Description

The theory of permutations touches several areas of mathematics such as discrete mathematics, group theory, representation theory and probability. Although permutations appear to be simple, they mask interesting mathematical structures underneath. For instance, suppose you collect hats from N individuals and randomly re-distribute the hats to them. The expected number of people who get their own hat back is about 1. We can even identify the (random) number of people who get their own hats back. How can we do this?

The aim of the project is for students to understand a topic about permutations according to their liking after we have covered some fundamental topics as a group. For instance, students interested in algebra can look into the RSK bijection and the connection between permutations and group representations. Students interested in analysis can investigate Poisson statistics of cycles in random permutations, or explore the notion of "the limit of a permutation".

Group project

The group project will revolve around learning basic concepts and results about permutations. By the end of the group project we would have learned:

• Permutations as a group
• Cycle structure of permutations
• Enumerating certain pattern avoiding permutations
• Permutations and generating functions

Mode of operation and evidence of leanring

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Individual project

The individual project will build on the knowledge you have gained in the group project and will explore additional advanced topics. A few examples of topics you will be able to investigate are:

• Poisson statistics of random permutations
• The RSK bijection and Young tableaux
• Pattern avoidance and the Stanley-Wilf conjecture
• Permutation limits and Permutons

Mode of operation and evidence of leanring

The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.

Prerequisites

Probability II is desirable but not required.

Resources

Students will be guided to lecture notes, textbooks or papers for various topics. The following textbooks will be helpful.

• M. Bona, Combinatorics of Permutations.
• R. Stanley, Algebraic Combinatorics.
• H. Wilf, Generatingfunctionology.

Get in touch by email if you have questions.