Introduction to Enumerative Combinatorics (Fall 2010)
Pak, MS 6240 (subject to change), pak@math.
Class Schedule: MWF 1:00-1:50, MS 5217.
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 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.
- 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.
1) H.S. Wilf, "generatingfunctionology". Downloadable here.
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
- 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
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.
- HA1 (due Oct 8).
- HA2 (due Oct 25).
- HA3 (due Nov 12).
to return to Igor Pak Home Page.
To e-mail me click
here and delete .zzz
Last updated 9/20/2010