2025/12/13 by Tianyi Yu, Chou, Jack Chen-An, Yu, Tianyi
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2512.12185
Lam, Lee, and Shimozono introduced the double Stanley symmetric functions in their study of the equivariant geometry of the affine Grassmannian. They proved that the associated double Edelman--Greene coefficients, the double Schur expansion coefficients of these functions, are positive, a result later refined by Anderson. They further asked for a combinatorial proof of this positivity. In this paper, we provide the first such proof, together with a combinatorial formula that manifests the finer positivity established by Anderson. Our formula is built from two combinatorial models: bumpless pipedreams and increasing chains in the Bruhat order. The proof relies on three key ingredients: a correspondence between these two models, a natural subdivision of bumpless pipedreams, and a symmetry property of increasing chains.