Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 246: Algebraic Combinatorics
Approved: 2009-03-01, Anne Schilling and Jesus De Loera
Units/Lecture:
Fall, alternate years; 4 units; lecture/extensive problem solving
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
Enumerative Combinatorics, Vol. II, R. P. Stanley ($140). See supplemental texts below.
Search by ISBN on Amazon: 978-0-521-78987-5
Search by ISBN on Amazon: 978-0-521-78987-5
Prerequisites:
MAT 245 or consent of instructor
Course Description:
Algebraic and geometric aspects of combinatorics. The use of structures such as groups, polytopes, rings, and simplicial complexes to solve combinatorial problems.
Suggested Schedule:
| Lectures | Sections | Topics/Comments |
|---|---|---|
| Depending on the instructor, different emphasis may be given to the various topics. | ||
| 1 | Chapter 7.1 | Ring of symmetric functions |
| 2 | Chapter 7.2 | Partitions and their orderings |
| 3 | Chapters 7.3 - 7.7 | Various bases of the ring of symmetric functions |
| 4 | Chapter 7.9 | Scalar product |
| 5 | Chapter 7.10 | Combinatorial definition of the Schur functions |
| 6 | Chapter 7.11 | RSK algorithm |
| 7 | Chapter 7.12 | Cauchy identity |
| 8 | Chapter 7.14 | Dual RSK |
| 9 | Chapter 7.15 | Classical definition of Schur functions |
| 10 | Chapter 7.15 | Littlewood-Richardson coefficients and Pieri rule |
| 11 | Chapter 7.16 | Jacobi-Trudi identity |
| 12 | Appendix A1.1, Sagan 3.4 | Knuth Relations |
| 13 | Appendix A1.2, Sagan 3.7 | Jeu de Taquin |
| 14 | Chapter 7.17 | Murnaghan-Nakayama rule |
| 15 | Chapter 7.18 | Characters of symmetric group |
| Further topics (as time allows) |
The following lectures are Jesus De Loera's recommendation for the second half of the course. The text is "Combinatorics and Commutative Algebra", by R. P. Stanley.
| Lectures | Sections | Topics/Comments |
|---|---|---|
| 16-20 | Chapter 0 | Basic commutative and homological algebra |
| 21 | Chapter 2 | Simplicial complexes and face lattices |
| 22-23 | Chapter 2 | F-vectors and h-vectors, Dehn Sommerville Equations |
| 24-25 | Chapter 2 | Face ring of a simplicial complex |
| 26-27 | Chapter 3 | Two special examples: simplicial polytopes; matroid complexes |
| 28 | Chapter 3 | Gorenstein face rings |
| 29 | Chapter 3 | Toric varieties and g-vectors |
| 30 | Chapter 3 | A proof of the Upper Bound Theorem |
Additional Notes:
"The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions," by Bruce Sagan ($55). "Combinatorics and Commutative Algebra," by R. P. Stanley ($80).