  We introduce the quantum multi-Schur functions, quantum factorial Schur
functions and quantum Macdonald polynomials. We prove that for restricted
vexillary permutations the quantum double Schubert polynomial coincides with
some quantum multi-Schur function and prove a quantum analog of the
Nagelsbach-Kostka and Jacobi-Trudi formulae for the quantum double Schubert
polynomials in the case of Grassmannian permutations. We prove, also, an analog
of the Billey-Jockusch-Stanley formula for quantum Schubert polynomials.
Finally we formulate two conjectures about the structure of quantum double and
quantum Schubert polynomials for 321-avoiding permutations.
