From Grassmannians to Catalan numbers
The binomial coefficients have a well-studied q-analogue known as Gaussian polynomials. These polynomials appear as Poincare polynomials (or point counts) of the Grassmannian of k-planes in C^n (or F_q^n).
Another family of important combinatorial numbers is the Catalan numbers, and they have two well-studied q-analogues from the 1960s, due to Carlitz and Riordan and to MacMahon respectively. I will explain how these q-analogues appear as the Poincare polynomial and point count, respectively, of an open (non-compact) subvariety of the Grassmannian known as the top positroid variety. The story involves connections to knot theory and to the geometry of flag varieties.
The talk is based on joint work with Pavel Galashin.