The Combinatorics Of Moments Of Ranks And Cranks Of Partitions

The theory of partitions is a fruitful field of number theory and combina-torics. Dyson’s rank and the Andrews-Garvan-Dyson crank are two fundamental statistics in the theory of partitions. These two statistics can be used to interpret the famous Ramanujan congruences. In2003, Atkin and Garvan introduced the moments of ranks and cranks of partitions in order to prove further identities and partition congruences. From the symmetry of ranks and cranks, it is clear that the odd moments are all zero. To study the odd moments of ranks and cranks, Andrews, Chan and Kim defined the positive moments of ranks and cranks in2013.The main object of this thesis is to provide combinatorial interpretations of moments of ranks and cranks. In2007, Andrews gave a combinatorial inter-pretation for the2k-th symmetrized moment η2k(n) of ranks by introducing the k-marked Durfee symbol. In this thesis, we first consider the k-th symmetrized positive moment ηk(n) of ranks which is defined as the truncated sum over pos-itive ranks of partitions of n. In this notation, the odd moments are nontrivial. The first result of this thesis is to give a combinatorial interpretation for ηk(n), which implies the interpretation of η2k(n) given by Andrews. The second result of this thesis is to give a combinatorial interpretation of the2k-th symmetrized moment μ2k(n) of cranks by introducing the k-marked Dyson symbol based on a representation of ordinary partitions given by Dyson. We prove that μ2k(n) equals the number of (k+1)-marked Dyson symbols of n. Besides, we show that there exists an infinite family of congruences for μ2k(n) and prove some congruences for μ2k(n) modulo5,7and11. By defining the pd-rank for the partitions with designated summands, we give a combinatorial interpretation of a congruence given by Andrews, Lewis and Lovejoy.This thesis is organized as follows. The first chapter is devoted to an intro-duction to the theory of partitions and the background of ranks and cranks. We present two of the most basic techniques for dealing with partitions:graphical representation of partitions and infinite product generating functions.In Chapter2, we consider the k-th. symmetrized positive moment ηk(n) of ranks of partitions of n. We give combinatorial interpretations of η2κ(n) and η2k-1(n). We show that for fixed k and i with1≤i≤k+1, ηak-1(n) equals the number of (k+1)-marked Durfee symbols of n with the i-th rank being zero and η2k(n) equals the number of (k+1)-marked Durfee symbols of n with the i-th rank being positive. In view of the symmetry of ranks, η2k(n) also equals the number of (k+1)-marked Durfee symbols of n with the i-th rank being negative. These results imply the interpretation of η2k(n) given by Andrews since η2k(n) equals η2k-1(n) plus twice of η2k(n). Moreover, we obtain the generating functions of η2k(n) and η2k-1(n).In Chapter3, we give a combinatorial interpretation of μ2k(n). We introduce the k-marked Dyson symbol generalized from a representation of ordinary parti-tions given by Dyson. We show that μ2k(n) equals the number of (k+1)-marked Dyson symbols of n. We then define the first full crank and the second full crank of a k-marked Dyson symbol. We show that there exists an infinite family of congruences for the first full crank function of k-marked Dyson symbols. This implies that for fixed prime p≥5and positive integers r and k≤(p-1)/2, there exist infinitely many non-nested arithmetic progressions An+B such that μ2k(An+B)=0(mod pr). We then use the second full crank to give combina-torial interpretations for congruences of μ2k(n) mod5,7and11.In Chapter4, we consider the partition function PD(n) which counts parti-tions of n with designated summands introduced by Andrews, Lewis and Lovejoy. Relying on Chan’s identity on Ramanujan’s cubic continued fraction and some identities on cubic theta functions, we derive the generating functions of PD(3n), PD(3n+1) and PD(3n+2). By introducing the pd-rank on the partitions with designated summands, we give a combinatorial interpretation of the congruence for PD(3n+2) proved by Andrews, Lewis and Lovejoy.