Combinatorial Theory (Math 206, Fall 2013)

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.

Content and Schedule

Additional reading:

• 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).

Course notes:

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.