Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of flag varieties G/P are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of ``smaller" flag varieties. We consider the partial flag varieties associated to the groups G=SL(n),Sp(2n) and find that H^*(G/P) exhibits Horn recursion on a certain deformation of the cup product. If a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written as a product of structure constants coming from induced maximal flag varieties.