Generating self-complementary uniform hypergraphs

In 2007, Szymański and Wojda proved that for positive integers n,kn,k with k≤nk≤n, a self-complementary kk-uniform hypergraph of order nn exists if and only if nk is even. In this paper, we characterize the cycle type of a kk-complementing permutation in Sym(n) which has order equal to a power of 2. This yields a test for determining whether a finite permutation is a kk-complementing permutation, and an algorithm for generating all self-complementary kk-hypergraphs of order nn, up to isomorphism, for feasible nn. We also obtain an alternative description of the necessary and sufficient conditions on the order of a self-complementary kk-uniform hypergraph, in terms of the binary representation of kk. This extends previous results for the cases k=2,3,4k=2,3,4 due to Ringel, Sachs, Suprunenko, Kocay and Szymański.