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We enumerate smooth and rationally smooth Schubert varieties in the classicalfinite types A, B, C, and D, extending Haiman's enumeration for type A. To dothis enumeration, we introduce a notion of staircase diagrams on a graph. Thesecombinatorial structures are collections of steps of irregular size, forminginterconnected staircases over the given graph. Over a Dynkin-Coxeter graph,the set of "nearly-maximally labelled" staircase diagrams is in bijection withthe set of Schubert varieties with a complete Billey-Postnikov (BP)decomposition. We can then use an earlier result of the authors showing thatall finite-type rationally smooth Schubert varieties have a complete BPdecomposition to finish the enumeration.

Staircase diagrams and enumeration of smooth Schubert varieties