Enumerating Smooth-Like Permutations Via the Geometry of Schubert Varieties

In 1984, Ryan showed that any smooth Schubert variety in type A is an iterated fiber-bundle of Grassmannian varieties. Later, Haiman calculated the generating function for the number of smooth permutations (equivalently smooth Schubert varieties) of size n using this fiber-bundle structure. In this talk, I will discuss how several other families of "smooth-like" permutations can be enumerated using similar methods. Each of these families can be characterized by pattern avoidance similar to smooth permutations. This talk includes joint work with W. Slofstra and F. Azam.