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 structure. For instance, suppose you collect hats from N individuals and randomly re-distribute the hats back. The expected number of people who get their own hat back is 1, and the probability that no one gets their own hat back is about 1/e(!).

• Permutations and the Symmetric group
• Cycle structure of permutations and Poisson statistics
• The RSK bijection and Young tableaux
• Pattern avoidance and the Stanley-Wilf conjecture
• Permutation limits and Permutons

The aim of the project is for students to understand a topic about permutations according to their liking. 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".

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.