Arithmetic properties of overpartition functions with combinatorial explorations of partition inequalities and partition configurations

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2017.In this thesis, various partition functions with respect to `-regular overpartitions, a special partition inequality and partition con gurations are studied. We explore new combinatorial properties of overpartitions which are natural generalizations of integer partitions. Building on recent work, we state general combinatorial identities between standard partition, overpartition and `-regular partition functions. We provide both generating function and bijective proofs. We then establish an in nite set of Ramanujan-type congruences for the `-regular overpartitions. This signi cantly extends the recent work of Shen which focused solely on 3{regular overpartitions and 4{regular overpartitions. We also prove some of the congruences for `-regular overpartition functions combinatorially. We then provide a combinatorial proof of the inequality p(a)p(b) > p(a+b), where p(n) is the partition function and a; b are positive integers satisfying a+b > 9, a > 1 and b > 1. This problem was posed by Bessenrodt and Ono who used the inequality to study a maximal multiplicative property of an extended partition function. Finally, we consider partition con gurations introduced recently by Andrews and Deutsch in connection with the Stanley-Elder theorems. Using a variation of Stanley's original technique, we give a combinatorial proof of the equality of the number of times an integer k appears in all partitions and the number of partition con- gurations of length k. Then we establish new generalizations of the Elder and con guration theorems. We also consider a related result asserting the equality of the number of 2k's in partitions and the number of unrepeated multiples of k, providing a new proof and a generalization.MT201

Asymptotic formulas for stacks and unimodal sequences

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions. These results describe combinatorial properties, generating functions, and asymptotic formulas for the enumeration functions. We also prove several new asymptotic results that fill in the notable missing cases from the literature, including an open problem in statistical mechanics due to Temperley. Furthermore, we explain the combinatorial and asymptotic relationship between partitions, Andrews' Frobenius symbols, and stacks with summits.Comment: 19 pages, 4 figure

A unifying combinatorial approach to refined little G\"ollnitz and Capparelli's companion identities

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G\"ollnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called $k$-strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, we provide a unified combinatorial treatment of their results and shed more lights on the intriguing conditions of their companion to Capparelli's identities.Comment: This is the second revision submitted to JCTA in June, comments are welcom

A Schur function identity related to the (-1)-enumeration of self-complementary plane partitions

We give another proof for the (-1)-enumeration of self-complementary plane partitions with at least one odd side-length by specializing a certain Schur function identity. The proof is analogous to Stanley's proof for the ordinary enumeration. In addition, we obtain enumerations of 180-degree symmetric rhombus tilings of hexagons with a barrier of arbitrary length along the central line.Comment: AMSLatex, 14 pages, Parity conditions in Theorem 3 corrected and an additional case adde