  Horn recursion is a term used to describe when non-vanishing products of
Schubert classes in the cohomology of complex flag varieties are characterized
by inequalities parameterized by similar non-vanishing products in the
cohomology of "smaller" flag varieties. We consider the type A partial flag
variety and find that its cohomology exhibits a Horn recursion on a certain
deformation of the cup product defined by Belkale and Kumar in \cite{BK06}. We
also show that if a product of Schubert classes is non-vanishing on this
deformation, then the associated structure constant can be written in terms of
structure constants coming from induced Grassmannians.
