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In this talk, I will introduce the order  polynomial $\Omega(P,x)$ and its computations. I will also present our latest  result on a new expression for $\Omega(P,x)$. By applying this new result and  Stanley's work on the relation between order polynomials and chromatic  polynomials $\chi(G,x)$ of graphs $G$, a new expression for $\chi(G,x)$ follows  directly.