Constructing maximal pipedreams of double Grothendieck polynomials

Pechenik, Speyer and Weigandt defined a statistic $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(\cdot )$ on permutations which characterizes the leading monomial in top degree components of double Grothendieck polynomials. Their proof is combinatorial: They showed there exists a unique pipedream of a permutation $w$ with row weight $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(w)$ and column weight $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(w^{-1})$. They proposed the problem of finding a ``direct recipe’’ for this pipedream. We solve this problem by providing an algorithm that constructs this pipedream via ladder moves.