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We enumerate the number of staircase diagrams over classically finite$E$-type Dynkin diagrams, extending the work of Richmond and Slofstra(Staircase Diagrams and Enumeration of smooth Schubert varieties) andcompleting the enumeration of staircase diagrams over finite type Dynkindiagrams. The staircase diagrams are in bijection to smooth and rationallysmooth Schubert varieties over $E$-type thereby giving an enumeration of thesevarieties.

Enumerating Staircase Diagrams and Smooth Schubert Varieties over type $E$ Dynkin Diagrams