Asymptotic enumeration of sparse uniform hypergraphs with given degrees

Let $r\ge 2$ be a fixed integer. For infinitely many $n$, let $\mathit{k}=(k_1,\mathrm{.}\mathrm{.}\mathrm{.},k_n)$ 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 $k_{\mathrm{m}\mathrm{a}\mathrm{x}}$ satisfies $k_{\mathrm{m}\mathrm{a}\mathrm{x}}^3=o(M)$.