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数学学科离散数学研究所学术报告(董峰明 南洋理工大学)

发布者:付慧娟   发布时间:2019-12-17  浏览次数:74

报告题目:  New expressions for order polynomials and chromatic polynomials

报 告  人:董峰明,新加坡南洋理工大学副教授(终身职位), 博导



摘要: In 1970, Stanley introduced the order polynomial and the strict order polynomial of a poset (i.e. partially ordered set). Let $P$ be a poset on $n$ elements with a binary relation $\preceq$. For $u,v \in P$, let $u \prec v$ mean that $u \preceq v$ but $u \neq v$. A mapping $\sigma :P \rightarrow[m]$ is said to be {\it order-preserving}  (resp., {\it strictly order-preserving}) if $u \preceq v$ implies that  $\sigma(u) \leq \sigma(v)$ (resp., $u \prec v$  implies that $\sigma(u) < \sigma(v)$). Let $\Omega(P, x)$ (resp., $\bar{\Omega}(P, x)$)  be the function which counts the number of order-preserving (resp., strictly order-preserving) mappings $\sigma :P \rightarrow[m]$ whenever $x = m$ is a positive integer. Both $\Omega(P, x)$ and $\bar{\Omega}(P, x)$ are polynomials in $x$ of degree $n$ and are respectively called the {\it order polynomial} and the {\it strict order polynomial} of $P$.


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.


 个人概况:董峰明,南洋理工大学副教授(终身职位), 博导。 主要从事图论与组合数学方面的研究,解决了包括牛津大学 Dominic Welsh 教授提出的 the Shameful Conjecture在内的若干公开问题。在Journal of Combinatorial Theory, Series A,Journal of Combinatorial Theory, Series B,Journal of Graph Theory,European Journal of Combinatorics, SIAM Journal on Discrete Mathematics, Electronic Journal of Combinatorics等杂志上发表学术论文70余篇。





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