Correlation for Permutations

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 Sn, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on Sn, 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.