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Overview

Course Prescription

Combinatorics is a branch of mathematics that studies collections of objects that satisfy specified criteria. An important part of combinatorics is graph theory, which is now connected to other disciplines including bioinformatics, electrical engineering, molecular chemistry and social science. The use of combinatorics in solving counting and construction problems is covered using topics that include algorithmic graph theory, codes and incidence structures, and combinatorial complexity.

Course Overview

This course is intended for students who have enjoyed basic combinatorics and discrete mathematics presented in MATHS 254 and COMPSCI 225. 

The course covers key methods in combinatorics including advanced counting techniques such as generating functions, and proof techniques such as the probabilistic method. Many of the covered topics have real-world applications and connections to other fields, in particular computer science. We cover graph connectivity which plays an important role in algorithmic considerations for real-world problems, and duality which is one of the cornerstones of combinatorial optimisation. The syllabus also includes incidence structures such as block designs, which play a role in coding theory and experimental design.

After successfully completing this course on combinatorics, students will have a solid foundation in this field of growing importance and be familiar with its connections to other fields, as well as some important applications. Students will also be well prepared for further graduate courses on the topic, such as MATHS 715 or MATHS 782.

Key Topics

  • Generating functions: formal power series, basic manipulations, symbolic method to count recursively defined combinatorial objects, solving recursions, transfer theorems for coefficient asymptotics
  • Graph connectivity: k-connectedness, structure of 2- and 3-connected graphs, min-cut max-flow and related results (Menger's theorem, König's and Hall's theorems, Dilworth's theorem), tree decompositions and tree-bramble duality
  • Combinatorial algorithms: the greedy heuristic (and how badly it may fail), minimum spanning tree, Huffman coding,  bipartite maximum matching, graph colouring (including the 5-colour theorem), efficiently solvable special cases of difficult algorithmic problems (leading up to parametrised complexity and Courcelle's theorem)
  • Extremal and probabilistic combinatorics: basic probabilistic methods (first moment method, alterations), Turan's theorem, Ramsey's theorem, Erdös-Ko-Rado theorem, Kraft's inequality and limits of data compression
  • Incidence structures: balanced incomplete block designs, necessary conditions including Fisher's inequality and the Bruck-Ryser-Chowla theorem, Steiner systems, affine and projective planes, mutually orthogonal latin squares

Workload Expectations

This course is a standard 15-point course and students are expected to spend 10 hours per week involved in each 15-point course that they are enrolled in. While different students will allot their time in different ways that match their skills and preferences, a common breakdown of this time per week is as follows: 

  • 4 hours for in class activities, such as lectures and team tasks, 
  • 3 hours for revision and studying with peers, and 
  • 3 hours for working on practice problems and preparing for the test and exam.

Course Prerequisites, Corequisites and Restrictions

Prerequisite

Locations and Semesters Offered

LocationSemester
City

Teaching and Learning

Campus Experience

  • The activities for the course are scheduled as a standard weekly timetable.
  • Attendance is required at scheduled activities such as readiness assurance tests and team tasks.
  • Attendance in lectures is expected.
  • Lectures will be available as recordings. Other learning activities will not be available as recordings.

Learning Resources

Taught courses use a learning and collaboration tool called Canvas to provide students with learning materials including reading lists and lecture recordings (where available). Please remember that the recording of any class on a personal device requires the permission of the instructor.

Additional Information on Learning Resources

Notes and lecture slides will be made available on Canvas. 

There is a broad collection of books in the library on these topics, as well as plenty of online resources; every section of the course notes contains recommendations of textbooks for further reading.

Copyright

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.


You may copy the course content for the purposes of private study or research, but you may not upload onto any third-party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

Other Information

This course uses a team based learning approach. After a preliminary topic devoted to revision of the basics, the course is divided into five main topics. Each topic has the same structure: 

  1. preliminary reading, available in advance,  
  2. a readiness assurance test (RAT) based on the reading at the beginning of the topic; this is a multi-choice quiz which students first complete individually, and then discuss in teams of 3-4 students,
  3. another RAT based on the lecture contents towards the end of each topic,
  4. a team task, where students are given a problem to solve in teams of 3-4 students.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Similarly, research students must meet the University’s expectations of good research practice. This requires:

  • Honesty - in all aspects of research work
  • Accountability - in the conduct of research
  • Professional courtesy and fairness – in working with others
  • Good stewardship – on behalf of others
  • Transparency – of research process and presentation of results
  • Clarity - communication to be understandable, explainable and accessible

For more information on the University’s expectations of academic integrity, please see the Academic Conduct section of the University policy hub.

Disclaimer

Elements of this outline may be subject to change. The latest information about taught courses is made available to enrolled students in Canvas.

Students may be asked to submit assessments digitally. The University reserves the right to conduct scheduled tests and examinations online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.


Assessment and Learning Outcomes

Course Learning Outcomes

CLO #OutcomeProgramme Capability Link
1
2
3
4
5
6

Assessments

Assessment TypeAssessment PercentageAssessment Classification

Additional Information on Assessment

There are no marked homework assignments for this paper.

The coursework component consists of readiness assurance tests, team tasks, and an engagement component. To get full marks on this engagement component, you are required to submit attempts to at least 5 of the practice questions in each chapter, or demonstrate engagement in another way such as asking/answering questions in class or on EdDiscussions, or discussing partial solutions to practice problems during office hours.

Assessment to CLO Mapping

Assessment Type123456

Student Feedback, Support and Charter

Student Feedback

Feedback on taught courses is gathered from students at the end of each semester through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback and respond with summaries and actions. Your feedback helps teachers to improve the course and its delivery for future students. In addition, class Representatives in each class can take feedback to the department and faculty staff-student consultative committees.

Additional Information on Student Feedback

The course contents have been significantly updated since last year. 

Class representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.

Tuākana

Tuākana is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at Tuakana Learning Communities.

Additional Information on Tuākana

Whanaungatanga and manaakitanga are fundamental principles of Tuākana Maths, a community of learning for Māori and Pasifika students taking mathematics courses. The Tuākana Maths programme provides workshops, drop-in times, and a space where Māori and Pasifika students can work alongside our Tuākana tutors and other Māori and Pasifika mathematics students.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website.

Wellbeing

We all go through tough times during the semester, or see our friends struggling. There is lots of help out there - please see the Support Services page for information on support services in the University and the wider community.

Special Circumstances

If your ability to complete assessed work is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due. If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page. This should be done as soon as possible and no later than seven days after the affected test or exam date.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter.

Student Academic Complaints and Disputes

Students with concerns about teaching including how a course is delivered, the resources provided, or supervision arrangements, have the right to express their concerns and seek resolution. The university encourages informal resolution where possible, as this is quicker and less stressful. For information on the informal and formal complaints processes, please refer to the Student Academic Complaints Statute in the Student Policies and Guidelines section of the Policy Hub.