Correlation for permutations

TL;DR

This paper explores correlation inequalities for up-sets of permutations under different partial orders, establishing positive correlation under the strong Bruhat order and counterexamples under the weak Bruhat order, with implications for non-uniform measures.

Contribution

It proves positive correlation of up-sets under the strong Bruhat order and demonstrates failure under the weak Bruhat order, extending results to non-uniform measures like Mallows.

Findings

Up-sets are positively correlated under the strong Bruhat order.

Counterexamples show no correlation under the weak Bruhat order.

Results extend to measures including Mallows measures.

Abstract

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 $\pi$, the event that no point is displaced by more than a fixed distance $d$ and the event that $\pi$ 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.