Enumeration与超树的随机构造
Enumeration and randomized constructions of hypertrees
摘要：
三十多年前，卡莱证明了一个美丽的$d$维的凯莱公式的模拟数的$n$顶点树。他列举了$d$维超树，这些超树由它们的$(d-1)$维同调群的平方大小加权.然而，这并没有回答$d$ -超树的未加权枚举的更基本的问题，这是我们在这里关注的问题。我们的主要结果，定理1.4，显着提高了$d$ -超树数量的下界。此外，我们还研究了$d$ -复合物的随机$1$ -out模型，其中每个$(d-1)$ -维面选择一个包含它的随机$d$ -面，并证明了它具有可忽略的$d$ -维同调.
英文摘要：
Over thirty years ago, Kalai proved a beautiful $d$-dimensional analog of Cayley's formula for the number of $n$-vertex trees. He enumerated $d$-dimensional hypertrees weighted by the squared size of their $(d-1)$-dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of $d$-hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of $d$-hypertrees. In addition, we study a random $1$-out model of $d$-complexes where every $(d-1)$-dimensional face selects a random $d$-face containing it, and show it has a negligible $d$-dimensional homology.
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