Universal and Near-Universal Cycles of Set Partitions

Abstract

We study universal cycles of the set $\mathcal{P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with $\mathcal{P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or ucycles) exist, since vertices represent equivalence classes of partitions. We use this result to prove, however, that ucycles of $\mathcal{P}(n,k)$ exist for all $n \geq 3$ when $k=2$. We reprove that they exist for odd $n$ when $k = n-1$ and that they do not exist for even $n$ when $k = n-1$. An infinite family of $(n,k)$ for which ucycles do not exist is shown to be those pairs for which ${{n-2}\brace{k-2}}$ is odd ($3 \leq k < n-1$). We also show that there exist universal cycles of partitions of $[n]$ into $k$ subsets of distinct sizes when $k$ is sufficiently smaller than $n$, and therefore that there exist universal packings of the partitions in $\mathcal{P}(n,k)$. An analogous result for coverings completes the investigation.