New combinatorics of the weak and strong Bruhat orders

Abstract

This thesis describes a line of work, much of it joint with Yibo Gao, which began with our proof of Stanley's conjecture that the weak order on the symmetric group is Sperner. Further developments---either directly related or related in our thinking at the time---involve weighted path enumeration in the weak and strong Bruhat orders, specializations of Schubert polynomials, separable elements in finite Weyl groups, and inequalities for linear extensions of finite posets.