Unlabeled Acyclic Hypergraph Count

LetⅤbe a finite set,ξbe a family subsets of X and E_i≠φ(1≤i≤q), U_i=1^q E_i=V. Then H=(V,ξ) is called a hypergraph with vertext setⅤand edges setξ. Applying Polya's Enumeration theorem (PET), we obtain the counting series for uniform linear k-graphs with loops, consistent genuine d -k graphs with loops, genuine acyclic hypergraphs and genuine k-graph.