Combinatorial Proof Of Binomial Identities And Partition Identities

In combinatorics, there are many different kinds of identities, which together constitute an integral part of combinatorial mathematics. there are many experts and scholars studying their properties and proof and so on, among which the proof of identities has always been a very hot research topic. As we know, there are so many identities in combinatorics and their proof methods are also so many. This master thesis mainly study the proof of binomial identities and partition identities, unlike in the past, here we use combinatorial method to study the proof of some common identities of these two kinds, and it is necessary for us to have a good understand of combinatorial proof ideas. Specifically, this paper has done the following work:In the preface, we mainly introduces the domestic and foreign research status about the proof of combinatorial identities and we review some previous work done by predecessors and some important results obtained by them.In the second chapter, we mainly expound some fundamental concepts, properties and theorems, such as mapping, the binomial theorem, binomial coefficients, combination, combination with repetitions, integer partition, partition identities, combinatorial proof.In the third chapter, the combinatorial proof of binomial identities is studied systematically. To be specific, we divide the research work into three categories, namely, combinatorial proof of identities of combination without repetitions, combinatorial proof of identities of combination with repetitions, as well as combinatorial proof of binomial identities with alternating sign. In addition, we use a proven binomial identity with alternating sign to prove the famous principle of inclusion and exclusion, this proof can be considered an important application of binomial identities.In the fourth chapter, we study the combinatorial proof of some common integer partition identities, we divide them into two research content, wherein the first section mainly introduces combinatorial proof of some basic properties and theorems about integer partition, the second section gives the combinatorial proof of some other common partition identities.In the fifth chapter, we summarize some main work of this thesis, and review the conclusion or original work that we make, and present the prospects on further study of combinatorial proof of identities.