Bijection between modified path-diagram terms and RC-graphs/pipe dreams
Establish a weight-preserving bijection between the modified path-diagram objects arising from computing the coefficients c_{1,w_0(n)}^{v^{-1}w_0(n)}(x;y) by multiplying factorial elementary symmetric polynomials in increasing order of degree (and sorting columns of the resulting diagram) and the RC-graphs/pipe dreams that index terms in the pipe dream formula for the double Schubert polynomial sch_v(x;y), proving that the sets of terms and their weights coincide.
References
We suspect from empirical evidence, though have not proved, that a modification of our formula for $\sch_v(x;y)$ to computing $c_{1,w_0(n)}{v{-1}w_0(n)}(x;y)$ but applying Proposition \ref{proposition:minipieri} in increasing order of degree instead of decreasing order yields the same terms as the definition of pipe dream polynomials with the corresponding RC-graph obtained directly by sorting the columns of our diagram.