How Often Are Two Permutations Comparable?
Transactions of the American Mathematical Society
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.
Permutations, Bruhat order
Hammett, Adam J. and Pittel, Boris, "How Often Are Two Permutations Comparable?" (2008). Science and Mathematics Faculty Publications. 320.