  We define the acyclic orientation polynomial of a graph to be the generating
function for the sinks of its acyclic orientations. Stanley proved that the
number of acyclic orientations is equal to the chromatic polynomial evaluated
at $-1$ up to sign. Motivated by this link between acyclic orientations and the
chromatic polynomial, we develop "acyclic orientation" analogues of theorems
concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky.
As an application, we provide a new proof for Stanley's sink theorem for
chromatic symmetric functions $X_G$. This theorem gives a relation between the
number of acyclic orientations with a fixed number of sinks and the
coefficients in the expansion of $X_G$ with respect to elementary symmetric
functions.
