The Combinatorics On Partition Functions

Partition function is an important research field in Combinatorics. The main objectof this thesis is researching some famous partition functions and combinatorial identi-ties via algebraic and combinatorial methods, mainly including the ordinary partitionfunction p n, Stanley’s partition function t(n), Euler’s partition theorem, q-Catalannumbers, Ramanujan’s theta functions and so on. The algebraic method is mainly touse elementary q-series manipulations, and the combinatorial method is mostly to usethe language of integer partition to explain the sets and the identities, then building themap between the partitions via the combinatorial tools.This thesis consists of five chapters. The first chapter is devoted to an introduc-tion to the background, the basic concepts and the usual notations of integer partition.Meanwhile, we will introduce some algebraic and combinatorial tools which shouldbe used during the research, such as Jacobi triple product identity, q-binomial theo-rem, Young diagram and so on. And we also give the definition and an example ofcombinatorial proof. At last, we shall introduce the outline of this thesis.In Chapter2, we will research the distributing property of some partition func-tions. Stanley defined a new partition function t(n)in2002. Then Andrews researchedthis function and raised several open problems in2004. In this chapter, we use ele-mentary q-series manipulations to obtain the generating function of the complementarypartition function u(n). Then we show that t(n)has the same parity as the ordinarypartition function p n. Furthermore, we present the combinatorial explanation of thisfact and show some conclusions concerning hook lengths. Yee obtained the generatingfunctions of the refined forms for partitions and answered the open problem of An-drews. In the last section of this chapter, we will provide several inequalities for Yee’sgenerating functions via combinatorial method.In Chapter3, we will particularly discuss Boulet’s four-parameter formula on gen-erating function for partitions in2006. In this chapter, we rectify the typos in Boulet’sformulas and obtain the generalizations which could be used to give the algebraic proofs of our partition results in Chapter4. In the last section of this chapter, we present therelation between our generalizations and q-Catalan numbers which we shall dwell onin Chapter5.In Chapter4, we will research the classic Euler’s partition theorem. Many famousmathematicians, such as Andrews, Bessenrodt, Fine, Glaisher and Sylvester, delvedinto Euler’s partition theorem and obtained the refinements and generalizations. Fur-thermore, we obtain a new partition theorem including Bessenrodt’s refinement andAndrews’s generalization relying on the results in Chapter3and elementary q-seriesmanipulations. We also give a combinatorial proof of our partition theorem which im-plies a stronger partition theorem. Analogously, we will provide another new partitiontheorem concerning alternating sum and the number of odd parts when the multiplici-ties of even parts is bounded by an odd number.In Chapter5, we shall study the combinatorial identities involving q-Catalan num-bers. We obtain some new identities involving q-Catalan numbers. And we also providethe combinatorial proof of one identity relying on the techniques about q-Catalan num-bers which is raised by Andrews. In2011, Andrews provided three identities involvingq-Catalan numbers which are separately q-analog of the three identities involving Cata-lan numbers in Koshy’s monograph, and also gave the partial combinatorial proofs. Inthis chapter, we present another partial combinatorial proofs of these identities.