MATHS 326 : Combinatorics
Science
2025 Semester One (1253) (15 POINTS)
Course Prescription
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.
Capabilities Developed in this Course
| Capability 3: | Knowledge and Practice |
| Capability 4: | Critical Thinking |
| Capability 5: | Solution Seeking |
| Capability 6: | Communication |
| Capability 7: | Collaboration |
Learning Outcomes
- Display a high level of knowledge about combinatorial structures and their properties. (Capability 3)
- Understand and use important combinatorial tools such as generating functions and the probabilistic method. (Capability 3, 4 and 5)
- Demonstrate an understanding of duality and its uses in identifying optimal solutions to optimisation problems (Capability 3, 4 and 5)
- Understand and critically evaluate combinatorial aspects of a problem instance which allow an efficient algorithmic solution. (Capability 3 and 4)
- Develop and demonstrate an understanding of applications and connections to other disciplines. (Capability 3, 4 and 5)
- Be able to model and solve combinatorial problems both individually and as part of a team. (Capability 3, 5, 6 and 7)
Assessments
| Assessment Type | Percentage | Classification |
|---|---|---|
| Final Exam | 50% | Individual Examination |
| Test | 30% | Individual Test |
| Coursework | 20% | Group & Individual Coursework |
| 3 types | 100% |
| Assessment Type | Learning Outcome Addressed | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | |||||
| Final Exam | ||||||||||
| Test | ||||||||||
| Coursework | ||||||||||
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 decomositions 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.
Delivery Mode
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
Course materials are made available in a learning and collaboration tool called Canvas which also includes 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.
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.
Student Feedback
During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.
At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.
Your feedback helps to improve the course and its delivery for all students.
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:
- preliminary reading, available in advance,
- 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,
- another RAT based on the lecture contents towards the end of each topic,
- 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, tests and examinations 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. A student's assessed work may be reviewed against electronic source material using computerised detection mechanisms. Upon reasonable request, students may be required to provide an electronic version of their work for computerised review.
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.
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.
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 http://disability.auckland.ac.nz
Special Circumstances
If your ability to complete assessed coursework 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 https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.
This should be done as soon as possible and no later than seven days after the affected test or exam date.
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 your assessment is fair, and not compromised. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator, and if disruption occurs you should refer to the University Website for information about how to proceed.
The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.
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 https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html.
Disclaimer
Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.
In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.