Generalized chromatic polynomials of graphs from Heaps of pieces

Let $G$ be a simple graph and let $\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\textbf k$-chromatic polynomial of $G$ in terms of dimensions of the grade spaces of $\mathcal{L}(G)$. This will give us a new interpretation for the chromatic polynomials in terms of multilinear heaps and Lyndon length. The classical results of Stanley, and Greene and Zaslavsky regarding the acyclic orientations of $G$ are obtained as corollaries. A heap with a unique minimal piece is said to be a pyramid and our main theorem is proved using the properties of pyramids. For this reason, we prove two important properties of pyramids namely the pyramid proportionality lemma, and the pyramid and Lyndon heap lemma. In the last section, we will introduce the $(m,\lambda)$-labelling on acyclic orientations which is similar to Stanley's $\lambda$-compatible pairs. As an application of our main theorem, using this $(m,\lambda)$-labelled acyclic orientations, we will prove the reciprocity theorem for the derivatives of the chromatic polynomials.