Math 172—Combinatorics—Spring 2010
Professor Haiman's Office: 855 Evans Hall. Office hour
Mondays 12:30-2pm.
Catalog description: Basic combinatorial principles, graphs,
partially ordered sets, generating functions, asymptotic methods,
combinatorics of permutations and partitions, designs and
codes. Additional topics at the discretion of the instructor.
Lectures: MWF 11-12pm, Room 71 Evans
Course Control Number: 54557
Prerequisites: Math 55
Text: Miklos Bona, A Walk Through Combinatorics: An
Introduction to Enumeration and Graph Theory, Second Edition (World
Scientific, 2006).
Homework: Weekly problem sets due each Wednesday
Exams:
- Midterm exam in class, Friday, March 12.
- Final exam Tuesday, May 11, 7-10pm (exam group 8), Room 71 Evans
Grading: Homework 35%, Midterm exam 20%, Final exam 45%.
Syllabus:
- Counting basics: sum and product principles, pigeonhole principle.
Review of mathematical induction.
- Elementary enumeration problems: words, permutations, subsets,
distributions
- Binomial and multinomial theorems
- Set partitions; Stirling and Bell numbers
- Integer partitions
- The "twelve-fold way"
- Permutations and their cycle structure
- Ordinary generating functions and examples; Catalan numbers
- Generating functions for elementary enumeration problems
- Partition identities
- Sieve principle (via generating functions)
- Exponential generating functions and examples; exponential structures.
- Graphs and trees. Generating functions for counting trees.
Matrix-tree theorem.
- Polya-Burnside theorem.
- Cycle index generating function for species of structures;
counting connected graphs up to isomorphism
- Additional topics if time; may include graph coloring and
matchings, planar graphs, posets.
Reading and Homework Assignments
- Reading for Lectures 1-3: Bona, Chapters 1-3. Problem Set 1, due Jan. 27 (corrected 1/22; the previous
version was based on exercise numbers from the old edition of the
book); PS 1 Solutions.
- Reading for Lectures 4-6: Chapters 4 and 5.1-5.2. Problem Set 2, due Feb. 3; PS 2 Solutions. Note: problem 4.37 was graded
but not counted, because it turned out that the problem was misprinted
in some versions of the book.
- Reading for Lectures 7-9: Chapter 5 continued. Problem Set 3, due Feb. 10 (problem D corrected
2/8); PS 3 Solutions.
- Reading for Lectures 10-11: Come to class! (or get notes from
another student who did, on "12-fold way"). Also, Chapter 6. Problem Set 4, due Feb. 17. This homework is a
bit shorter than usual, since Monday is a holiday. PS 4 Solutions.
- Reading for Lectures 12-14: Chapter 6 continued, Chapter 8 up
through 8.1.2, additional Notes on Ordinary
Generating Functions. Problem Set 5, due
Feb. 24 (Note: Problem A(b) was wrong, fixed 1/22. I also
changed Problem C, since we didn't get as far in the lecture as I
hoped when I made up the problems.) PS 5
Solutions.
- Reading for Lectures 15-17: Chapter 8 through 8.1.3, Notes on
Ordinary Generating Functions. Problem Set 6,
due March 3. PS 6 Solutions.
- Reading for Lectures 18-20: Notes on
Partitions and their Generating Functions. Problem Set 7, due March 10 (typo in Problem D
corrected 3/7); PS 7 Solutions.
Reminder: Midterm Exam is Friday, Mar. 12, in class, covering
the material on Problem Sets 1-6.
- Midterm Exam Solutions
- Reading for Lectures 21-22: Chapter 7. Problem Set 8, due March 17 (a short one, since
March 12 is the midterm exam); PS 8
Solutions.
- Announcement: Mar. 15 through Apr. 2, our class will move
to Room 118 Barrows
Hall so that equipment can be installed in 71 Evans.
- Reading for Lectures 23-25: Chapter 9 (for definitions), 10.1, and
Notes on Ordinary Generating Functions, sections 3 and 4. Problem Set 9, due March 31. (Correction
3/29: problems C and D only require an algebraic identity for
V(x), not a closed formula.) PS 9
Solutions.
- Reading for Lectures 26-28: Notes on
Matrix-Tree theorem and Cayley's tree enumerator, sections 1 and 2.
Problem Set 10, due April 7; PS 10 Solutions.
- Reading for Lectures 29-31: 10.3-10.4; Matrix-Tree and Cayley
Notes, sections 3 and 4. Problem Set 11, due
April 14; PS 11 Solutions.
- Reading for Lectures 32-34: 8.2, Notes
on exponential generating functions and structures. (Caveat: the
example of alternating permutations discussed in Section 7 of
the notes is not actually a combinatorial species, since the
definition of the structure depends on the specific set
X={1,...n}, or more precisely, on the intrinsic ordering
of X.) Problem Set 12, due April 21;
PS 12 Solutions.
- Reading for Lectures 35-39: Additional notes
on exponential generating functions and Notes on cycle generating functions. You
will probably find this last set of notes harder reading than the
others, but I'll give you some clues in the lectures.
Problem Set 13, due Friday, April 30.
Note: since some of the homework problems are on material that
I'll finish covering in Friday's lecture, you may turn in PS 13 under
my office door (855 Evans) at any time on Friday. I'll pick it up
Saturday morning. PS 13 Solutions.
- I'll have RRR week office hours MW 11-1. I expect to have the graded
last homework back by the Wednesday office hour.
- Final Exam is Tuesday, May 11, 7-10pm, in 71 Evans. Any
topics covered in class and/or homework may be on the exam. Rules and
format are the same as on the midterm. You may bring one sheet of
notes (written on both sides). There will be space for answers on the
exam itself, but you should bring your own scratch paper.
- Final Exam Solutions (correction: in
the solution to Problem 7 I left out 4 edges from the picture, and the
matrix should have 4's on the diagonal, not 3's).
[Top of page | Calendar | Prof. Haiman's home page]