## 18.211     COMBINATORIAL ANALYSIS

class meets:   Mondays, Wednesdays, Fridays; 2 - 3 pm; room 4-145

instructor:   Alexander Postnikov

office hours: Monday 3 - 4 pm or by appointment

description:
Combinatorial problems and methods for their solution. Enumeration, generating functions, recurrence relations, construction of bijections. Introduction to graph theory. Prior experience with abstraction and proofs is helpful.

topics:
pigeon-hole principle, mathematical induction, permutations, binomial theorem, compositions, partitions, Stirling numbers, inclusion-exclusion principle, recurrence relations, generating functions, Catalan numbers, graphs, trees, Eulerian walks, Hamiltionian cycles, matrix-tree theorem, electrical networks, graph colorings, chromatic polynomials, (and if time allows) Polya counting, Ramsey theory, pattern avoidance, probabilistic method, partial orders, combinatorial algorithms ...

recommended textbook:
*  Miklos Bona, A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory, 4th edition, World Scientific. (Previous editions of the textbook are also fine for the course.)

additional reading: There are many great textbooks on combinatorics. You don't need the following books for this class. But, if you want to learn more, you are welcome to take a look at them.
*  Richard P. Stanley, Algebraic Combinatorics: Walks, Trees, Tableaux, and More. This book was written for 18.212 Algebraic Combinatorics, which is a continuation of this course.
*  Richard P. Stanley, Enumerative Combinatorics, Vol 1 and Vol 2. This is a famous book on enumerative combinatorics. It is a graduate level textbook. It covers many topics from this course on a deeper level.

grading:   Problem sets due every two weeks 50% + 3 inclass quizes 50%. There will be no final exam.

problem sets:

• Problem Set 1 (due Wednesday, September 18, 2019)
• Problem Set 2 (due Friday, October 4, 2019)
• Problem Set 3 (due Friday, October 18, 2019)
• Problem Set 4 (due Monday, October 28, 2019)
• Problem Set 5 (due Monday, November 18, 2019)
• Problem Set 6 (due Wednesday, November 27, 2019)
• Optional Problem Set 7 (turn in any number of solutions before Wednesday, December 11, 2019). Please turn in your solutions by email to zhenkun@mit.edu (cc to apost@math.mit.edu) or hand in your handwritten solutions to Zhenkun Li.

Notice: It is perfectly fine if you discuss the problems from problem sets with each other. But, if you collaborated on a problem, you should acknowledge this and explicitly list the names of your collaborator(s) in your solutions. You should write your solutions by youself and in your own words. Copying other students' solutions or using latex files with other students' solutions, might be considered a case of cheating and plagiarism, see What is Academic Integrity?

practice for quizes:

• Quiz 1 will cover the intersection of the material covered in Lectures 1-9 and Chapters 1-5 of [Bona]. Here are

4 problems and 5 more problems from past years, and their answers.

• Quiz 2 will cover the intersection of Lectures 11-23 and Chapters 6-9 of [Bona].

Here are some practice problems from old quizes:

• Quiz 3 will cover the material of Lectures 25-35, except, matroids, Tutte polynomial, and chip-firing games. It will be mostly about graph theory. The material might include: graphs, spanning trees, adjacency and Laplacian matrices, the matrix tree theorem, minimial weight spanning trees, matchings in graphs, Hall's marriage theorem, graph colorings, chromatic polynomial, and chromatic number, acyclic orientations, deletion-contraction, planar graphs, Euler's formula, parking functions, etc.

Here are some practice problems from old quizes:

(Problem 2 in the last practice quiz is about resistance. We have not discussed electrical networks in class. So don't worry about this kind of problem.)

These problems should give you some idea about the difficulty level of the quiz. Note that the actual quiz may not have exactly the same format and involve the same same number of problems as in these practice quizes.

In order to prepare for the quiz, you should

• Review the material covered in Lectures 24-35, including definitions, theorems, and examples.
• Solve the above practice problems.
• Solve as many problems from Chapters 10-12 of [Bona] as you like.

Average scores for problem sets and quizes:     P1: 96.88/100,   Q1: 37.94/40,   P2: 95.47/100,   P3: 91.76/100,   P4: 44.97/50,   Q2: 33.15/40,   P5: 72.09/80,   P6: 45.5/50,   Q3: 36.33/40.

lectures (with suggested reading from [Bona]):

1. W 09/04/2019. Introduction. What is combinatorial analysis?

2. F 09/06/2019. Pigeon-hole principle. Ramsey's and Erdos-Szekeres theorems. [Bona, Chapter 1].

3. M 09/09/2019. Mathematical induction. [Bona, Chapter 2].

4. W 09/11/2019. Permutations. [Bona, Chapter 3].

5. F 09/13/2019. Binomial theorem. Binomial and multinomial coefficents. [Bona, Chapter 4].

6. M 09/16/2019. Length and the number of inversions of permutations. q-factorial.

7. W 09/18/2019. q-binomial coefficients. Compositions. [Bona, Section 5.1]. Problem Set 1 is due.

F 09/20/2019. Student holiday - no classes.

8. M 09/23/2019. Compositions (cont'd), set partitions, and integer partitions. Fibonacci numbers. [Bona, Chapter 5].

9. W 09/25/2019. Set partitions and integer patitions (cont'd). Bell and Stirling numbers.

10. F 09/27/2019. Quiz 1.

11. M 09/30/2019. Integer portitions (cont'd).

12. W 10/02/2019. Cycles in permutations. Stirling numbers of 1st kind vs Stirling numbers of 2nd kind. [Bona, Chapter 6].

13. F 10/04/2019. Stirling numbers of 1st kind (cont'd). Records of permutations. Intro to inclusion-exclusion principle. Problem Set 2 is due.

14. M 10/07/2019. Inclusion-exclusion principle. Derangements. [Bona, Chapter 7].

15. W 10/09/2019. Ordinary generating functions. Examples: Generating functions for partitions numbers and Fibonacci numbers. [Bona, Chapter 8].

16. F 10/11/2019. Generating functions (cont'd). From recurrence relations to generating functions. Catalan numbers.

M 10/14/2019. Columbus Day - vacation.

17. W 10/16/2019. Generating functions (cont'd). Exponential generating functions. Exponential formula.

18. F 10/18/2019. Generating functions (cont'd). Problem Set 3 is due.

19. M 10/21/2019. Generating functions (cont'd). Ordinary generating functions vs exponential generating functions. Recurrence relations and differential equations. The Catalan numbers and the reflection method.

20. W 10/23/2019. The Catalan numbers (cont'd): Cyclic shifts, binary trees, plane trees & depth-first search.

21. F 10/25/2019. The Catalan numbers (cont'd): queue-sortable & stack-sortable permutations, pattern avoidance [Bona, Chapter 14]. Graph theory: Euler's Königsberg bridge problem & Eulerian trails [Bona, Chapter 9].

22. M 10/28/2019. Eulerian trails and Hamiltonian cycles. Cayley's formula for the number of trees and Prüfer's codes [Bona, Chapter 10]. Problem Set 4 is due.

23. W 10/30/2019. Spanning trees of graphs.

24. F 11/01/2019. Quiz 2.

25. M 11/04/2019. Minimal-weight spanning trees. Kruskal's Greedy Algorithm. Matroids.

26. W 11/06/2019. Graphs and matrices. Proof of Matrix-Tree Theorem.

27. F 11/08/2019. Matchings in graphs. Hall's Marriage Theorem. [Bona, Chapter 11].

M 11/11/2019. Veterans Day - holiday.

28. W 11/13/2019. Graph colorings. The chromatic polynomial. Deletion-contraction recurrence.

29. F 11/15/2019. Acyclic orientations of graphs. Chordal graphs.

30. M 11/18/2019. The Tutte dichromat polynomial. Problem Set 5 is due.

31. W 11/20/2019. Planar graphs. Euler's formula. Kuratowksi' theorem. Polytopes. [Bona, Chapter 12].

32. F 11/22/2019. Parking functions. The tree inversion polynomial.

33. M 11/25/2019. Chip-firing game on graphs.

34. W 11/27/2019. Directed Eulerian tours and arborescences. BEST theorem. Matrix-tree theorem for directed graphs. Problem Set 6 is due.

F 11/29/2019. Thanksgiving vacation.

35. M 12/02/2019. Systems of distinct representatives. Eigenvalues of the adjacency matrix vs eigenvalues of the Laplacian matrix. The number of spanning trees in the d-cube graph.

36. W 12/04/2019. Quiz 3.

37. F 12/06/2019. Domino tilings.

38. M 12/09/2019. Guest lecture #1 by Prof. Thomas Lam (U Michigan & MIT): Electrical networks.

39. W 12/11/2019. Guest lecture #2 by Thomas Lam: Electrical networks (cont'd).
Turn in any number of solutions for (optional) Problem Set 7 before this date.