Universal and Near-Universal Cycles of Set Partitions

Universal and Near-Universal Cycles of Set Partitions

We study universal cycles of the set P(n, k) of k-partitions of the set [n] := {1, 2, . . . , n} and prove that the transition digraph associated with 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 P(n, k) exist for all n > 3 when k = 2. We re...

Near-Universal Cycles for Subsets Exist

Let S be a cyclic n-ary sequence. We say that S is a universal cycle ((n, k)-Ucycle) for k-subsets of [n] if every such subset appears exactly once contiguously in S, and is a Ucycle packing if every such subset appears at most once. Few examples of Ucycles are known to exist, so the relaxation to packings merits investigation. A family {Sn} of (n, k)-Ucycle packings for fixed k is a near-Ucycl...

Products of Universal Cycles

Universal cycles are generalizations of de Bruijn cycles to combinatorial patterns other than binary strings. We show how to construct a product cycle of two universal cycles, where the window widths of the two cycles may be different. Applications to card tricks are suggested.

On Universal Cycles for k-Subsets of an n-Set

Universal cycles for k-subsets of an n-set

Generalized from the classic de Bruijn sequence, a universal cycle is a compact cyclic list of information. Existence of universal cycles has been established for a variety of families of combinatorial structures. These results, by encoding each object within a combinatorial family as a length-j word, employ a modified version of the de Bruijn graph to establish a correspondence between an Eule...