Combinatorial Theory (Math 206, Fall 2013)
Instructor: Igor Pak
firstname.lastname@example.org (add .edu at the end)
Class schedule: MWF 3:00 - 3:50 pm, MS 5148
Office Hours: M 4-5, MS 6125
- Richard Stanley, Enumerative Combinatorics (EC), vol 1 (second edition), and vol. 2.
Download vol 1 from the author's website.
- Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (AC)
Download the book website.
- Andrew Odlyzko, Asymptotic enumeration methods (AEM), in Handbook of Combinatorics, vol. 2,
Elsevier, 1995, pp. 1063-1229.
Download the survey here.
Additional reading will be posted below.
Grading: The grade will be 100% based on homeworks.
Difficulty: This is a graduate class in Enumerative Combinatorics.
Students are assumed to be familiar
with undergraduate Combinatorics and Graph Theory.
Content and Schedule
We will follow various sections from the above survey and monographs.
- H. Wilf, What is an answer?, Monthly article.
- I. Pak, Partition bijections survey (see esp. Section 9.6).
- J. Dixon, The probability of generating the symmetric group, Math. Z. (1969).
- my old lectures on Dixon's theorem here (see lectures 3-6).
- J.W. Moon, Counting labelled trees, Canadian Mathematical Congress, 1970.
- D.B. Wilson, Generating random spanning trees more quickly than the cover time, original article using the LERW.
- O. Bernardi, On the spanning trees of the hypercube and other products of graphs, a bijective proof.
- E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, TCS (2003).
- R. Stanley, Log-Concave and Unimodal Sequences (1989), a survey.
- B. Sagan, Inductive proofs of q-log concavity, DM (1992).
- C. Krattenthaler, Combinatorial proof of the log-concavity of the sequence of matching numbers, JCTA (1996).
- J. De Loera, J. Rambau and F. Santos, Triangulations, Section 6.2 on triangulation of product of simplices.
(see table of content)
- I. Macdonald, An Elementary Proof of a q-Binomial Identity (1989), a followup on D. Zeilberger's One line proof... (expect the paper to be longer than one line).
- S. DeSalvo and I. Pak, Log-concavity of the partition function (2013).
- R. Stanley, Increasing and Decreasing Subsequences of Permutations and Their Variants (2005).
- J.M. Hammersley, A few seedlings of research (1972).
- A.M. Odlyzko and H. Wilf, The Editor's Corner: n Coins in a Fountain, an asymptotic formula whose proof uses Rouché's theorem.
- R. Proctor, Solution of Two Difficult Combinatorial Problems with Linear Algebra,
a simple proof of Sylvester's theorem in the Monthy
- J. Sylvester, Proof of the hitherto undemonstrated Fundamental Theorem of Invariants, Phil. Mag. (1878), the original article proving unimodality of q-bin coeff.
- A. Cayley, On a problem in the partition of numbers, Phil. Mag. (1857), the original theorem enumerating Cayley's compositions.
- M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections (2012), a bijective proof of Cayley's theorem.
- K. Mahler, On a special functional equation, J. LMS (1940), asymptotics for partitions into distinct powers; see also Don Knuth, An almost linear recurrence (1966), for comparison.
- N.G. de Bruijn, On Mahler's partition problem (1948).
Available at this Dropbox folder curtesy of Ted Dokos.
Here are all lectures in a single file.
Note: they are neither checked nor edited. Possibly, no one ever read them. The alleged jokes which appear in the notes never happened.
to return to Igor Pak Home Page.
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Last updated 11/29/2013.