Combinatorics of Posets (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).


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: Individual lecture notes are posted on CCLE website if and when they come available.

Update: All lecture notes can now be download in one large file, 188 pages, 92 Mb.
Warning: these are neither checked nor edited. Possibly, completely useless to anyone.


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).


None of these are required, all are recommended. Selected relevant chapters will be posted on CCLE.


Lecture by lecture background reading

  1. Basic notions

  2. Dilworth's theorem

  3. Gallai-Milgram theorem

  4. Chains and antichains in the Boolean lattice

  5. Gray codes and universal sequences

  6. Extremal Combinatorics

  7. Perfect graphs

  8. Subsets of distinct numbers via Sperner's property (using LA)

  9. Greene-Kleitman theorems and permutations

  10. Greene-Kleitman theorems via Combinatorial Optimization

  11. Operations on posets and distributive lattices

  12. Linear extensions, labeled trees and standard Young tableaux

  13. Bounds on the number of linear extensions

  14. Asymptotics of the number of linear extensions

  15. Schützenberger's promotion and its applications

  16. Schützenberger's evacuation and its applications

  17. Domino and P-domino tabelaux

  18. Poset sorting and HLF

  19. P-partition theory

  20. Poset polytopes

  21. Applications of poset polytopes via Ehrhart polynomial and Alexandrov-Fenchel inequalities

  22. More applications and generalizations of poset polytopes

  23. Correlation inequalities

  24. The FKG and XYZ inequalities

  25. Proof of the XYZ inequality

  26. Comparisons via Linear extensions

  27. Kahn-Saks theorem via Grünbaum's theorem

  28. Sorting probability, duality and complexity

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.

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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You must begin your email with "Dear Professor Pak," and nothing else in the first line.
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 12/11/2020.