Lam, Lee, and Shimozono recently introduced backstable double Grothendieck polynomials to represent K-theory classes of the infinite flag variety. From these, they defined double \(\beta \) -Stanley symmetric functions, which expand into double symmetric Grothendieck functions with polynomial coefficients known as double \(\beta \) -Edelman–Greene coefficients. Anderson proved that these coefficients are \(\beta \) -LLS positive using geometric methods, while a positive combinatorial formula for them remains open. We resolve this problem in the vexillary case, where this problem is equivalent to a positivity statement for skew flagged double \(\beta \) -Grothendieck functions. In this setting, we obtain a tableau formula for double \(\beta \) -Edelman–Greene coefficients that is manifestly \(\beta \) -LLS positive. In addition, our result exhibits a strictly stronger form of positivity not previously known.
