Given some positive integers $n,e$ and $c$, I would like to know the number of spanning subgraphs of $K_n$ having $e$ edges and $c$ connected components.

Essentially, what I am asking for here is the coefficient of $v^eq^c$ in $Z_{K_n}(q,v)$ (the form of the Tutte polynomial of $K_n$ which is obtained by setting all "edge variables" in the multivariate version to the same variable $v$). I would be happy with any information about the coefficients of Tutte polynomials of complete graphs in general though.

What is known about these numbers? Is finding an arbitrary one as hard as computing the whole Tutte polynomial of the corresponding complete graph? In some sense there seems to be quite a lot known about them, such as the link with Stirling numbers of the first kind (via the chromatic polynomial) and the fact that for a given $n$ they must sum to the total number of subgraphs $2^{n(n-1)/2}$. But the closest I have found in the literature to an actual formula or even approximation is an exponential generating function for the whole family of polynomials, from which it is does not seem to be possible to extract any specific numbers.

What about if we forget about numbers of edges, and just ask how many spanning subgraphs of $K_n$ with a given number of connected components there are?