Bijective proofs of partition identities and covering systems
type: CreativeWork
description: This dissertation involves two topics. The first is on the theory of partitions, which is discussed in Chapters 2 − 5. The second is on covering systems, which are considered in Chapters 6 − 8. In 2000, Farkas and Kra used their theory of theta functions to establish a beautiful theorem on colored partitions, and they asked for a bijective proof of it. In Chapter 2, we give a bijective proof of a more general partition identity, with the Farkas and Kra partition theorem being a special case. We then derive three further general partition identities and give bijective proofs of these as well. The quintuple product identity is one of the most famous and useful identities in the theory of theta functions and q-series, and dates back to 1916 or earlier. In his recent survey paper on this identity, Shaun Cooper remarked that there does not exist a bijective proof of it. In Chapter 3, employing bijective proofs of Jacobi’s triple product identity and Euler’s pentagonal number theorem, we provide the first bijective proof of the quintuple product identity. In a recent paper, The parity in partition identities, George Andrews investigated parity questions in partition identities and listed 15 open problems at the end of his paper. In Chapter 4, we provide solutions to the first two open problems suggested by Andrews. More precisely, we provide combinatorial proofs of two partition identities which were derived by comparing Andrews’ new identity with G¨ollnitz-Gordon identities or certain generalizations thereof. In our last chapter on partitions, Chapter 5, we give a combinatorial proof of a companion to Euler’s famous recurrence formula for the sum of divisors function. Euler’s recurrence formula had previously been combinatorially proved using a double counting argument, but its equally famous companion has not heretofore been established combinatorially. We not only provide such a combinatorial proof, but we also give a combinatorial proof of a vast generalization as wel…
abstract: This dissertation involves two topics. The first is on the theory of partitions, which is discussed in Chapters 2 − 5. The second is on covering systems, which are considered in Chapters 6 − 8. In 2000, Farkas and Kra used their theory of theta functions to establish a beautiful theorem on colored partitions, and they asked for a bijective proof of it. In Chapter 2, we give a bijective proof of a more general partition identity, with the Farkas and Kra partition theorem being a special case. We then derive three further general partition identities and give bijective proofs of these as well. The quintuple product identity is one of the most famous and useful identities in the theory of theta functions and q-series, and dates back to 1916 or earlier. In his recent survey paper on this identity, Shaun Cooper remarked that there does not exist a bijective proof of it. In Chapter 3, employing bijective proofs of Jacobi’s triple product identity and Euler’s pentagonal number theorem, we provide the first bijective proof of the quintuple product identity. In a recent paper, The parity in partition identities, George Andrews investigated parity questions in partition identities and listed 15 open problems at the end of his paper. In Chapter 4, we provide solutions to the first two open problems suggested by Andrews. More precisely, we provide combinatorial proofs of two partition identities which were derived by comparing Andrews’ new identity with G¨ollnitz-Gordon identities or certain generalizations thereof. In our last chapter on partitions, Chapter 5, we give a combinatorial proof of a companion to Euler’s famous recurrence formula for the sum of divisors function. Euler’s recurrence formula had previously been combinatorially proved using a double counting argument, but its equally famous companion has not heretofore been established combinatorially. We not only provide such a combinatorial proof, but we also give a combinatorial proof of a vast generalization as wel…
author: Kim, Sun
language: en
keywords: partitions, bijective proofs, covering systems
date published: 2010-05-14T20:51:58Z
date modified: 2026-07-22T22:25:08Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/15597