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.

我们将图的非循环定向多项式定义为其非循环定向的汇的生成函数。Stanley证明，无环取向的数量等于在-1到正负号处评估的色多项式。受非循环取向和色多项式之间这种联系的激励，我们开发了与Birkhoff，Whitney和Greene-Zaslavsky的色多项式有关的定理的“非循环取向”类似物。作为一种应用，我们为Stanley的色对称函数宿定理提供了新的证明$X_G$。该定理给出了具有固定凹陷数的无环取向的数量与扩张的系数之间的关系。$X_G$ 关于基本对称函数。