A Pieri-type formula for the K-theory of a flag manifold
  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. 
Cristian Lenart and Frank Sottile