Combinatorial Proofs Of Partition Identities

The objective of this thesis is to give direct combinatorial proofs of partition identities, including bijective proof of Euler's partition theorem, involutive proof of the Lebesgue identity. Moreover, we use the conception of labeled partitions to get some results of some indices of colored permutations.This thesis is organized as follows. In Chapter 1, we give the background of partition theory, and introduce some definitions and notations that are used throughout the thesis. Meanwhile, we will present two of the most elemental tools for treating partitions: graphical representation of partitions and infinite product generating functions.Chapter 2 is concerned with Euler's partition theorem. In 1748, Euler gave his famous theorem on partitions with odd parts and distinct parts. Sylvester presented his celebrated bijective proof and gave Sylvester's refinement of Euler's partition theorem. There are other refinements of Euler's partition theorem due to Fine, Glaisher, Bousquet-Mélou and Eriksson. Furthermore, Bessenrodt and Zeng gave a full summary of Euler's partition theorem. Moreover Zeng found a generating function proof of a three-parameter refinement of Euler's partition theorem. By utilizing a variant of Bessenrodt's insertion algorithm, we present a new combinatorial proof of the unification of the refinements of Glaisher and Bousquet- Mélou- Eriksson.Chapter 3 is devoted to the Lebesgue identity. The Lebesgue identity can be obtained by Euler's formulae and Heine's transformation. It is a special case of q-Kummer (Bailey-Daum) sum which also can be obtained by Heine's transformation. Alladi and Gordon gave a bijective proof of the original Lebesgue identity. After that, Bessenrodt presented a bijective proof of an equivalent form of the Lebesgue identity by using Sylvester's bijection. Recently, Fu gives a combinatorial proof based on the insertion algorithm of Zeilberger. We will present an involution to prove the Lebesgue identity by generalizing Vahlen's involution. In Chapter 4, we apply the partition theory to the enumeration of statistic indices of colored permutations. By using labeled partitions, we derive the generating functions of the fmaj_k indices of colored permutations. By generalizing Chen-Xu's method, we get a combinatorial treatment of a relation on the q-derangement numbers. Finally, we obtain involutive proofs of Gessel-Simon formula and Adin-Gessel-Roichman formula.We conclude this thesis with the proofs of the Euler formula and Heine's transformation in the Appendix.