  Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a
combinatorial expression for Schubert polynomials. The duality between the set
of reduced pipe dreams and certain antidiagonals has important geometric
implications [A. Knutson and E. Miller, Gr\"obner geometry of Schubert
polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was
roundabout, relying on the algebra of certain monomial ideals and a recursive
characterization of reduced pipe dreams. This paper provides a direct
combinatorial proof.
