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

TL;DR

This paper counts staircase diagrams over finite type E Dynkin diagrams, linking them to smooth Schubert varieties, thereby completing their enumeration over all finite type Dynkin diagrams.

Contribution

It extends previous work by enumerating staircase diagrams over E-type Dynkin diagrams, connecting them to smooth Schubert varieties and completing the enumeration for finite types.

Findings

Number of staircase diagrams over E-type Dynkin diagrams is explicitly enumerated.

Bijection established between staircase diagrams and smooth Schubert varieties.

Completes enumeration of staircase diagrams for all finite type Dynkin diagrams.

Abstract

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.