Littlewood--Richardson rules from quivers for two-step flag varieties

Let ${\bigwedge }_1$ and ${\bigwedge }_2$ be two symmetric function algebras in independent sets of variables. We define vector space bases of ${\bigwedge }_1{\otimes }_{\mathbb{Z}}{\bigwedge }_2$ coming from certain quivers, with vertex sets indexed by pairs of partitions. We use these vector space bases to give a positive tableau formula for Littlewood–Richardson coefficients for the product of Schubert polynomials with certain Schur polynomials in two-step flag varieties, in the spirit of the Remmel-Whitney rule for the product of two Schur polynomials in Grassmannians. This in particular covers the cases considered by the Pieri rule.