The average number of spanning hypertrees in sparse uniform hypergraphs

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 et al. (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=(k1,…,kn). Our formula holds when r5kmax3=o((kr−k−r)n), where k is the average degree and kmax is the maximum degree.