报告题目： A Chung-Feller theorem for lattice paths with

respect to cyclically shifting boundaries

主 讲 人：郭军伟 教授

报告邀请人：孙怡东 教授

会议时间：2021年5月13日15:30-17:00

点击链接直接加入会议：https://meeting.tencent.com/s/nTGgdHXaa3zv

腾讯会议ID：198 706 477

报告摘要：

Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line x = ky (e.g. the Dyck paths when k = 1). On the other hand, the classical Chung–Feller theorem tells us that the number of lattice paths from (0, 0) to (n, n) with exactly 2k steps above the line x = y is independent of k, and is therefore the Catalan number 1/(n+1){2n\choose n}. In this talk, we study the number of lattice path boundary pairs (P, a) with k flaws, where P is a lattice path from (0, 0) to (n, m), a is a weak m-part composition of n, and a flaw is a horizontal step of P above the boundary ∂a. We prove bijectively, for a given a, that summing these numbers over all cyclic shifts of the boundary ∂a is equal to {n+m\choose m−1} . That is, we generalize the Irving– Rattan formula to a Chung–Feller type theorem. We also give a refinement of this result by taking the number of double ascents of lattice paths into account.

个人简介：

郭军伟，淮阴师范学院教授。1995年就读于南开大学数学试点班，1999年毕业，同年保送直接攻读博士学位，导师为陈永川教授，2004年获南开大学应用数学博士学位。随后去法国里昂第一大学做了一年半的博士后研究，合作导师为曾江教授。后在薛定谔国际数学物理研究所访问了三个月。曾任华东师范大学数学系教授，博士生导师。主要从事组合数学，q-级数和数论的研究。迄今为止共发表SCI论文110余篇，先后主持三项国家自然科学基金，以及上海市教育发展基金会晨光计划，上海市科委青年科技启明星计划，江苏省自然科学基金等项目，并入选江苏省教育厅"青蓝工程"中青年学术带头人。