A decomposition of a complete uniform hypergraph is {\em cyclic} if the parts are permuted transitively by a permutation of the vertex set. If there are $t$ parts in the decomposition, each part is called a $t$-{\em complementary hypergraph}. We characterize the cycle type of the associated permutations, and consequently determine the feasible orders of a $t$-complementary $k$-uniform hypergraph, and an algorithm for generating all of these structures of a given order. We also construct cyclic partitions of complete multipartite uniform hypergraphs.
Joint work with A. Pawel Wojda and Artur Szymanski, AGH University of Science and Technology, Krakow, Poland.