**Fall 2007**

·
**Instructor: Sergi Elizalde**

·
**Lectures: **TuTh 10:00-11:50 in 201 Kemeny

·
**X-period:** W 3:00-3:50

·
**Office Hours: **Tu 11:50-12:30, Th 1:30-3:30,
and by appointment

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**Office:**
332 Kemeny

·
**Email:**

·
**Phone:**
646-8191

**Course
description**

This is an introductory
course in algebraic and enumerative combinatorics. You will learn how to apply
techniques from algebra to solve enumeration problems, and to use combinatorial
methods to solve questions arising in other areas of mathematics. No prior
knowledge of combinatorics is expected, but some familiarity with linear
algebra and finite groups is preferable.

**Homework**

·
Problem Set 1, due on
10/11/07.

·
Problem Set 2, due on
10/25/07.

·
Problem Set 3, due on
11/8/07.

·
Problem Set 4, due on
11/27/07.

The **final exam** will
be handed out on Thursday, November 29, and due on Tuesday, December 4.

**Recommended
texts**

There is **no textbook**
required for this course.

Most of the material in the
course will be taken from these three sources:

- [Aig]
Martin Aigner, *A Course in Enumeration*, Graduate Texts in Mathematics
238, Springer, 2007.

- [dM]
Anna de Mier, *Lecture notes for "Enumerative Combinatorics".* [Download]

- [St]
Richard P. Stanley, *Topics in algebraic combinatorics*, course notes,
preliminary version. I will hand out these notes in class.

Other useful books that you
can look at are:

- [EC1]
[EC2] Richard P. Stanley, *Enumerative Combinatorics, Vols. I and II*,
Cambridge University Press, 1997/1999.

- [Wf]
Herb Wilf, *Generatingfunctionology*, Academic Press, 1990. Also available
online at http://www.math.upenn.edu/~wilf/DownldGF.html

- [vLW]
J.H. van Lint, R.M. Wilson, *A course in Combinatorics*, Cambridge
University Press, Cambridge, 1992.

- [Bo]
Miklos Bóna, *A walk through combinatorics*, World Scientific, 2002.

- [FS]
P. Flajolet, R. Sedgewick, *Analytic Combinatorics*. The preliminary
version of this forthcoming book is available online at http://algo.inria.fr/flajolet/Publications/books.html

- [Bg]
Kenneth P. Bogart, *Enumerative Combinatorics Through Guided Discovery*.
It is available at http://www.math.dartmouth.edu/~kpbogart/ComboNotes3-20-05.pdf

- [BS]
A. Björner, R. Stanley, *A Combinatorial Miscellany* [Download].

**Topics**

Here is a tentative list of
the topics that will be covered, together with the corresponding references.
The main references for each topic are in bold.

- Fundamental coefficients. Sets and
multisets. Compositions.

-**[Aig]**Sections 1.1, 1.2.

-**[dM]**Chapter 1.

- [EC1] Section 1.2. - Integer and set partitions. Stirling
numbers. Permutations.

-**[Aig]**Sections 1.3-1.5.

-**[dM]**Chapter 3.

- [vLW] Chapter 13.

- [EC1] Section 1.3. - Inclusion-Exclusion.

-**[Aig]**Section 5.1.

-**[dM]**Chapter 2.

- [EC1] Sections 2.1-2.3.

- [vLW] Chapter 10. - Generating functions. Recurrences. Formal
power series.

-**[dM]**Chapter 4.

- [Aig] Sections 2.1, 2.2, 3.1.

- [Wf] Section 2.1.

- [vLW] Chapter 14. - The symbolic method. Unlabelled
structures. Ordinary generating functions.

-**[dM]**Chapter 5.

- [FS] Chapter 1.

- [Wf] Section 2.2. - Labeled structures. Exponential generating
functions.

-**[dM]**Chapter 6.

- [Aig] Section 3.3.

- [FS] Chapter 2.

- [Wf] Section 2.3. - Partially ordered sets. Chains and
antichains. Sperner’s theorem.

-**[St]**Section 4. - Group actions on boolean algebras.

-**[St]**Section 5. - Young diagrams and
*q*-binomial coefficients.

-**[St]**Section 6.

- [Aig] Section 1.6. - Enumeration under group action. Pólya's
theorem.

-**[St]**Section 7.

-**[Aig]**Sections 6.1-6.3. - Young tableaux. The RSK algorithm.

-**[St]**Section 8.

- [EC2] Section 7.11.

- [BS] Section 4. - Lattices. Möbius Inversion.

-**[Aig]**Section 5.2.

- [EC1] Sections 3.3-3.8.

**Homework,
exams, and grading**

The course grade will be
based on

- homework (40%)
- a final exam (25%)
- class participation (10%)
- a final project (25%).

The homework will consist of
a problem set every two weeks. Collaboration is permitted, but you are not
allowed to copy someone else's work. The solutions must be written
individually. You have to mention on your problem set the names of the students
that you worked with, and also which books or articles you used.

The final will be a take-home exam. You must work on the problems on your own. **No
collaboration permitted** in the exam**.**

Class participation
involves attending lectures, as well as asking and answering questions in
class.

The final project will
consist of preparing a topic and presenting it in class. Students should work
in groups of 2 or 3. Here are possible topics
for the project.

**Students with
disabilities: **Students with disabilities enrolled in this course that may need
disability-related classroom accommodations are encouraged to make an office
appointment to see me before the end of the second week of the term. All
discussions will remain confidential,
although the Student Accessibility Services office may be consulted to discuss appropriate
implementation of any accommodation requested.

*Last
modified on Nov 12, 2007. *