Structure constants of Peterson Schubert calculus

We give an explicit, positive, and type-uniform formula for all equivariant structure constants of the Peterson Schubert calculus in arbitrary Lie types, using only the Cartan matrix of the corresponding root system $\mathrm{\Phi }$. This solves an open problem originally asked by Harada–Tymoczko in type A for all Lie types. As an application, we derive a type-uniform formula for the mixed $\mathrm{\Phi }$-Eulerian numbers.