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

Staircase diagrams and enumeration of smooth Schubert varieties

We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A, B, C, and D, extending Haiman's enumeration for type A. To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin-Coxeter graph, the set of "nearly-maximally labelled" staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey-Postnikov (BP) decomposition. We can then use an earlier result of the authors…

阶梯图、模式避免与光滑置换

Staircase diagrams, pattern avoidance, and smooth permutations

本文研究阶梯图偏序结构与光滑置换模式避免的联系，并枚举了包括抛光置换在内的若干光滑置换子类。

Lozenge tilings of a halved hexagon with an array of triangles removed from the boundary, part II

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later extended this tiling enumeration to a halved hexagon with a triangle cut off from the boundary. In the previous paper, the author proved a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon in the case an array of an arbitrary number of triangles has been removed from a non-staircase side. In this paper, we consider the other case when the array of triangles has been removed from the staircase…