We can derive an explicit formula:
\begin{split}
F(N,M) &= [y^M]\frac{M!}{(N-1)!}\sum_{m=0}^{N-1} \binom{N-1}m (-1)^{N-1-m} e^{(m+1)y} \\
&= [y^M]\frac{M!}{(N-1)!} e^y (e^y-1)^{N-1} \\
&=[y^M]\frac{M!}{(N-1)!} \left( (e^y-1)^N + (e^y-1)^{N-1} \right) \\
&=N\cdot S(M,N) + S(M,N-1),
\end{split}
where $S(\cdot,\cdot)$ are [Stirling numbers of the second kind](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind). Particular values $F(N,N-1)=1$ and $F(M,N)=0$ for $M<N-1$ easily follow from their definition.