Self-complementing permutations of k-uniform hypergraphs

A k-uniform hypergraphH = (V ;E) is said to be self-complementary whenever it is isomorphic with its complement H = (V ; ` V k ́ −E). Every permutation σ of the set V such that σ(e) is an edge ofH if and only if e ∈ E is called selfcomplementing. 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 λ(n) the unique integer such that n = 2c, where c is odd. In the paper we prove that a permutation σ of [1, n] with orbits O1, . . . , Om 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 + s, a is odd, 0 ≤ s < 2 and the following two conditions hold: