Combinatorial Theory (Math 206A, Fall 2020)
Instructor: Igor Pak
(see email instructions on the bottom of the page).
CCLE Website: is here.
Class schedule: MWF 2:00 - 2:50 pm, via Zoom.
Zoom meeting link and password will be sent by email from the course my.ucla site.
Office Hours: M 3-4, via Zoom (stay after class).
Grading: The grade will be based on attendance, class participation (20%),
and homeworks (80%) which will be posted below.
Difficulty: This is a graduate class in Combinatorics.
Students are assumed to be fully familiar
with undergraduate Combinatorics and Graph Theory
(see Math 180
and Math 184).
Content
Much of the course will be dedicated to the study of partially ordered sets,
their properties, examples and many applications. We will follow various sections from the textbooks
and surveys below.
Lecture notes: Will be posted on CCLE website if and when they come available.
Reading
These are very preliminary, the exact list will be updated and expanded weekly with specific
sections indicated. Note: the readings not linked below will be available from the
CCLE website (see "Additional Reading" section).
Textbooks:
- Richard P. Stanley, Enumerative Combinatorics (EC), vol 1 (second edition), and vol. 2.
Download vol 1 from the author's website.
- Douglas B. West, Combinatorial Mathematics, see Ch. 12 on CCLE website.
- Stasys Jukna, Extremal combinatorics. With applications in computer science
(second edition), Springer, 2011, see Ch. 9 on CCLE website.
- William T. Trotter, Combinatorics and partially ordered sets.
Dimension theory, Johns Hopkins University Press, 1992, see Ch. 1 on CCLE website (to be posted).
- Ph. Flajolet and R. Sedgewick, Analytic Combinatorics, CUP, 2009, available from here.
- R. Diestel, Graph Theory, Springer, 2016 (5th edition).
None of these are required, all are recommended. Selected relevant chapters will be posted on CCLE.
Surveys:
- C. Greene and D.J. Kleitman,
Proof techniques in the theory of finite sets, in Studies in combinatorics,
MAA, 1978, 22-79 (huge file on CCLE).
- G. Brightwell, Models of random partial orders, in Surveys in combinatorics, BCC, 1993, 53-83.
- G. Brightwell and D. West, Partially Ordered Sets, Chapter 11 in
Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000, 717-752.
- W.T. Trotter, Partially ordered sets, Chapter 8 in Handbook of combinatorics, Elsevier, 1995, 433-480.
Articles:
- R.P. Stanley, Two poset polytopes, Discrete and Computational Geometry, 1986.
- R.P. Stanley, Two combinatorial applications of the Aleksandrov-Fenchel inequalities, JCTA, 1981.
- G. Brightwell and P. Winkler, Counting linear extensions,
Order, 1991.
- S. Dittmer and I. Pak, Counting linear extensions of restricted posets, 2019.
- S.H. Chan, I. Pak and G. Panova, Sorting probability for large Young diagrams, 2020.
- S.H. Chan, I. Pak and G. Panova, Sorting probability of Catalan posets, 2020.
- J. Fox and B. Sudakov, Paths and stability number in digraphs,
Electron. J. Combin., 2009.
Lecture by lecture background reading
- Basic notions
- See West (§12.1), Trotter's survey (§1), or Jukna (§9.1.1)
- Dilworth's theorem (see West, Ch. 12)
- R.P. Dilworth, A Decomposition Theorem for Partially Ordered Sets, Ann. of Math., 1950.
- M.A. Perles, A proof of Dilworth's decomposition theorem for partially ordered sets, Israel J. Math., 1963
- E.C. Milner, Dilworth's decomposition
theorem in the infinite case, in The Dilworth theorems, Birkhauser, 1990, 30-35.
- Gallai-Milgram theorem (see West, Ch. 12)
- Chains and antichains in the Boolean lattice
- de Bruijn-Tengbergen-Kruyswijk theorem (see West, Ch. 12, Sym chains decomposition section).
- Greene-Kleitman bracket sequences construction (ibid.)
- Fubini numbers (see Flajolet-Sedgewick, "surjection numbers" on p. 110, 243-245, 259-260).
- Hansel theorem (see West, Ch. 12)
- Gray codes and universal sequences
- Gray codes
and De Bruijn sequences
(see J. Matousek and J. Nesetril, Invitation to Discrete Mathematics, §4.5).
- Universal sequences and the Lipski Theorem (see Jukna, Thm 9.4).
- F. Chung, P. Diaconis and R. Graham, Universal cycles for combinatorial structures,
Discrete Math., 1992.
- F. Ruskey, Combinatorial Gray Code,
in Encyclopedia of Algorithms, Springer, 2016.
- Extremal Combinatorics
- Perfect graphs (see Diestel, §5.5; see also earlier sections for notations and some background)
- Subsets of distinct numbers via Sperner's property (using LA, see Proctor's paper and Stanley's talk slides)
- R. Proctor, Solution of Two Difficult Combinatorial Problems
with Linear Algebra, Amer. Math. Monthly, 1982.
- R.P. Stanley, Weyl
groups, the hard Lefschetz theorem, and the Spemer property, SIAM J. Algebraic
Discrete Methods, 1980.
- R.P. Stanley, The Erdős-Moser Conjecture, talk slides, 2009.
- Sequence A025591, OEIS.
- B.D. Sullivan, On a Conjecture of Andrica and Tomescu,
Journal of Integer Sequences, 2013.
Home assignments
These will be posted here. The solutions will need to be uploaded to via
Gradescope,
which is also linked to the course CCLE website.
- HA1 is here, due Oct 19.
- HA2 is here, due Nov 2.
Collaboration policy:
For the home assignments, you can form discussion groups of up to 3 people each. In fact, I would like
to encourage you to do that. You can discuss problems but have to write your own separate solutions.
You should write the list of people in you group on top of each HA.
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Last updated 10/14/2020.