Teaching

Math 155R

Meets: Wednesdays and Fridays during the Fall 2024 semester from 9am to 10:15am at Science Center 304.
Office hours: Tuesday 11:00-12:00, Friday 10:30-11:30 and by appointment

Course

Math 155R will be an introduction to enumerative and algebraic combinatorics. The course will introduce several classes of combinatorial objects (permutations, Dyck paths, trees…) as well as some classical methods used to enumerate them. The course will then cover topics related to the combinatorics of the representation theory of the symmetric group such as Young tableaux, the hook-length formula and the Robinson-Schensted correspondence.

References

The main references will be:

• Stanley: Enumerative Combinatorics I & II
• Bruce Sagan: The Symmetric Group
• Philippe Flajolet, Robert Sedgewick: Analytic Combinatorics.

Prerequisites: Basic linear algebra and also basic abstract algebra (such as from Math 122).

Lectures

• Lecture 1: Outline of the course
• Lecture 2: Combinatorial classes, generating functions
• Lecture 3: Some classical combinatorial objects
• Lecture 4: Tree structures
• Lecture 5: Lagrange Inversion formula
• Lecture 6: Cartier Foata monoids
• Lecture 7: The Transfer-matrix Method
• Lecture 8: Viennot's theorem
• Lecture 9: Labelled structures, exponential generating functions
• Lecture 10: Cayley's formula
• Lecture 11: Lindström–Gessel–Viennot lemma
• Lecture 12: Planar maps and Tutte recursion
• Lecture 13: Representation theory of finite groups
• Lecture 14: Examples of representations
• Lecture 15: Schur’s lemma, Maschke’s theorem
• Lecture 16: Conjugacy classes, orthogonality of characters
• Lecture 17: Character table
• Lecture 28: Specht modules
• Lecture 19: The Robinson–Schensted algorithm
• Lecture 20: Shadow diagrams, jeu de taquin
• Lecture 21: The Hook-length formula