Introduction to Enumerative Combinatorics (Fall 2010)

Instructor: Igor Pak, MS 6240 (subject to change), pak@math.

Class Schedule: MWF 1:00-1:50, MS 5217.

Brief outline

This course is aimed to be a standard introduction to the subject. I will cover a large range of topics including enumeration of trees, linear extensions of posets, Young tableaux, integer partitions, etc. I will try to show a number of tools and ideas to give a basic picture of the field.

The prerequisites for the course are standard undergraduate linear algebra and combinatorics. I will assume that you have seen basic generating functions and some combinatorial bijections before, or are willing to learn on the fly. The rest will be self-contained.

Content:

• Enumeration using generating functions. Lagrange inversion.
• Enumeration using bijections. Trees, Catalan numbers, permutation statistics.
• Inclusion-exclusion, derangements, rook polynomials.
• Posets, Mobius inversion, linear extensions, distributive lattices, Fundamental Theorem.
• Integer points in polyhedra, Ehrhart polynomial, Stanley's P-partition theorem.
• Partitions, Euler's Pentagonal Theorem, Jacobi's Triple Product Identity, Rogers-Ramanujan identities, involution principle.
• Plane partitions, MacMahon's formula.
• Young tableaux, Robinson-Schensted-Knuth correspondence, hook-length formula.

If you are taking this course for credit, there will be a number of homeworks, which you will have to do. The homeworks will be posted on this page.

Course textbooks:

2) R.P. Stanley, "Enumerative Combinatorics", vol 1 and 2, Cambridge University Press. See also new version of vol 1.
3) B. Sagan, "The Symmetric Group", Springer.

We will not use Wilf's book - it is listed here to give you some background in g.f. Stanley's and Sagan's books are really important and you will need them in more advanced courses as well.

• Wilf's What is an answer? Monthly article.
• Either the second half of section 1.3 in Stanley, or section 4.1 in my When and how n choose k popular article.
• For more on parking functions, read here (see also Exc. 5.49 in Stanley)
• Stanley's lecture notes on hyperplane arrangements.
• The proof of unimodality of Gaussian coefficients is based on this paper. A readable outline of a combinatorial proof is given here. A stronger (q-log-concavity) result is proved here A general introduction to unimodality and log-concavity, especially in the enumerative combinatorics context, is given in this survey
• Tutte polynomial and its various applications is very well explained in "Modern Graph Theory" by Béla Bollobás (Chapter X). A survey by Ellis-Monaghan and Merino is also nice and quite readable. If you feel like learning what happens in a more general context of general matroids, read the "The Tutte polynomial and its applications" survey by Brylawski and Oxley. The proofs I sketched of the recurrence relations for inversion and Tutte polynomials, and (to be explained in the next class) the application to g.f. follows these papers by this classical paper by Mallows and Riordan, and this more recent paper by Gessel.

Home Assignments:

• HA1 (due Oct 8).
• HA2 (due Oct 25).
• HA3 (due Nov 12).