  The Murnaghan-Nakayama rule expresses the product of a Schur function with a
Newton power sum in the basis of Schur functions. We establish a version of the
Murnaghan-Nakayama rule for Schubert polynomials and a version for the quantum
cohomology ring of the Grassmannian. These rules compute all intersections of
Schubert cycles with tautological classes coming from the Chern character.
