New expressions for order polynomials and chromatic polynomials

Abstract

Let $G=(V,E)$ be a simple graph with $V=\{1,2,\cdots,n\}$ and $\chi(G,x)$ be its chromatic polynomial. For an ordering $\pi=(v_1,v_2,\cdots,v_n)$ of elements of $V$, let $\delta_G(\pi)$ be the number of $i$'s, where $1\le i\le n-1$, with either $v_i<v_{i+1}$ or $v_iv_{i+1}\in E$. Let ${\cal W}(G)$ be the set of subsets $\{a,b,c\}$ of $V$, where $a<b<c$, which induces a subgraph with $ac$ as its only edge. We show that ${\cal W}(G)=\emptyset$ if and only if $(-1)^n\chi(G,-x)=\sum_{\pi} {x+\delta_G(\pi)\choose n}$, where the sum runs over all $n!$ orderings $\pi$ of $V$. To prove this result, we establish an analogous result on order polynomials of posets and apply Stanley's work on the relation between chromatic polynomials and order polynomials.