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.