Generating self-complementary uniform hypergraphs

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