Combinatorics of Posets (Math 206A, Winter 2024)
Instructor: Igor Pak
(see email instructions on the bottom of the page).
Gradescope Website: is here. Entry code: PWE8VW.
Class schedule: MWF 2:00 - 2:50 pm, MS 5203.
Office Hours: M 3-3:50.
Grading: The grade will be based on attendance, class participation (75%),
and homeworks (25%) 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. In the first half of the course we will
follow our Fall 2020 lecture notes. We will also follow various sections from the textbooks
and surveys below.
Fall 2020 lecture notes are available here in one large file,
188 pages, 92 Mb.
Warning: these are neither checked nor edited.
Reading
These are very preliminary, the exact list will be updated and expanded weekly with specific
sections indicated.
Textbooks:
- R.P. Stanley, Enumerative Combinatorics (EC), vol. 1 and vol. 2 (second edition).
- D.B. West, Combinatorial Mathematics, Cambridge Univ. Press, 2020, Ch. 12.
- S. Jukna, Extremal combinatorics. With applications in computer science
(second edition), Springer, 2011, see Ch. 9.
- W. T. Trotter, Combinatorics and partially ordered sets.
Dimension theory, Johns Hopkins University Press, 1992, see Ch. 1.
- R. Diestel, Graph Theory, Springer, 2016 (5th edition).
- A. Schrijver, Combinatorial Optimization, volumes A, B, and C, about 2000 pages, Springer, 2004.
- N. Alon and J. Spencer, The Probabilistic Method, Wiley, 2016, see Ch. 6.
None of these are required, all are recommended.
Surveys:
- C. Greene and D.J. Kleitman,
Proof techniques in the theory of finite sets, in Studies in combinatorics,
MAA, 1978, 22-79.
- 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.
- S.H. Chan and I. Pak, Linear extensions of finite posets, see here.
Lecture by lecture background reading
- Basic notions
- See West (§12.1), Trotter's survey (§1), or Jukna (§9.1.1)
- Dilworth's theorem
- 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
- Chains and antichains in the Boolean lattice
- For chains in Bn see Fubini numbers (see also
Flajolet-Sedgewick book "surjection numbers" on p. 110, 243-245, 259-260).
- For antichains in Bn see Dedekind number (see also this article in Quanta Magazine).
- de Bruijn-Tengbergen-Kruyswijk theorem (see West, Ch. 12, Sym chains decomposition section).
- Greene-Kleitman bracket sequences construction (ibid.)
- Hansel theorem (see West, Ch. 12)
- Subsets of distinct numbers via Sperner's property (using LA)
- 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.
- H.H. Nguyen, A new approach to an old problem of Erdos and Moser, 2011.
- Greene-Kleitman theory
- Operations on posets and distributive lattices
- Schützenberger's promotion and its applications
- Jeu de taquin
- M.P. Schützenberger, Promotion
des morphismes d'ensembles ordonnés (in French), Discrete Math., 1972.
- P. Edelman, T. Hibi and R.P. Stanley, A
recurrence for linear extensions, Order, 1989.
- See §2 in R.P. Stanley, Promotion and evacuation, Electronic Journal Combin., 2008.
- A. Bjorner and M. Wachs, q-hook length formulas for forests, JCTA, 1989.
- Corolary 2 on p. 4564 in
A. Hammett and B. Pittel, How
often are two permutations comparable?, Trans. AMS, 2008.
- Poset sorting and HLF
- Karamata inequality and applications
Home assignments
These will be posted here. The solutions will need to be uploaded to via Gradescope.
- HA1 is here, due Feb 5.
- HA2 is here, due Feb 19.
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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Any and all grade discussion must be done from your official UCLA email. Enclose your UCLA id number and full name as on the id on the bottom.
Last updated 2/28/2024.