2 research outputs found
Universal and Near-Universal Cycles of Set Partitions
We study universal cycles of the set of -partitions of the
set and prove that the transition digraph associated
with 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 exist for
all when . We reprove that they exist for odd when and that they do not exist for even when . An infinite family
of for which ucycles do not exist is shown to be those pairs for which
is odd (). We also show that there exist
universal cycles of partitions of into subsets of distinct sizes when
is sufficiently smaller than , and therefore that there exist universal
packings of the partitions in . An analogous result for
coverings completes the investigation.Comment: 22 page