A Study of Integer Partitions and their Derivations

This M.Sc. thesis studies the partitions of integers, mainly restricted integers and how to derive them methodically. Analysis is made of different theories of calculating integers which are generating functions, Euler’s identity, McMahon’s recurrence, Sylvester’s approach, Frobenius partitions and generalized partitions. Chapter 1 shows how to obtain partition identities using Ferrer’s diagram, Durfee square and Jacobi’s triple product identity. The basic generation of partition of integers is considered first. This is followed by the expression of partitions using Ferrer’s diagram in chapter 2. In chapter 3, the number of partitions in a set of integers is calculated using the method of function generation. Using the preceding chapters, partition identities are obtained and further explained them in chapter 5 using Durfee squares and its relation to Ferrer’s diagram. Euler’s identity is proven combinatorically by means of bijection in chapter 6 and Euler’s pentagonal number is used to represent a special case of Jacobi’s triple product identity in chapter 7. When the pattern of a pentagonal number is notable, McMahon’s approach is used to generate functions to calculate partitions in restricted integers as discussed in chapter 8. The first Sylvester wave is defined which is an explicit formula for the polynomial part of a restricted partition function. The last three chapters looks at special cases in generalized partitions and use Euler’s result for identically distributed partitions.