For n→∞ and m=o(r^{-3}n^{3/2}), the number of linear r-uniform hypergraphs with n vertices and m edges is asymptotically binom(binom(n,r),m) exp(-[r]_2^2m^2/(4n^2)-[r]_3^2(3r^2-15r+20)m^3/(24n^4)+O(r^6m^2/n^3)).
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Asymptotic enumeration of linear hypergraphs with given number of vertices and edges
For n→∞ and m=o(r^{-3}n^{3/2}), the number of linear r-uniform hypergraphs with n vertices and m edges is asymptotically binom(binom(n,r),m) exp(-[r]_2^2m^2/(4n^2)-[r]_3^2(3r^2-15r+20)m^3/(24n^4)+O(r^6m^2/n^3)).