Enumeration of unlabeled uniform hypergraphs

We consider the enumeration problem of unlabeled hypergraphs by using Pólya’s counting theory and Burnside’s counting lemma. Instead of characterizing the cycle index of the permutation group acting on the edge set E , we treat each cycle in the cycle decomposition of a permutation ρ acting on E as an equivalence class (or transitive set) of E under the operation of the group generated by ρ . Compared to the cycle index-based method, our method is more effective to deal with the enumeration problem of hypergraphs. Using this method we establish an explicit counting formula for unlabeled k -uniform hypergraphs of order n , where k is an arbitrary integer with 1 ≤ k ≤ n − 1 . Based on our counting formula, the asymptotic behavior for the number of unlabeled uniform hypergraphs is also analyzed.