Geodesic
Duality of Antidiagonals and Pipe Dreams
Séminaire lotharingien de combinatoire, Tome 58 (2007-2008)

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Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. 161, 1245-1318]. The original proof of the duality was roundabout, relying on the algebra of certain monomial ideals and a recursive characterization of reduced pipe dreams. This paper provides a direct combinatorial proof. 

@article{SLC_2007-2008_58_a4,
     author = {Ning Jia and Ezra Miller},
     title = {Duality of {Antidiagonals} and {Pipe} {Dreams}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {58},
     year = {2007-2008},
     url = {http://geodesic.mathdoc.fr/item/SLC_2007-2008_58_a4/}
}
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Ning Jia; Ezra Miller. Duality of Antidiagonals and Pipe Dreams. Séminaire lotharingien de combinatoire, Tome 58 (2007-2008). http://geodesic.mathdoc.fr/item/SLC_2007-2008_58_a4/