Double orthodontia formulas and Lascoux positivity

We give a new formula for double Grothendieck polynomials based on Magyar’s orthodontia algorithm for diagrams. Our formula implies a similar formula for double Schubert polynomials ${\mathfrak{S}}_w(\mathbf{x};\mathbf{y})$. We also prove a curious positivity result: for vexillary permutations $w\in S_n$, the polynomial $x_1^n\dots x_n^n{\mathfrak{S}}_w(x_n^{-1},\dots ,x_1^{-1};1,\dots ,1)$ is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all $w\in S_n$. This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.