Universal cycles of -partitions of an -set

In 1992 Chung, Diaconis and Graham generalized de Bruijn cycles to other combinatorial families with universal cycles. Universal cycles have been investigated for permutations, partitions, k -partitions and k -subsets. In 1990 Hurlbert proved that there exists at least one Ucycle of n − 1 -partitions of an n -set when n is odd and conjectured that when n is even, they do not exist. Herein we prove Hurlbert’s conjecture by establishing algebraic necessary and sufficient conditions for the existence of these Ucycles. We enumerate all such Ucycles for n ≤ 13 and give a lower bound on the total number for all n . Additionally we give ranking and unranking formulae. Finally we discuss the structures of the various solutions.