MIT   Fall 2019


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

instructor:   Alexander Postnikov

office hours: Monday 3 - 4 pm or by appointment

grader:   Zhenkun Li

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.

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 ...

course level:   undergraduate

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:

practice for quizes:

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.