SCHWEITZER 1979 Problem 3

Let $g(n,k)$ denote the number of strongly connected, $\textit{simple}$ directed graphs with $n$ vertices and $k$ edges. ( $\textit{Simple}$ means no loops or multiple edges.) Show that $$\sum_{k=n}^{n^2-n}(-1)^kg(n,k)=(n-1)!.$$

A. A. Schrijver