It is well-known that the number of acyclic orientations of $K_n$ is $n!$. Does anybody know of a combinatorial argument for this fact which uses the identity: $$n!=\sum_{k=1}^ns(n,k),$$ where the $s(n,k)$ are Stirling numbers of the first kind? If such a thing exists; what do the different Stirling numbers correspond to exactly?

I would also be interested in any other information or references linking Stirling numbers with acyclic orientations. And failing all this, if anybody knows of any other nice combinatorial arguments (that is, not involving the evaluation of the chromatic polynomial at $-1$) for the numbers of acyclic orientations of complete graphs then I would be interested to hear them.

anygraph, due to Greene and Zaslavsky. See Corollary 7.4 of their paper at vulcan.math.binghamton.edu/zaslav/Tpapers/iwn.tams1983.pdf. – Richard Stanley Jun 25 '12 at 23:09