Enumeration of Totally Positive Grassmann Cells

Abstract

Postnikov (Webs in totally positive Grassmann cells, in preparation) has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted (Gr_{kn ^+}) and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in (Gr_{kn ^+}) according to their dimension. As a corollary, we give a new proof that the Euler characteristic of (Gr_{kn ^+}) is 1. Additionally, we use our result to produce a new (q)-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.