Q-Analogues Of Binomial Coefficient Identities

Let B_n be the Boolean lattice of subsets of an n-element set {1,2,…,n},and V_n(q) a n-dimensional vector space over the finite field GF(q) with q elements,L_n(q) its lattice of subspaces.The q-analogue between L_n(q) and B_n means that some qualities and identities on the Boolean lattice B_n are extended onto the lattice of subspaces L_n(q),then their q-analogues are discovered on the lattice of subspaces L_n(q),where q is an parameter.While taking the limit q→1,the q-analogues become corresponding qualities on the Boolean lattice B_n.By the combinatorial proof,the identity is equipped with certain count meaning.The most general way in the combinatorial proofs is to count two sides of the identity by two different methods.Generally,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 one-to-one property of bijection,the identity is proved.This thesis just applies this method to give the combinatorial proof of identities.In the thesis some classical identities with binomial coefficient are given with their combinatorial proof,and the q-analogues of some identities are offered with corresponding combinatorial proofs on the lattice of subspaces.Especially,one remarkable result is that q-analogues of three binomial coefficient identities are obtained with their combinatorial proof on the vector space.The main content of this thesis can be summarized as follows:1.Introduce some basic knowledge about the q-analogue,such as poset,lattice,combinatorial proof,binomial coefficient and so on.2.Some classical identitis are provided with their combinatorial proof on the Boolean lattice B_n.3.Introduce the concept of q-analogue,q-analogues of some classical identities are given with their combinatorial proofs on the vector space.We also offer a general method of subsetsubspace analogy and introduce the multiset Mahonian statistics.4.q-Analogues of three identities are obtained with its combinatorial proofs on the vector space.