Teodor Petter Johannesson: Integer Partitions
Time: Wed 2021-12-15 09.00 - 10.00
Respondent: Teodor Petter Johannesson
Abstract: This thesis looks into some selected elements in the theory of integer partitions. The goal of the thesis is to explore methods to select partitions at random. Before these can be developed some preliminary theory is introduced. Ferrers diagrams are presented, a graphical representation for integer partitions. Then the theory of generating functions related to integer partitions is expanded on, which are useful for proving partition identities. Euler's pentagonal number theorem is presented and proven bijectively. This theorem is then used to derive a recurrence relation for the partition function. Other properties of the partition function \(p(n)\) are explored and the partition function \(p(n, k)\) is introduced. Different ways of ordering partitions are considered. Finally, the problem of selecting integer partitions at random is studied and algorithms for generating random partitions are developed. Difficulties in selecting partitions at random are tied to the elusive nature of the partition function \(p(n)\). A better understanding of algorithms generating random partitions could give insight into \(p(n)\).