**论文题目：**Proof of the Andrews–Dyson–Rhoades conjecture on the spt-crank

**论文作者：**William Y.C. Chen , Kathy Q. Ji , Wenston J.T. Zang

**发表杂志：**Advances in Mathematics 270 (2015) 60–96

**文章内容：**

The spt-crank of a vector partition, or an S-partition, was introduced by Andrews, Garvan and Liang. Let denote the net number of S-partitions of n with spt-crank m, that is, the number of S-partitions (π1 , π2 , π3 )of n with spt-crank m such that the length of π1 is odd minus the number of S-partitions (π1 , π2 , π3 )of n with spt-crank m such that the length of π1 is even. Andrews, Dyson and Rhoades conjectured that is unimodal for any n, and they showed that this conjecture is equivalent to an inequality between the rank and crank of ordinary partitions. They obtained an asymptotic formula for the diﬀerence between the rank and crank of ordinary partitions, which implies for suﬃciently large n and ﬁxed m. In this paper, we introduce a representation of an ordinary partition, called the m-Durfee rectangle symbol, which is a rectangular generalization of the Durfee symbol introduced by Andrews. We give a proof of the conjecture of Andrews, Dyson and Rhoades. We also show that this conjecture implies an inequality between the positive rank and crank moments obtained by Andrews, Chan and Kim.

**所属实验室或研究中心：**教育部核心数学与组合数学重点实验室

**论文：**PDF