**Instructor:** Igor Pak

pak@math.ucla (add .edu at the end)

**Class schedule:** MWF 3:00 - 3:50 pm, MS 5148

**Office Hours:** M 4-5, MS 6125

**Textbooks:**

- 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.

- 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).

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.

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*Last updated 11/29/2013*.