  We derive explicit Pieri-type multiplication formulas in the Grothendieck
ring of a flag variety. These expand the product of an arbitrary Schubert class
and a special Schubert class in the basis of Schubert classes. These special
Schubert classes are indexed by a cycle which has either the form
(k-p+1,k-p+2,...,k+1) or the form (k+p,k+p-1,...,k), and are pulled back from a
Grassmannian projection. Our formulas are in terms of certain labeled chains in
the k-Bruhat order on the symmetric group and are combinatorial in that they
involve no cancellations. We also show that the multiplicities in the Pieri
formula are naturally certain binomial coefficients.
