University of St Andrews University of St Andrews

MT5861: Advanced Combinatorics

Please email me with questions, or request a
meeting on Teams if you think it would help.

Definition of basis
I am grateful to Jiaping Lu for pointing out
a mistake in the definition in the slides
for Week 5. This is now fixed.
In fact the two definitions are equivalent:
can you prove this?

The next presentation is scheduled for 2020-21, Semester 2 (commencing January 2021).

There are three syllabuses for this module, available here. In 2020-21, the second syllabus, focussing on combinatorial polynomials, will be covered.

Course material from previous presentations of the module can be found here.

As you are aware, teaching (including both lectures and tutorials) will be on-line for the foreseeable future. I am something of a beginner at on-line teaching. I prefer to give lectures live, and to give you the possibility of interacting with me during the lectures. But occasionally it may be necessary to pre-record a lecture.

Assessed coursework

Marked solutions to Coursework 2 have been uploaded to MMS today (13 April). Sample solutions are available here.

Marked solutions to Coursework 1 have been uploaded to MMS. Sample solutions are here.


Lectures are 12:00–13:00 on Mondays (odd weeks), Wednesdays and Fridays. The current plan is that the topics covered will be as in the table below. This may change according to audience requests!

1Coding theoryhere here 1, 2, 3
2Countinghere here 1, 2
3Graph colouringhere here 1, 2, 3
4Symmetryhere here 1, 2
5Matroidshere here 1, 2, 3
6Tutte polynomialhere here 1, 2
7Matroids and codeshere here 1, 2
8Cycle indexhere here 1, 2, 3
9IBIS groupshere here 1, 2, 3
10Miscellaneahere here 1, 2

Notes: The captions on the videos have not been edited. Please don't take them too seriously! I think only "Team" members can see them – let me know if this is a problem.

Supplementary material

This is not an examinable part of the module, but is there for your interest, if you want to learn a little more about the topics covered.

I'm sorry that these ground to a halt; life got busy. I talked briefly about the Moonshine Conjectures on 14 April: you can find some notes here.


Coursework sheets will be put here. You are welcome to hand in written work to be marked. I suggest on each sheet a maximum number of questions you should do. My solutions will be posted after the hand-in date.

I hope to post a sheet of miscellaneous problems here shortly; these might be useful for your revision. There will be no specific hand-in date but I am as usual happy to look at your solutions.

Tutorials will start in Week 2. At present there are two times reserved for tutorials: Mondays 15:00–16:00 and Tuesdays 11:00–12:00. Please come to the tutorials prepared to ask questions about anything related to lectures, coursework, or combinatorics more generally.

Exam resources

You requested some indication of what would be expected in the exam. Here is what I have in the way of past exam papers from Syllabus 2 of the module. Please remember that when these papers were set the module was worth 20 credits; the content has been reduced for the current 15-credit version. Also, the 2015 exam paper has gone missing, but I have provided a preliminary version of questions, some of which were on the paper. Finally, in the current circumstances, we have been told to set exams which will test understanding rather than memory, so the style may be a little bit different.


Unlike my normal practice, I will make lecture notes available in advance of the lectures, so as to give you maximum flexibility in planning your time. But as usual I strongly advise you to attend the lectures, and then make use of the notes or slides to clarify things that were not clear from the lectures. Warning: The notes or slides may change, subject to correction of mistakes and comments from the audience!

Unlike in previous years, assessment will be 20% coursework and 80% examination.There will be coursework sheet provided every week. These are optional except for two which will be clearly indicated in advance, which will be each worth 10% of the overall mark for the module.

Please do not hesitate to contact me by email if you have any questions.

Peter J. Cameron
16 April 2021