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On universal cycles for new classes of combinatorial structures
 SIAM J. Discret. Math
, 2011
"... A universal cycle (ucycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of ucycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For sub ..."
Abstract

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A universal cycle (ucycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of ucycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For subsets, we show that a ucycle exists for the ksubsets of an nset if we let k vary in a non zero length interval. We use this result to construct a “covering ” of length (1 + o(1)) n
Universal and NearUniversal Cycles of Set Partitions
"... We study universal cycles of the set P(n, k) of kpartitions 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 ..."
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We study universal cycles of the set P(n, k) of kpartitions 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 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 n−2 k−2 is odd (3 6 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 P(n, k). An analogous result for coverings completes the investigation. 1