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

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) and completing the enumeration of staircase diagrams over finite type Dynkin diagrams. The staircase diagrams are in bijection to smooth and rationally smooth Schubert varieties over $E$-type thereby giving an enumeration of these varieties.

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