  The notion of set-valued Young tableaux was introduced by Buch in his study
of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson,
Miller and Yong showed that the double Grothendieck polynomials of
2143-avoiding permutations can be generated by set-valued Young tableaux. In
this paper, we introduce the structure of set-valued Rothe tableaux of
permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued
Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of
positive integers into the squares of $D(w)$ such that the rows are weakly
decreasing and the columns are strictly increasing. We show that the double
Grothendieck polynomials of 1432-avoiding permutations can be generated by
set-valued Rothe tableaux. When restricted to 321-avoiding permutations, our
formula specializes to the tableau formula for double Grothendieck polynomials
due to Matsumura. Employing the properties of tableau complexes given by
Knutson, Miller and Yong, we obtain two alternative tableau formulas for the
double Grothendieck polynomials of 1432-avoiding permutations.
