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 𝔖 w ( x ; y ) . We also prove a curious positivity result: for vexillary permutations w ∈ S n , the polynomial x 1 n ⋯ x n n 𝔖 w ( x n - 1 , ⋯ , x 1 - 1 ; 1 , ⋯ , 1 ) is a graded nonnegative sum of Lascoux polynomials. We conjecture that this positivity result holds for all w ∈ S n . This conjecture would follow from a problem of independent interest regarding Lascoux positivity of certain products of Lascoux polynomials.