The totally nonnegative part of a Grassmannian,
and a q-analogue of the Eulerian numbers

The totally nonnegative part of a real Grassmannian, denoted Gr_k,n^≥0(R), is a certain CW complex contained in the Grassmannian which has some amazing combinatorial properties. For example, Alex Postnikov has shown that the poset of cells of Gr_k,n^≥0(R) is isomorphic (as a graded poset) to the poset of decorated permutations, and the poset of L-diagrams (certain tableau). Additionally, the poset of cells of Gr_k,n^≥0(R) contains the Bruhat order of the symmetric group S_k.

In my talk I will give an explicit generating function which enumerates the cells in Gr_k,n^≥0(R) according to their dimension. As a corollary, we obtain a new proof that the Euler characteristic of Gr_k,n^≥0(R) is 1. Another corollary is the discovery of a q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients. If time permits, I may explain some partial results related to shellability of the poset of Gr_k,n^≥0(R), or connections to cluster algebras.