题目：Overpartitions and Bressoud’s conjecture
报告人：季青教授，天津大学
时间：2020/07/15 周三 10：00-11：00
腾讯会议 ID：629 623 584
摘要：In a memoir in 1980, Bressoud obtained an analytic generalization of the Rogers-Ramanujan-Gordon identities based on Andrews' generalization of Watson's q-analogue of Whipple's theorem. Let j = 0 or 1, Bressoud also defined two partition functions $A_j$ and $B_j$ depending on multiple parameters as combinatorial counterparts of his identity, where the function $A_j$ can be viewed as the generating function of partitions with certain congruence conditions and the function $B_j$ can be viewed as the generating function of partitions with certain difference conditions. Bressoud conjectured $A_j$=$B_j$ which specializes to most of the well-known theorems in the theory of partitions, including Euler's partition theorem, the Rogers-Ramanujan-Gordon identities and the Andrews-Gollnitz-Gordon identities. Special cases of his conjecture have been subsequently proved by Bressoud, Andrews, Kim and Yee. Recently, Kim proved that Bressoud's conjecture is true for the general case when j = 1.
In this talk, we introduce a new partition function $\overline{B}_j$ which could be viewed as an overpartition analogue of the partition function $B_j$ introduced by Bressoud. By constructing bijections, we showed that there is a relationship between $\overline{B}_1$ and ${B}_0$ and a relationship between $\overline{B}_0$ and ${B}_1$. Based on these two relations, we could obtain overpartition analogues of many classical partition theorem including Euler's partition theorem. In particular, we prove Bressoud's conjecture for j = 0 by establishing an overpartition analogue of Bressoud’s conjecture for $j =1$. The generating function of the overpartition analogue of Bressoud’s conjecture is also obtained with the aid of Bailey pairs.
报告人介绍：季青，天津大学教授，2007年于南开大学获得理学博士学位，主要从事q-级数和整数分拆理论的研究，先后在Adv. Math.、J. Reine Angew. Math.、Trans. Amer. Math. Soc.、J. Combin. Theory A等期刊上发表论文20篇。先后主持国家自然科学基金项目优秀青年基金项目和青年基金项目，参与创新群体项目和重点项目。
