报告题目：Distribution of maxima and minima statistics on alternating permutations, Springer numbers, and avoidance of flat POPs

报告人：张彪副教授，天津师范大学

报告时间：2024年10月14日（周一）14:30

报告地点：腾讯会议876 119 635

报告摘要：In this paper, we find distributions of the left-to-right maxima, right-to-left maxima, left-to-right minima and right-to-left-minima statistics on up-down and down-up permutations of even and odd lengths. For instance, we show that the distribution of right-to-left maxima on up-down permutations of even length is given by $(\sec (t))^{q}$. We also derive the joint distribution  of the maxima (resp., minima) statistics. To accomplish this, we generalize a result of Kitaev and Remmel by deriving joint distributions involving non-maxima (resp., non-minima) statistics. Consequently, we refine classic enumeration results of Andr\'e by introducing new $q$-analogues and $(p,q)$-analogues for the number of alternating permutations.

Additionally, we verify Callan's conjecture (2012) that the number of up-down permutations of even length fixed by reverse and complement equals the Springer numbers, thereby offering another combinatorial interpretation of these numbers. Furthermore, we propose two $q$-analogues and a $(p,q)$-analogue of the Springer numbers. Lastly, we enumerate alternating permutations that avoid certain flat partially ordered patterns (POPs), where the only minimum or maximum elements are labeled by the largest or smallest numbers.

报告人简介：张彪，天津师范大学副教授，研究方向为组合数学，主要从事单峰型理论和对称函数理论的研究工作。2015年6月博士毕业于南开大学。2018年8月至2019年8月赴美国宾夕法尼亚大学访问。在《Journal of Combinatorial Theory, Series A》、《Proceedings of the American Mathematical Society》、《Advances in Applied Mathematics》、《SIAM Journal on Discrete Mathematics》等期刊发表学术论文20余篇，先后主持国家自然科学基金数学天元基金项目、青年基金项目、面上项目。