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-Knuth 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).

Tentative schedule

• 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
• Lecture 6: Cartier Foata monoids
• Lecture 7: The Transfer-matrix Method
• Lecture 8: Lindström–Gessel–Viennot llima
• Lecture 9: Labelled structures , exponential generating functions
• Lecture 10: Cayley's formula
• Lecture 11: Planar maps and Tutte recursion
• Lecture 12: Representation theory of finite groups
• Lecture 13: Characters
• Lecture 14: Frobenius formula, generating function of bipartite maps
• Lecture 15: Specht Modules
• Lecture 16: RS correspondence
• Lecture 17: Some properties of the RS correspondence
• Lecture 18: Jeu de Taquin
• Lecture 19: Hook-length formula
• Lecture 20: The space of symmetric functions
• Lecture 21: Schur functions
• Lecture 22: RSK correspondence for semi-standard tableaux and Cauchy identity
• Lecture 23: Jacobi-Trudi identity