We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565-596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood-Richardson rule of Buch (Acta Math 189(1):37-78, 2002). The proof is based on a generalization of the Robinson-Schensted and Edelman-Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447-450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665-687, 1999) for the cohomological quiver polynomials.
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