Math 465: Introduction to Combinatorics

Fall 2011

Course meets: Tuesday and Thursday 1:10-2:30 in 232 Dennison.

Instructor: Sergey Fomin, 2858 East Hall, 764-6297, fomin@umich.edu

Office hours: Tuesday 3:10-4 PM and Thursday 4:10-5:30 in 2858 East Hall.

Grader: Qi Han, qihan@umich.edu.

Course homepage: http://www.math.lsa.umich.edu/~fomin/465f11.html

Level: undergraduate.

Prerequisites: Linear algebra (Math 214, 217, 256, 286, 296, 417, 419, or equivalent) or permission of instructor.

Student work expected: several problem sets.

Grade will be based on two 1.5-hour midterm exams, 25% each; and 50% homework. Your lowest homework set score will be dropped.

Exams will be held in the same room where the class meets. Tentative dates of exams: October 27 and December 13.

This course will not be graded on a curve, i.e., there are not a set number of each grade to be given out.
Every student with the total score of 90% (resp., 80%, 70%, 60%) is guaranteed the final grade of A (resp., B or higher, C or higher, D or higher).

Synopsis: This course introduces the fundamental notions, techniques, and theorems of enumerative combinatorics and graph theory.

Background: Combinatorics is the study of finite mathematical objects, including their enumeration, structural properties, design, and optimization. Combinatorics plays an increasingly important role in various branches of mathematics and in numerous applications, including computer science, statistics and statistical physics, operations research, bioinformatics, and electrical engineering.

Textbook:
R. Brualdi, Introductory combinatorics, 5th edition, Pearson Prentice Hall, 2010. Errata.
Where can I buy this book?

Other introductory textbooks:
M. Bóna, A walk through combinatorics, 2nd edition, World Scientific, 2006.
V. Bryant, Aspects of combinatorics, Cambridge University Press, 1993.
R. Merris, Combinatorics, 2nd edition, Wiley, 2003.

More advanced undergraduate textbooks:
P. J. Cameron, Combinatorics: topics, techniques, algorithms, Cambridge University Press, 1994.
J. H. van Lint and R. M. Wilson, A course in combinatorics, 2nd edition, Cambridge University Press, 2001.
J. Matoušek and J. Nešetřil, Invitation to discrete mathematics, 2nd edition, Oxford University Press, 2008.

Lectures (tentative plan)

1. The pigeonhole principle.
2. The Erdös-Szekeres Theorem. Basic Ramsey Theory.
3. Basic counting principles. Binomial coefficients.
4. Multinomial coefficients. Applications.
5. Pascal triangle. The binomial theorem. Binomial identities.
6. The multinomial theorem. Newton's binomial theorem. Generating functions.
7. Multiplication principle for generating functions. Counting triangulations.
8. Linear recurrences with constant coefficients.
9. Fibonacci numbers. Linear recurrences: the case of multiple roots.
10. Nonhomogeneous linear recurrences.
11. Catalan numbers.
12. Stirling numbers.
13. Inclusion-exclusion principle. Derangements.
14. Permutations with restricted positions.
15. First midterm exam.
16. Basic notions of graph theory. Trees. Independent cycles.
17. Planar graphs. Euler's formula.
18. Eulerian paths and circuits. Hamiltonian cycles.
19. Chromatic number and chromatic polynomial.
20. Spanning trees. The minimal spanning tree problem.
21. The shortest path problem.
22. Flows in networks. The Ford-Fulkerson theorem and algorithm.
23. Menger's theorems. Bipartite networks. Hall's theorem. SDR's.
24. Birkhoff's theorem. Partially ordered sets.
25. Chains and antichains. Sperner's theorem. Symmetric chain decompositions.
26. Minimal chain/antichain decompositions. Dilworth's theorem.
27. Second midterm exam.