New expressions for order polynomials and chromatic polynomials

TL;DR

This paper introduces new expressions linking order polynomials and chromatic polynomials of graphs, using combinatorial sums over permutations and poset orderings, extending Stanley's foundational work.

Contribution

It provides novel formulas connecting chromatic polynomials with order polynomials through permutation sums and poset theory, offering new combinatorial insights.

Findings

Established a new formula for chromatic polynomials involving permutation sums.

Linked the absence of certain subgraphs to polynomial identities.

Extended Stanley's work on order 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.