Q-Analogue And Combinatorial Proof Of Identities

By the combinatorial proof or combinatorial interpretation, the identity is equipped with certain count meaning. The most general way in the combinatorial proof is to count two sides of the identity by two different methods. Generaly, through building a bijection from one set to another one, the number of the two sets respectively represents the two sides of the identity. Because of the 1-1 property of bijection, the identity is proved. This thesis just applies this method to give identities their combinatorial proof. As a result, a general approach to combinatorial proof is illustrated.q-Identity is a certain class identity in combinatorial identity. Most q-identities are q-analogues of indentities with discrete variable, where q is a parameter. q-Analogue of identity has much count meaning. So giving combinatorial proof of q-identity is vary meaningful. In the thesis , we give combinatorial proof of some q-identities using integer partition.In the thesis some vital identities with binomial coefficient are given with their combinatorial proof, and the q-analogues of some identities are offered with corresponding combinatorial proof using integer partition. Especially, one remarkable result is that a new combinatorial proof of the square-sums identity is obtained. We also give combinatorial proof of the sums of even squares, odd squares and their q-analogues using integer partition.The main content of this thesis can be summarized as follows:1. Introduce relevant basic knowledge and concept about integer partition , such as Integer partition, Ferrer graph, Poset, Lattice, Combinatorial Proof, Binomial coefficient and so on.2. Some important identitis are provided with their combinatorial proof or combinatorial interpretation on the Boole Lattice B_n.3. Introduce the concept of q-analogue. q-Analogues of some important identities are given with their combinatorial proof using integer partition.4. We give a new combinatorial proof of the identity 。 We also give combinatorial proof of the sums of even squares, odd squares and their q-analogues using integer partitions.