报告人:张文静（湖南大学）

日  期: 2019年10月12日

时  间: 14:00

地  点: 理科楼 LD202

摘  要: Let N(m,n) denote the number of partitions of n with rank m, and let

M(m,n) denote the number of partitions of n with crank m. Chan and Mao proved that for any nonnegative integers m and n, N(m,n)>=N(m+2,n) and for any nonnegative integers m and n such that n>=12, n\neq m+2, N(m,n)>=N(m,n-1). Recently, Ji and Zang showed that for n>=44 and 1<=m<=n-1, M(m-1,n)>=M(m,n) and for n>=14 and 0<=m<=n-2, M(m,n)>=M(m,n-1). In this paper, we analogue the result of Ji and Zang to overpartitions. Note that Bringmann, Lovejoy and Osburn introduced two type of cranks on overpartitions, namely the first residue crank and the second residue crank. Consequently, for the first residue crank \overline{M}(m,n), we show that

\overline{M}(m-1,n)>=\overline{M}(m,n) for m>=1 and n>=3 and \overline{M}(m,n)>=\overline{M}(m,n+1) for m>=0 and n>=1. For the second residue crank \overline{M2}(m,n), we show that \overline{M2}(m-1,n)>=\overline{M2}(m,n)

for m>=1 and n>=0 and \overline{M2}(m,n)>=\overline{M2}(m,n+1) for m>=0 and n>=1. Moreover, let M_k(m; n) denote the number of k-colored partitions of n with k-crank m, which was defined by Fu and Tang. They conjectured that when k>=2, M_k(m-1,n)>=M_k(m,n) except for k = 2 and n = 1. With the aid of the inequality \overline{M}(m-1,n)>=\overline{M}(m,n) for m>=1 and n>=3, we confirm this conjecture. This talk is based on joint work with Wenston J.T. Zang.

报告人简介：张文静，博士毕业于天津大学应用数学中心，师从著名组合数学家陈永川院士。现于湖南大学工作，目前主要研究overpartition上各种秩的算术性质、单峰性及其与Mock Theta函数的关系等，已有研究工作发表在J. Number Theory、Ramanujan J.等数论与组合方向的权威期刊上。

公司联系人:傅士硕

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