Constructive Proofs Of Partition Identities And Q-Series Identities

The main goal of this dissertation is to constructively prove several well-known q-series identities and partition identities utilizing combinatorial method(i.e. bijection and involution), which contains a partial theta identity of Ramanu-jan, a partial theta identity of Andrews and its generalization, a partition theoremof Fine, two weight partition identity of Alladi, Ramanujan’s third order mocktheta functions identities and Andrews’ generalizations for them, and the alter-nating sum identity of Gauss coefcients. To start, we will translate both sides ofa q-series identity into generating functions of two types of partitions in the lan-guage of integer partitions, and represent these partitions graphically. We thenuse various combinatorial method to establish a bijection between the two typesof partitions, or construct an involution on partitions of the same type, whichwill give proofs of a q-series identity. Proofs in this way are called constructiveproofs, or combinatorial proofs.This dissertation is organized as follows. The first Chapter is devoted to in-troduce some definitions and notation in the theory of partitions and q-series, andthen describes two classic maps: Sylvester’s bijection and Franklin’s involution,which pioneer combinatorial proofs of partition identities and q-series identities.Chapter2is concerned with the partial theta series identities of Ramanu-jan and Andrews. Alladi deduced two weighted partition identities from them.In terms of parity sequence, Berndt, Kim and Yee gave a combinatorial proof ofRamanujan’s identity. By establishing a new involution Φ, we give another combi-natorial proof of Ramanujan’s identity. What’s more, based on this involution Φ,we obtain a Franklin type involution Ψ, which serves as a proof of the weightedpartition identity deduce by Alladi from Ramanujan’s identity and answer thequestion posed by Berndt, Kim and Yee in J. Combin. Theory Ser. A. Besides,this Franklin type involution Ψ also proves another weighted partition identity of Alladi derived from Andrews’ identity and Fine’s partition theorem. On theother hand, we construct a variant Υ of the Franklin type involution Ψ. Usingthis variant Υ, we provide a direct combinatorial proof of Andrews’ identity andanswer a problem of Andrews in Amer. Math. Monthly. We also find a moregeneral weighted partition identity.In Chapter3, we study Ramanujan’s third order mock theta functions identi-ties and Andrews’ generalizations for them. Fine firstly explored the connectionsbetween third order mock theta functions and integer partitions, and he obtaineda partition identity about the third order mock theta function f (q). We provideda proof of the partition identity of Fine through an involution. In addition, weconstruct another two involutions, and deduce two partition identities concerningthe third order mock theta functions φ(q) and ψ(q), respectively. In light of thethree partition identities, we prove two classical identities of Ramanujan aboutthe three third order mock theta functions. Furthermore, the three involutionsapply to Andrews’ generalizations for the two Ramanujan’s identities.Chapter4deals with the alternating sum identity of Gauss coefcients.Through studying Alladi’s sliding operation and hook operation on partitions,we obtain a weighted partition identity relating unrestricted partitions and par-titions into distinct odd parts. Based on this weighted identity, we prove thealternating sum identity of Gauss coefcients, and provide a combinatorial inter-pretation of the multiplicity κ(1n)of the irreducible representation S(1n)in theconjugate representation of the symmetric group Sn, which answers one questionof Stanley in a special case, posed in his book: Enumerative Combinatorics vol2. To end this chapter, we introduce two combinatorial proofs of the Gauss iden-tity. By reformulating the Gauss identity, Chen, Hou and Lascoux provided acombinatorial proof and derived two generalizations. Joon Yop Lee gave anothercombinatorial proof by involution.