Congruence Properties Related To Partition Statistics

In this thesis,we mainly study a variation of the Andrews-Stanley partition function,the weighted generalized crank moment of the k-colored partitions,and we also give a combinatorial proof of the two beautiful identities of Chern.The research contents are as follows:In Chapter 1,we introduce the research background on partitions congruence and the two basic statistics in the theory of partitions,namely,rank and crank.Then we go on expound the research status,the main contents of this thesis and research methods applied in this thesis.In Chapter 2,based on the Andrews-Stanley partition function,we introduce a new partition function related to partition statistic lrank(?)=O(?)+O(?'),where ?' is the conjugate of ?.Let pi+(n)denote the number of partitions of n with lrank?i(mod 4),where i=0.2.We obtain the generating functions of p0+(n)and p2+(n)and show that they satisfy similar properties to pi(n).In addition,we also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences p0+(5n+4)?p2+(5n+4)?0(mod 5).In Chapter 3,two beautiful identities of Chern are simply established again by com-binatorial method.In Chapter 4,we extend the combinatorial method of proving the results of Chern to the study of k-colored partitions.As a consequence,we derive a large number of new Andrews-Beck type congruences for k-colored partitions.