Science and Mathematics Faculty Publications

Title

How Often Are Two Permutations Comparable?

Document Type

Article

Publication Date

2008

Journal Title

Transactions of the American Mathematical Society

Volume

360

Issue

9

First Page

4541

Last Page

4568

Abstract

Two permutations of are comparable in the Bruhat order if one is closer, in a natural way, to the identity permutation, , than the other. We show that the number of comparable pairs is of order at most, and at least. For the related weak order, the corresponding bounds are and , where . In light of numerical experiments, we conjecture that for each order the upper bound is qualitatively close to the actual number of comparable pairs.

Keywords

Permutations, Bruhat order

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