Combinatorial Theory (Math 206B, Winter 2023)

Instructor: Igor Pak
(see email instructions on the bottom of the page).

Class schedule: MWF 3:00 - 3:50 pm, MS 5137

Office Hours: M 4-4:50, MS 6125

Grading: The grade will be based on attendance, class participation (100%).

Difficulty: This is a graduate class in Combinatorics. Students are assumed to be fully familiar with undergraduate Combinatorics and Graph Theory (see Math 180 and Math 184 here). On top of that, I will assume fluency with undergraduate Probability and Linear Algebra.

Content

Much of the course will be dedicated to the study of Enumerative and Asymptotic Combinatorics, as well as some applications. Our main emphasis is on bijective and direct combinatorial proofs.

Reading

Textbooks:

Richard Stanley, Enumerative Combinatorics. Download vol. 1 here.
Richard Stanley, Catalan numbers, CUP
Philippe Flajolet and Robert Sedgewick (FS), Analytic Combinatorics. Download the book here.
Andrew Odlyzko, Asymptotic enumeration methods (AEM), in Handbook of Combinatorics, vol. 2, Elsevier, 1995, pp. 1063-1229. Download the survey here.
Sergey Kitaev, Patterns in Permutations and Words, Springer.
Caroline Klivans, The Mathematics of Chip-firing, CRC Press.

Articles:

M. Apagodu, D. Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences.
I. Pak, History of Catalan numbers, appendix to Stanley's book.
V. Vatter, Permutation classes, a survey.
C.H. Yan, Parking functions, a survey.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, parking functions rep.
D. Foata and M.-P. Schutzenberger, Major index and inversion number of permutations, symmetry of Inv and Maj.
I. Pak, When and how n choose k, expository article, 1998.
I.M. Gessel and D.L. Wang, Depth-first search as a combinatorial correspondence, 1979.
I.M. Gessel and B.E. Sagan, The Tutte Polynomial of a Graph, Depth-First Search, 1996.
N.L. Biggs, Chip-firing and the critical group of a graph, 1999.
C. Merino, Chip firing and the Tutte polynomial, 1997.
D. Chebikin and P. Pylyavskyy, A family of bijections between G-parking functions and spanning trees, 2005.
R.P. Stanley, Hyperplane arrangements, interval orders, and trees, PNAS, 1996.
R.P. Stanley, An introduction to hyperplane arrangements, survey, 2004 (also errata).

Websites and collections:

Slides:

I. Gessel, Combinatorial proofs of congruences, talk slides.
I. Pak, 321- vs. 312-avoidance.
I. Pak, Tree bijections, from labeled trees to parking functions.

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Last updated 1/23/2023.