Mixed Eulerian Numbers and Peterson Schubert Calculus.

The article focuses on the study and computation of mixed \(\Phi\)-Eulerian numbers, which generalize classical combinatorial numbers such as Eulerian numbers, Catalan numbers, and binomial coefficients, and arise as coefficients in the volume polynomial of weight polytopes associated with crystallographic root systems \(\Phi\). It establishes a deep connection between these numbers and the geometry and cohomology of Peterson varieties \(\operatorname{Pet}_{\Phi}\), a singular subvariety of the flag variety \(G/B\) linked to a semisimple algebraic group \(G\). The main result expresses mixed \(\Phi\)-Eulerian numbers as intersection numbers (integrals) of products of first Chern classes \(\varpi_i\) over \(\operatorname{Pet}_{\Phi}\), and provides explicit recursive formulas for their computation via Peterson Schubert calculus, involving connected subsets of simple roots and coefficients \(m_{i,K}^J\) determined by the Dynkin diagram types. The paper also extends known combinatorial models from type \(A\) to arbitrary Lie types, offers a combinatorial interpretation in terms of left–right diagrams, and tabulates explicit values of these coefficients for all Lie types, thereby enabling efficient calculation of mixed \(\Phi\)-Eulerian numbers through geometric and combinatorial methods.