The Combinatorial Study Of Hooks And Cranks For Partitions

The theory of partitions is an interesting and important field which is re-lated to group representation theory, probability theory and combinatorics. S-tatistics of partitions are main subjects in partition theory, especially "hooks" and "Andrews-Garvan-Dyson crank". Hooks are widely studied in representa-tion theory and algebraic combinatorics, and attract numerous mathematicians’ attention including the Fields medalist A. Okounkov. Cranks are brought out by F. Dyson and discovered by G.E. Andrews and F. Garvan. The crank is also a fundamental partition statistic which was originally used to interpret the famous Ramanujan congruences.The main objective of this thesis is to give direct combinatorial proofs for generating function formulas of partition functions concerning hook lengths and cranks. The methods used here primarily focus on transformations and invo-lutions on diagrams of partitions. We first present the generating function on the maximum number of κ-hooks of partitions of n. The second main result of the thesis is to give a combinatorial derivation of the generating function for the number of partitions of n with crank no more than a given nonnegative integer t. Besides, we combinatorially prove that the number of partitions of n with crank no more than t equals the number of partitions of n that have t-subpartitions with even lengths.The thesis is organized as follows. The first chapter is devoted to the in-troduction of the theory of partitions. We present the graphical representations and infinite product generating functions of partitions which are two of the most elementary tools for treating partitions, as well as some basic notation and useful concepts of integer partitions.The objective of Chapter2is to derive a generating function formula for the maximum number of κ-hooks in the Young diagrams of partitions of n. Let ακ(λ) denote the number of κ-hooks in a partition A and let b(n,κ) be the maximum value of ακ(λ) among partitions of n. Pak obtained the generating function for the statistic α1(λ) and Han showed a hook formula involving ακ(λ) respectively. Amdeberhan posed a conjecture on the generating function of b(n,1). We give a proof of this conjecture. In general, we obtain a formula that can be used to determine b(n, k). This leads to a generating function formula for b(n,k). We introduce the notion of nearly κ-triangular partitions. We also show that for any n, there is a nearly κ-triangular partition which can be transformed into a partition of n that attains the maximum number of κ-hooks. The operations for the transformation enable us to compute the number b(n, k).In Chapter3, we give a combinatorial proof of the generating function on the number of partitions of n with crank no more than a given nonnegative integer t. Let M(≤t;n) denote the number of partitions of n with crank no more than t and q(t;n) be the number of partitions of n whose rank-sets contain t. Dyson showed that M(≤t; n)=q(t;n) and obtained their generating functions algebraically. By constructing an involution on partitions, we give a direct combinatorial proof of the generating function of q(t; n) and then we obtain the generating function of M(≤t;n) immediately. Meanwhile, by a similar involution, we present a s-traightforward combinatorial treatment for an identity containing partial theta function of Andrews which was also investigated by Kim. Finally, we introduce the definition of the m-subpartition which is a generalization of the subpartition with gap1introduced by Kim. By m-subpartitions, we give a combinatorial in-terpretation of M(≤t;n) as the number of partitions of n having m-subpartitions with even lengths. In addition, we obtain the generating function of this partition function.