In this note we investigate correlation inequalities for ‘up-sets’ of permutations, in the spirit of the Harris–Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris–Kleitman sense). Thus, for example, for a (uniformly) random permutation π, the event that no point is displaced by more than a fixed distance d and the event that π is the product of at most k adjacent transpositions are positively correlated. In contrast, under the weak Bruhat order we show that this completely fails: surprisingly, there are two up-sets each of measure 1/2 whose intersection has arbitrarily small measure.

We also prove analogous correlation results for a class of non-uniform measures, which includes the Mallows measures. Some applications and open problems are discussed.

在本文中，我们本着哈里斯-克莱特曼不等式的精神，研究了排列“集”的相关不等式。我们专注于两个经过充分研究的部分订单${\mathrm{小号}}_{ñ}$，引起了不同的意念。我们的第一个结果表明，在强Bruhat阶上${\mathrm{小号}}_{ñ}$，情绪波动呈正相关（在Harris–Kleitman的意义上）。因此，例如，对于（均匀）随机排列π，没有点移位超过固定距离d的事件与π是最多k个相邻换位的乘积的事件正相关。相反，在弱的Bruhat阶下，我们证明这完全失败了：令人惊讶的是，每个小节1/2都有两个扰动，它们的交点任意小。

我们还证明了一类非均匀度量（包括Mallows度量）的相似相关结果。讨论了一些应用程序和未解决的问题。