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 EE, we treat each cycle in the cycle decomposition of a permutation ρρ acting on EE as an equivalence class (or transitive set) of EE under the operation of the group generated by ρρ. 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 kk-uniform hypergraphs of order nn, where kk is an arbitrary integer with 1≤k≤n−11≤k≤n−1. Based on our counting formula, the asymptotic behavior for the number of unlabeled uniform hypergraphs is also analyzed.