MTH5153 - Combinatorics - Monash University

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Overview

Combinatorics is the study of arrangements and combinations of discrete objects. Combinatorial problems arise in many areas of pure mathematics, (e.g. algebra, probability, topology, and geometry), and in many applied areas as well (e.g. communications, operations research, experiment design, genetics, statistical physics etc). This unit will cover a selection of topics from the following list: combinatorial enumeration, ordinary and exponential generating functions, asymptotic enumeration, counting via matrix functions or group actions, the principle of inclusion-exclusion, Mobius inversion, permutations, partitions, compositions, combinatorial designs, Latin squares, Steiner triple systems, block designs, Hadamard matrices, finite geometries, algebraic combinatorics, strongly regular graphs, symmetric functions, Young tableaux, additive combinatorics and combinatorial geometry.

Rules

Enrolment Rule

Contacts

Unit Coordinator(s)

Dr Daniel Horsley

Professor Ian Wanless

Notes

This unit is offered in alternate years commencing Semester 1, 2020

Learning outcomes

On successful completion of this unit, you should be able to:

Formulate complex problems using appropriate combinatorial terminology.

Demonstrate a profound understanding of the benefits and challenges unique to working with discrete mathematical objects.

Recognise certain features of combinatorial problems which indicate their level of difficulty.

Apply sophisticated combinatorial arguments in a variety of settings.

Appreciate the role of combinatorics in other areas of mathematics.

Understand several real-world applications of combinatorics.

Teaching approach

Peer assisted learning

Problem-based learning

Active learning

Primary content is delivered through traditional lectures (which will be recorded, and available for review via the moodle page). Applied classes will be active learning. Small groups of students will work together around a whiteboard to solve problems. A key aspect is that this component involves writing your solutions on the board and critiquing the solutions of others, meaning that both written and spoken communication of mathematical ideas are developed (with input from your tutor). You also develop strategies for solving the questions together, meaning this is peer-assisted learning. Assignments will be problem-based learning. The questions will involve techniques from recent lectures and applied classes, but sometimes in settings or formats that are not immediately familar. Thus part of the challenge is adapting what you previously knew or have just learnt to a new setting.

Assessment

1 - Continuous assessment

This unit is offered at both Level 4 and Level 5, differentiated by the level of the assessment. If you are enrolled in MTH5153 you will be expected to demonstrate a higher level of learning in this subject than those enrolled in MTH4153. The assignments and exam in this unit will use some common items from the MTH4153 assessment tasks, in combination with several higher level questions and tasks.

Value %:40

2 - Examination (3 hours and 10 minutes)

Value %:60

Hurdle type:Threshold

Hurdle description:

If you would otherwise have passed the unit but you do not achieve at least 45% of the marks available for the end-of-semester examination you will receive a Hurdle Fail (NH) grade and a mark of 45 on your transcript.

Workload requirements

Workload

Learning resources

Recommended resources

Availability in areas of study

Master of Mathematics