Congruence Properties For Three Kinds Of Restricted Partition Functions

The theory of congruence properties for restricted partition functions is one of the most active fields in contemporary combinatorics.It has received much attention as it is strongly connected to diverse areas of mathematics such as q-series,number theory,algebra,automated proving.Moreover,it plays an important role in the field of mathematics,physics,probability theory and computer science.In recent years,although a large number of congruences for restricted partition functions have been discovered,there are still many problems need to be solved.In this thesis,we establish the congruences of generalized Frobenius partitions with six colors,restricted binary partition and Overpartitions.The specific work is as follows:In Chapter 1,we introduce the research background,the research progress of restricted partition functions and the main work of this paper.In Chapter 2,we first establish the generating function for)13(6fnc(10)due to Hirschhorn.Then we prove some congruences modulo powers of 3 and confirm a conjecture on a congruence modulo 243 for)(6fnc by utilizing the block formula and q-series operation.In Chapter 3,by using the method of algebraic combination and the operation of q-series,we prove a large number of congruences modulo powers of 2 and 3 for nW)(.In particular,we solve the open problem presented by Lan and Sellers.In chapter 4,we first establish the generating function of (?)(5n)by using the method of computer algebra and the theta function identity,and then we establish the infinite family of congruences modulo 5 and 9 for Overpartitions by utilizing the theory of quadratic residue.Moreover,we generalize some results due to Treneer and Chen,Sun,Wang and Zhang respectively.