报告题目:  Quasi-Stirling Polynomials on Multisets

报 告 人：严慧芳 教授 (浙江师范大学)

时    间：2021年5月19号 上午 9:00-10:00

地    点：腾讯会议443 931 731

报告人简介：

严慧芳，2006年博士毕业于南开大学组合数学研究中心，现任浙江师范大学数学与计算机科学学院教授，硕士生导师。主要研究组合结构的计数以及组合统计量方面的问题， 在J. Combin. Theory Ser. A，  Adv in Appl. Math. ,  European J. Combin.等杂志上发表论文30余篇。 先后主持国家自然科学基金3项（面上2项， 青年1项）， 浙江省自然科学基金一项。

报告内容：

A permutation of a multiset is said to be a quasi-Stirling permutation if there does not exist four indices such that and . For a multiset M, denote by the set of quasi-Stirling permutations of . The quasi-Stirling polynomial on the multiset is defined by , where denotes the number of descents of . By employing generating function arguments, Elizalde derived an elegant identity involving quasi-Stirling polynomials on the multiset , in analogy to the identity on Stirling polynomials. In this paper, we derive an identity involving quasi-Stirling polynomials for any multiset ,  involving which is a generalization of the identity on Eulerian polynomial and Elizalde's identity on quasi-Stirling polynomials on the multiset . We provide a combinatorial proof the identity in terms of certain ordered labeled trees. Specializing implies a combinatorial proof of Elizalde's identity in answer to the problem posed by Elizalde. As an application, our identity enables us to show that the quasi-Stirling polynomial   has only real roots and the coefficients of are unimodal and log-concave for any multiset in analogy to Brenti's result for Stirling polynomials on multisets. This is a joint work with Xue Zhu.