Asymptotic enumeration of sparse uniform hypergraphs with given degrees

Let r ≥ 2 be a fixed integer. For infinitely many n, let k = (k1, . . . , kn) be a vector of nonnegative integers such that their sum M is divisible by r. We present an asymptotic enumeration formula for simple r-uniform hypergraphs with degree sequence k. (Here “simple” means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree kmax satisfies k 3 max = o(M).