Universal and Near-Universal Cycles of Set Partitions

We study universal cycles of the set \({\cal P}(n,k)\) of \(k\)-partitions of the set \([n]:=\{1,2,\ldots,n\}\) and prove that the transition digraph associated with \({\cal P}(n,k)\) 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 \({\cal P}(n,k)\) 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 \(S(n-2, k-2)\) 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 \({\cal P}(n,k)\). An analogous result for coverings completes the investigation.