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- W4298055815 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." @default.
- W4298055815 created "2022-10-01" @default.
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- W4298055815 date "2015-02-13" @default.
- W4298055815 modified "2023-09-26" @default.
- W4298055815 title "Universal and Near-Universal Cycles of Set Partitions" @default.
- W4298055815 doi "https://doi.org/10.48550/arxiv.1502.04076" @default.
- W4298055815 hasPublicationYear "2015" @default.
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