MATH5505 – Advanced Combinatorics

MATH5505 Advanced Combinatorics was given as an Honours level course in pure mathematics at the University of New South Wales in 1997. The lecturer was Norman Wildberger. David Angell attended the course, wrote up course notes and added some extra material. The course notes here provided are therefore due primarily to Norman and secondarily to David. They are posted with Norman's permission. Notes are also included of individual lectures by George Szekeres and Ben Hutchinson.

A similar course was given by David in 1999, some extra topics being added. These appear in Chapter 7.

The notes include a number of problems and "extra problems". The problems are due to Norman; all have well-defined solutions, and they were set as assessment for students taking the course. The extra problems are due to David; many of them are speculative and at least one of them is wrong (in the sense that it asks for a proof of something which is not in fact true).

The lecture notes are in the form of PostScript files. You will need Ghostview to access them. To download the version of Ghostview currently used at the School of Mathematics, go back to my home page.

• Chapter 1, Introduction: Hall's Marriage Theorem; pigeonhole principle; multisets; calculus of differences.
• Chapter 2, Graph Theory and Ramsey Theory: graph theory; algorithms in graphs; Ramsey theory.
• Chapter 3, Order Structures: partial orders; lattices; ideals; Möbius inversion.
• Chapter 4, Group Theory and Symmetric Functions: Stirling numbers; Pólya counting; tableaux; partitions; Catalan numbers; symmetric functions.
• Chapter 5 is omitted but may be posted at some stage.
• Chapter 6, Representations of Groups: characters of abelian groups; hypergroups.
• Chapter 7, Miscellaneous Topics: more on Hall's Marriage Theorem and the pigeonhole principle; generating functions; Rota's method of linear functionals; more on Ramsey theory.