Enumeration of hypergraphs

In this paper, we will study enumeration of hypergraphs. Let S-p be a symmetric group acting on a p-setX. It induces the permutation group S-p* acting on the set of all subsets of X. Our problem is reduced to finding a good formula for the cycle index of S-p* 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 of S-P* explicitly and concretely. We will find a relation between the cycle inventory of the induced permutation and the number of convex k-gons of a given regular p-gon and obtain the formula for its cycle inventory. (C) 2001 Academic Press.