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Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is …
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Combinatorial proof of the Cauchy identity for double Schubert polynomials

Yes this is proven in the paper RC-graphs and Schubert polynomials, using double rc-graphs (pipe dreams). See section 4 of the link.
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