整数分拆统计量的单峰性和单调性的研究

Integer partitions plays an important role in combinatorics and number theory. The statistics on integer partitions was first introduced by F. Dyson (who is known as a Fellow of Royal Society), G. Andrews (who is known as a fellow of the American Mathematical Society and a member of the National Academy of Sciences) etc. Since then, many other statistics on integer partitions were also well studied, such as rank, crank, the ospt-function, the spt-function and so on. But the monotonicity properties and unimodal properties on these statistics had been overlooked. Until 2013, Andrews, Dyson and Rhoades conjectured that the spt-crank has unimodality. This conjecture was confirmed by Chen, Ji and Zang in 2015. In 2014, Chan and Mao studied the monnotonicity property on rank, and using this property, Chan and Mao gave the upper-bound and lower-bnound on ospt(n). They also conjectured that the other statistics has similar properties. The main purpose of this project is to give the monotonicity properties on crank and the rank and crank on overpartitions. This yields an answer to Chan and Mao's conjecture. This project also mean to give an upper-bound and lower-bound on spt(n) and ospt(n).

整数分拆是组合数学以及数论中的重要的研究对象。整数分拆上统计量最早由英国皇家数 学会会员F.Dyson和美国数学会成员、美国科学院士G.Andrews等人提出。随后人 们陆续定义了许多分拆上的统计量，如秩、c-秩、ospt函数、spt 函数等等。但统计量的单调性这一重要性质却罕有问津。2013年Andrews和Dyson等人猜测spt-秩具有单调性 ，后在2015年由Chen，Ji和Zang给予证实。2014年，Chan和Mao研究了分拆秩的单调性，并利用这一单调性给出了ospt(n)的上下界估计，并猜想其他分拆统计量有类似结论。本项目主要目的是给出c-秩的单调性，以及overpartition上秩和c-秩的单调性，这样就回答了Chan和Mao的猜想。并可以利用这些单调性给出spt(n)和ospt(n)的精细估计。

整数分拆是组合数学的重要研究对象。组合序列的单峰性问题同样是组合学家的重点研究内容。本项目主要研究了整数分拆上的统计量的单调性和单峰性，得到了一系列整数分拆上统计量的单峰性和单调性，回答了Chan 和 Mao、 Zaleski 以及林丽双等人提出的公开问题。发表文章4篇，均发表在Adv. Math. 等高水平国际刊物上。