Sinks in Acyclic Orientations of Graphs

Greene and Zaslavsky proved that the number of acyclic orientations of a graph G with a unique sink at a given vertex is, up to sign, the linear coefficient of the chromatic polynomial. We give three proofs of this result using pure induction, noncommutative symmetric functions, and an algorithmic bijection. We also prove their result that if e=u0v0 is an edge of G then the number of acyclic orientations having a unique source at u0 and unique sink at v0 is Crapoʹs beta invariant.