For n ≥ 3 and r = r(n) ≥ 3, let k = k(n) = (k 1 , . . . , k n ) be a sequence of non-negative integers with sum M (k) = n j=1 k j . We assume that M (k) is divisible by r for infinitely many values of n, and restrict our attention to these values. Let X = X(n) be a simple r-uniform hypergraph on the vertex set V = {v 1 , v 2 , . . . , v n } with t edges. We denote by H r (k) the set of all simple r-uniform hypergraphs on the vertex set V with degree sequence k, and let H r (k, X) be the set of all hypergraphs in H r (k) which contain no edge of X. We give an asymptotic enumeration formula for the size of H r (k, X). This formula holds when r 4 k 3 max = o(M (k)), t k 3 max = o(M (k) 2 ) and r t k 4 max = o(M (k) 3 ). Our proof involves the switching method. As a corollary, we obtain an asymptotic formula for the number of hypergraphs in H r (k) which contain every edge of X. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in H r (k) in the regular case.

An r-uniform hypergraph H consists of a set of vertices V and a set of edges whose elements are r-subsets of V . We define a hypertree to be a connected hypergraph which contains no cycles. A hypertree spans a hypergraph H if it is a subhypergraph of H which contains all vertices of H. Greenhill, Isaev, Kwan and McKay (2017) gave an asymptotic formula for the average number of spanning trees in graphs with given, sparse degree sequence. We prove an analogous result for r-uniform hypergraphs with given degree sequence k = (k 1 , . . . , k n ). Our formula holds whenwhere k is the average degree and k max is the maximum degree.

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