Schubert structure coefficients via derivatives

Schubert structure coefficients $c_{u,v}^w$ describe the multiplicative structure of the cohomology rings of flag varieties. Much work has been done on the problem of giving combinatorial formulas for these coefficients in special cases. In particular, the Littlewood-Richardson rule computes $c_{u,v}^w$ in the case that $u$, $v$, and $w$ are all $p$-Grassmannian permutations for some common $p$. Building on work on Wyser (2013), we introduce backstable clans to prove a "dual" positive combinatorial rule that computes $c_{u,v}^w$ when $u^{-1}$ is $p$-Grassmannian and $v^{-1}$ is $q$-Grassmannian. We derive new families of linear relations among Schubert structure coefficients, which we then use to give a further positive combinatorial rule for $c_{u,v}^w$ in the case that $u^{-1}$ is $p$-Grassmannian and $v^{-1}$ is covered in weak Bruhat order by a $q$-Grassmannian permutation.Comment: 25 pages. Substantially rewritten; both conjectures from the previous version are now theorem