  Alex Postnikov 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 this paper 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.
