Correlation for permutations

Research output: Contribution to journalArticlepeer-review

Authors

Colleges, School and Institutes

External organisations

  • Univ London Queen Mary
  • University of Cambridge

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 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.

Details

Original languageEnglish
Article number105260
JournalJournal of Combinatorial Theory, Series A
Volume175
Early online date28 Apr 2020
Publication statusPublished - Oct 2020

Keywords

  • Correlation inequality, permutation, Bruhat order