**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.

- 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.

- 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).

- I. Pak, Catalan numbers.
- R. Stanley, Catalan exercises.
- R. Stanley, Bijective proof problems.

- 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*.