claim 10