We provide an explicit product formula for the number of spanning trees of the Bruhat graph of the symmetric group, that is, the graph where two permutations π and σ are connected with an edge if $\pi {\sigma }^{-1}$ is a transposition. We also give the number of spanning trees for the graph where the two permutations are connected if $\pi {\sigma }^{-1}$ is an r-cycle for r even. For r odd we obtain the similar result for the alternating group.

中文翻译：

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Bruhat图的生成树数

我们为对称组的Bruhat图的生成树的数量提供了一个明确的乘积公​​式，即当两个置换π和σ与边连接时的图$\pi {\sigma }^{-\mathrm{1个}}$是换位。我们还给出了两个排列相连接的图的生成树数，如果$\pi {\sigma }^{-\mathrm{1个}}$是- [R型单循环为[R偶数。为[R奇数我们获得用于交替分组的类似的结果。