Self-complementing permutations of k-uniform hypergraphs

A k-uniform hypergraph H = ( V; E) is said to be self-complementary whenever it is isomorphic with its complement (H) over bar = ( V; ((V)(k)) - E). Every permutation sigma of the set V such that sigma(e) is an edge of (H) over bar if and only if e is an element of E is called self-complementing. 2-self-comlementary hypergraphs are exactly self complementary graphs introduced independently by Ringel ( 1963) and Sachs ( 1962). For any positive integer n we denote by lambda(n) the unique integer such that n = 2(lambda(n)) c, where c is odd. In the paper we prove that a permutation sigma of [1, n] with orbits O-1,..., O-m O m is a self-complementing permutation of a k-uniform hypergraph of order n if and only if there is an integer l >= 0 such that k = a2(l) + s, a is odd, 0 (i)n = b2(l+1) + r,r is an element of {0,..., 2(l) - 1 + s}, and (ii) Sigma(i:lambda(vertical bar Oi vertical bar) For k = 2 this result is the very well known characterization of self-complementing permutation of graphs given by Ringel and Sachs.