  We investigate the longstanding problem of finding a combinatorial rule for
the Schubert structure constants in the $K$-theory of flag varieties (in type
$A$). The Grothendieck polynomials of A. Lascoux-M.-P. Sch\"{u}tzenberger
(1982) serve as polynomial representatives for $K$-theoretic Schubert classes;
however no positive rule for their multiplication is known outside the
Grassmannian case. We contribute a new basis for polynomials, give a positive
combinatorial formula for the expansion of Grothendieck polynomials in these
glide polynomials, and provide a positive combinatorial Littlewood-Richardson
rule for expanding a product of Grothendieck polynomials in the glide basis.
Our techniques easily extend to the $\beta$-Grothendieck polynomials of S.
Fomin-A. Kirillov (1994), representing classes in connective $K$-theory, and we
state our results in this more general context.
  A specialization of the glide basis recovers the fundamental slide
polynomials of S. Assaf-D. Searles (2016), which play an analogous role with
respect to the Chow ring of flag varieties. Additionally, the stable limits of
another specialization of glide polynomials are T. Lam-P. Pylyavskyy's (2007)
basis of multi-fundamental quasisymmetric functions, $K$-theoretic analogues of
I. Gessel's (1984) fundamental quasisymmetric functions. Those glide
polynomials that are themselves quasisymmetric are truncations of
multi-fundamental quasisymmetric functions and form a basis of quasisymmetric
polynomials.
