Alex Postnikov recently gave 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.