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.