Equalities And Inequalities For Some Partition Statistics

The theory of integer partitions is an important topic in combinatorics and number theory.By investigating statistics in integer partitions,properties of integer partitions are characterized by mathematicians.The main object of this thesis focuses on the following two aspects:（1）the properties of srank which was defined by Stanley,a members of the United States National Academy of Science;（2）the relations between some classical statistics in integer partitions and truncated sums of restricted partition functions.The aim of this thesis is to establish some equalities and inequalities for some partition functions.Let O（π）denote the number of odd parts of the partitionπ.In 2005,Stanley introduced a new integer partition statistic srank（π）=O（π）-O（π^′）,whereπ^′is the conjugate ofπ.Let p（r,m;n）denote the number of partitions of n with srank congruent to r modulo m.Stanley,Andrews,Swisher,Berkovich and Garvan established the generating functions,congruences and inequalities for p（0,4;n）and p（2,4;n）.Motivated by their works,in chapter 2,we establish the generating functions for p（r,m;n）with m=16,24 and deduce their asymptotic formulas.We also prove several equalities and inequalities which are refinements of some results due to Swisher.Recently,Merca established truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank.Motivated by Merca’s work,in Chapter 3,we deduce several new truncated series involving ranks,cranks and M₂-ranks for restricted partition functions.As corollaries,we obtain linear inequalities for ordinary partition functions and their ranks,cranks and pod functions and their M₂-ranks.In particular,we generalize some results proved by Merca.At the end of this thesis,we give a conclusion and pose two conjecures.