We give new formulas for Grothendieck polynomials of two types.
One type expresses any specialization of a Grothendieck polynomial in at least
two sets of variables as a linear combination of products Grothendieck
polynomials in each set of variables, with coefficients Schubert structure
constants for Grothendieck polynomials.
The other type is in terms of chains in the Bruhat order.
We compare this second type with other constructions of Grothendieck
polynomials within the more general context of double Grothendieck polynomials
and the closely related H-polynomials.
Our methods are based upon the geometry of permutation patterns.