广义欧拉多项式的实根性

Eulerian polynomials are a class of important polynomials in combinatorics, which are defined as the generating functions for descent statistic over the symmetric group. It is well known that these polynomials are real-rooted polynomials, a classic result in combinatorics. The notion of Eulerian polynomials has been extended by Neggers and Stanley to P-partitions and by Brenti to finite Coxeter groups. Dilks, Petersen, and Stembridge also studied the affine Eulerian polynomials for irreducible finite Weyl groups in 2007. Since the classical Eulerian polynomials are real-rooted, it is natural to ask whether these generalized polynomials still possess the property of real-rootedness, which has attracted interests of many combinatorists. Recently, Savage and Visontai proved the real-rootedness of s-Eulerian polynomials and confirmed a conjecture of Brenti on the real-rootedness of Eulerian polynomials of type D. In this project, we shall apply the theory of s-Eulerian polynomials and stable theory, which has been developed rapidly recently, to studying the real-rootedness of various generalized Eulerian polynomials. Furthermore, the project is focused on affine Eulerian polynomials, Eulerian polynomials on zig-zag posets and Eulerian polynomials on k-stack sortable permutations.

欧拉多项式是一类重要的组合多项式，其定义为对称群上关于降位统计量的生成函数。它的一个经典性质是它有且仅有实根。多项式的实根性问题是组合数学中单峰型问题的一个重要研究内容。Neggers和Stanley将欧拉多项式的概念推广到P-分拆上，Brenti将其推广到有限Coxeter群上。此外，Stembridge等人还研究了Weyl群上的仿射欧拉多项式。一个自然的问题就是这些广义欧拉多项式是否也具有实根性，这个问题吸引了很多组合学家的兴趣，并于近期取得了重大突破。但是，依然有很多相关的问题未能解决。本项目运用s-欧拉多项式性理论和近来迅速发展的稳定性理论研究组合数学中几类广义欧拉多项式的实根性。本项目具体针对D型仿射欧拉多项式、zig-zag偏序集上的欧拉多项式以及k次堆栈可排排列上的欧拉多项式展开研究。

欧拉多项式是组合数学中一类常见的多项式。它的一个重要性质是其所有的根都是实数，简称“实根性”。Stanely、Brenti、Stembridge等很多组合数学家注意到很多欧拉多项式的推广和细化也具有实根性。本项目主要研究通过交错性方法和稳定性理论去证明很多广义欧拉多项式的实根性质。借助Hermite-Biehler定理和Routh-Hurwitz判别法，我们肯定了Brenti关于D型q-欧拉多项式的实根性问题；我们证明了矩形Narayana多项式具有实根性，进而解决了Kirillov的一个单峰性猜想。