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 $\boldsymbol{k} = (k_1,\ldots, k_n)$. Our formula holds when $r^5 k_{\max}^3 = o((kr-k-r)n)$, where $k$ is the average degree and $k_{\max}$ is the maximum degree.

中文翻译：

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稀疏均匀超图中的平均生成超树数

一个 $r$-uniform hypergraph $H$ 由一组顶点 $V$ 和一组边组成，这些边的元素是 $r$-$V$ 的子集。我们将超树定义为不包含环的连通超图。如果超树是包含 $H$ 的所有顶点的 $H$ 的子超图，则超树跨越超图 $H$。Greenhill、Isaev、Kwan 和 McKay (2017) 给出了具有给定稀疏度序列的图中生成树平均数的渐近公式。我们证明了具有给定度数序列 $\boldsymbol{k} = (k_1,\ldots, k_n)$ 的 $r$-均匀超图的类似结果。我们的公式在 $r^5 k_{\max}^3 = o((kr-kr)n)$ 时成立，其中 $k$ 是平均度数，$k_{\max}$ 是最大度数。