Q-Analogues And Combinatorial Proof Of Binomial Coefficient Identities

The combinatorial proof of identities gives counting significance in identity. The most frequently used approach to prove identity is to build a bijection between two sets. The two sets respectively corresponding to the two ends of identity, according to the one to one correspondence of bijection, we can prove the identity. This paper used this method to prove identity.This paper gives some approaches to prove identity, as well as some combinatorial proofs of identities with binomial coefficient. One remarkable result is that q-analogues and combinatorial proof of two Norlund identities, which respectively generalizes the identity given by Bin Yang and Warnaar。The main content of this paper can be summarized as follows:(1) We introduce some basic knowledge about q-analogues of identities on the vector space, such as binomial coefficient, lattice, combinatorial proof, poset and so on.(2) In this paper, we present q-analogues of some classical identities with binomial coef-ficient. We also offer a general method of subset-subspace analogy and the multiset Mahonian statistics.(3) We give q-analogues of Norlund identity and transformed identity, and their com-binatorial proof on the Boolean lattice. Generalize the identity given by Bin Yang and Warnaar.