The Maximum Difference between the Number of Atoms and Number of Coatoms of a Bruhat Interval of the Symmetric Group

Abstract

We determine the largest difference between the number of atoms and number of coatoms of a Bruhat interval of the symmetric group $S_n$. We then pose the question of describing such extremal intervals $[u,v] \subset S_n$ and give a partial description by specifying the elements $v$ that can occur.