M501/2 Combinatorics I, Fall 2005

Instructor:

Dr. A. Betten, room 207, Weber building

When and where:

M  W  F     2:10 - 3:00 ENGRG E106

Texts:

M502: van Lint / Wilson: Combinatorics, Cambridge

M501: Richard P. Stanley: Enumerative Combinatorics, Volume I, Cambridge Studies in Advanced Mathematics 49, Cambridge

Prerequisites:

Topics (M502, Spring 06):

• Algebraic graph theory (eigenvalues, Bose-Mesner algebra, special classes of graphs and digraphs etc)

• Isomorph free exhaustive generation of graphs and other discrete structures using the method of canonical forms and partition backtracking.

• Hadamard matrices (construction methods, the new Hadamard matrix of order 428 constructed by Kharagani and Tayfeh-Rezaie in 2005)

• Error correcting codes, parameter, bounds, classes of codes, weights, duality, Reed-Muller, BCH and Reed-Solomon codes, codes from curves.

• Projective spaces, in particular finite projective planes, Desargues, Pappus, coordinates, Latin squares, collineations.

• Ovals in Desarguesian projective planes, their Combinatorics, new construction methods using geometric codes.

• Combinatorics of tableaux, the hook formula, symmetric polynomials (elementary, monomial, complete, forgotten, power, Schur) and their Combinatorics. Characters of S_n.

• Designs, Steiner systems, parameter, standard constructions, square designs, construction with prescribed automorphism group, codes and designs.

• Linear Spaces, in particular the recent theory of line transitive point imprimitive ones, their Combinatorics and how to generate them.

• Extremal Combinatorics: distinct representatives (Hall), Ramsey, Sperner, Erdos etc.

Topics (M501 Fall 05):

This is an introductory course on Combinatorics at the graduate level.

The student who wishes to study discrete structures (like graphs, codes, designs, geometries etc.) needs two basic ingredients. He needs a strong background in fundamental counting techniques and he should also aquire a good understanding of algebraic concepts.

This course is about the first of these, the fundamental counting techniques. The textbook by Stanley (Enumerative Combinatorics) gives an overwhelming amount of seemingly different but essentially connected counting results. Getting through this book is already an achievement, and it can only be done by picking a good collection of samples. This is what we shall do in the course.

The core topics are chapters 1-3 in the book, covering for example:

• Basic counting. The twelvefold way (in how many ways can that many balls be put into that many boxes provided we can distinguish among the balls / boxes etc. etc.)

• The principle of inclusion and exclusion

• Formal power series

• Permutations

Apart from that, we may choose further topics from chapters 4 and 5. In particular, we will discuss the transfer matrix method, which is concerned with counting paths in directed graphs (in a similar but more sophisticated fashion as in the usual Computer Science Algorithms class).

The course will be continued in Spring 05. Then we will deepen our understanding of essential counting methods and we will study further applications. It would be helpful if people who wish to continue the sequence take M566 in Fall 05 in order to be prepared for a more algebraic approach to combinatorics in Spring 06 (unless, of course, they already feel strong enough on their algebra side).

Grading Rules:

Class participation and homework are essential, as is the final exam. I expect that course participants present small subject topics towards the end of the semester. The mix is: final / subject topic presentation / homework / class participation as to 30 / 30 / 30 / 10.

For further information:

Please contact A. Betten:
betten@math.colostate.edu betten@math.colostate.edu

or visit the course webpage at
http://www.math.colostate.edu/ betten/courses/M501/FA05/index.html. http://www.math.colostate.edu/~betten/courses/M501/FA05/index.html