  We study the action of a differential operator on Schubert polynomials. Using
this action, we first give a short new proof of an identity of I. Macdonald
(1991). We then prove a determinant conjecture of R. Stanley (2017). This
conjecture implies the (strong) Sperner property for the weak order on the
symmetric group, a property recently established by C. Gaetz and Y. Gao (2018).
