Universal and Near-Universal Cycles of Set Partitions

TL;DR

This paper investigates the existence of universal cycles of set partitions, proving their existence in many cases, identifying when they do not exist, and extending the concept to partitions with distinct subset sizes, coverings, and packings.

Contribution

It establishes conditions for the existence of universal cycles of set partitions, including new cases and non-existence results, and extends the concept to partitions with distinct sizes, packings, and coverings.

Findings

Universal cycles exist for all n ≥ 3 when k=2.

Universal cycles exist for odd n when k=n-1, but not for even n.

Non-existence is linked to the parity of S(n-2, k-2).

Abstract

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.