Title of Project: Topics in Combinatorics

Code: RB2122A

Description: Combinatorics is the branch of mathematics concerned with the theory of arranging or selecting finitely-many objects according to specified rules. The objects can be material (such as people in a group or cards from a pack) or abstract (such as numbers, symbols or shapes). A primary aim of combinatorics, when applied to particular cases, is to determine the number of arrangements or selections, but without actually listing them. Accordingly, combinatorics primarily involves the theory of counting or enumeration. This project will first study the enumeration of various fundamental discrete mathematical structures, including permutations of finite sets and multisets, combinations of finite sets and multisets, partitions of finite sets, and partitions and compositions of integers. It will then study the enumeration of a range of further mathematical objects, such as plane partitions, tableaux, graphs and lattice paths. The methods for enumeration will involve bijective arguments, recurrence relations and generating functions. The project will include guided reading of relevant texts, completion of combinatorial exercises, and computer calculations and simulations. There will be some flexibility, based on the preferences of the student, in the choice of the combinatorial topics and methods which are studied.

Type: Single 10 credit module, Autumn semester

Supervisor: Prof. R. Behrend

Prerequisite 2nd year modules:

MA2008/2058 Linear Algebra II

MA2011 Introduction to Number Theory I

MA2013 Algebra 1: Groups

Prerequisite 3rd year modules for concurrent study:

MA3013 Algebra 2: Rings

Maximum number of students: 2