Enumeration of Hypergraphs

In this paper, we will study enumeration of hypergraphs. Let Spbe a symmetric group acting on ap-setX. It induces the permutation group Sp * acting on the set of all subsets ofX. Our problem is reduced to finding a good formula for the cycle index ofSp * . A crucial point is to calculate the cycle inventory of the permutation induced by a single cycle. Their formulas are given inductively. We will give the cycle index ofSp * explicitly and concretely. We will find a relation between the cycle inventory of the induced permutation and the number of convexk-gons of a given regularp-gon and obtain the formula for its cycle inventory.