Students who are auditing this course are also expected to submit all homework sets and maintain at least 60% attendance overall.
Course Description
Permutations are fundamental objects in discrete mathematics, with deep connections to algebra, probability, and computer science. This graduate-level course explores the rich combinatorial structure of the symmetric group, focusing on both classical enumeration and modern structural theories. By integrating geometric, algebraic, and analytic techniques, students will investigate how permutations can be decomposed, restricted, and quantified. The course highlights two major themes: the study of permutation statistics and q-analogs, and the theory of pattern avoidance.
Prerequisites
This is a graduate-level course in mathematics. A solid foundation in undergraduate discrete mathematics, combinatorics, and abstract algebra is required. Familiarity with generating functions and basic group theory is highly recommended.
Course References
M. Bóna, Combinatorics of permutations, third edition, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2022.
D. Foata and G-N. Han, The q-Series in Combinatorics; Permutation Statistics, unpublished notes, 2011.
D. E. Knuth, The Art of Computer Programming: Sorting and Searching. Vol. 3, second edition, Addison-Wesley, Reading, MA, 1998.
Grading
Assignments: There will be biweekly assignments. This will contribute 40% to the total score.
Mid-Semester Examination: There will be a 120-minute written examination, which will contribute 30% to the total score.
End-Semester Examination: There will be a 180-minute written examination, which will contribute 30% to the total score.
To get an A grade, you must score atleast 80% overall. However, this doesn’t automatically guarantee an A.
Lectures
Homework Sets
The problem set will be updated over the semester. I will announce in the class or via email when I post a set here.
You should only submit the designated problem(s), but you are encouraged to try the rest as well.
To get the most out of this course, you are expected to spend at least thrice the amount of lecture hours on your own.