  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.
