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

Fall 2020 lecture notes are available here in one large file, 188 pages, 92 Mb.
Warning: these are neither checked nor edited.

Reading

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

1. Basic notions
   • See West (§12.1), Trotter's survey (§1), or Jukna (§9.1.1)

2. 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.

3. Gallai-Milgram theorem
   • See West, Ch. 12.
   • T. Gallai and A.N. Milgram Verallgemeinerung eines graphentheoretischen Satzes von Rédei, Acta Sci. Math., 1960
   • J.M. Brochet and M. Pouzet, Gallai-Milgram properties for infinite graphs, Discrete Math., 1991.

4. Chains and antichains in the Boolean lattice
   • For chains in B_n see Fubini numbers (see also Flajolet-Sedgewick book "surjection numbers" on p. 110, 243-245, 259-260).
   • For antichains in B_n 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)

5. 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.

6. Greene-Kleitman theory
   • Greene-Kleitman survey
   • Greene's theorem in Stanley, EC2, Appendix A.
   • Ford-Fulkerson algorithm, Wikipedia article and original paper
   • T.E. Harris and Gen. F.S. Ross (Ret.), Fundamentals of a Method for Evaluating Rail Net Capacities, classified report, 1955 (unclassified 1999).

7. Operations on posets and distributive lattices
   • See Stanley, §3.1-4
   • Distributive lattice
   • Fundamental Theorem for Finite Distributive Lattices

8. 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.

9. Poset sorting and HLF
   • See §3.10 in Sagan's The Symmetric Group and this NPS website (in French)
   • J-C Novelli, I. Pak and A.V. Stoyanovsky, A direct bijective proof of the hook-length formula, DMTCS, 1997.
   • I. Fischer, A bijective proof of the hook-length formula for shifted standard tableaux, 2001.
   • B. Beáta, Bijective proofs of the hook formula for rooted trees, Ars Comb, 2012.

10. Karamata inequality and applications
    • See proof in Karamata inequality, Wikipedia
    • I. A. Bochkov and F. V. Petrov, The bounds for the number of linear extensions via chain and antichain coverings, 2020.
    • I. Pak, F.V. Petrov and V. Sokolov, Hook inequalities, Math. Intelligencer, 2020.

Home assignments

• 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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Last updated 2/28/2024.