Two-row Schur times Schubert via parallel classes of positroid varieties

Schubert structure constants are intersection numbers of Schubert varieties. Finding positive combinatorial formulas for these constants remains a major open problem. We resolve this problem when one of the indexing permutations corresponds to a two-row partition. Our approach uses increasing chains, introduced by Sottile for the one-row case, and builds on the skew Schubert functions studied by Bergeron and Sottile. The key step is to connect increasing chains and these functions with the parallel classes of positroid varieties.