I need to find an exponential generating function for the number of way to split $n$ people into nonempty committees with at least 1 chairperson. I am struggling greatly with EGFs...
Anyways if we removed the requirement of needing one chair person I can see that the EGF could just be defined as $G(x) = \sum_{0}^{\infty}S(n,k) \frac{x^n}{n!}$ which has a well known closed form (S(n,k) here are the stirling numbers of the first kind. I am unsure how to incorporate the requirement of a chairperson. I see for any committee with $k$ people there are $k$ possibilities for chairperson, but what would we do? Multiply it into the EGF?
This is an application of the exponential formula. Consider the EGF of the same question expect you insist on just one committee. For $n$ people there are $n$ ways to do it, so the answer is $G(x)=\sum_{n=1}^\infty nx^n/n!=xe^x$. So the exponential generating function you seek is the exponential of this, to wit, $F(x)=\exp(xe^x)$.