Advanced Combinatorics – 3450:636-001 12:05-12:55 Kolbe Hall 215.

INSTRUCTOR: Dr. Stefan Forcey EMAIL:

OFFICE:CAS 275PHONE: 972-6779

OFFICE HOURS:MTuWF 2:00-3:00

Advanced combinatorics will both extend the material covered in an introductory course and highlight some of the current developments in the field. This course will be run in a less formal, more collaborative seminar style. There are individual tests and homework grades, but the work may all be done collaboratively. The tests will feel more like investigative projects to take home.

Website

Texts: Download these soon, in case they aren’t available online all semester!

N. Bergeron et.al.: Introduction to Species

R. Stanley:The Catalan addendum:

H. Wilf:generatingfunctionology

R. Thomas: Lectures in Geometric Combinatorics

(This last one was still online last I checked, also inexpensive at

Ziegler et.al. : BASIC PROPERTIES OF CONVEX POLYTOPES

R. Stanley: Enumerative Combinatorics

Supplementary material:

Order theory glossary

Monoidal Functors, Species and Hopf Algebras by M. Aguiar and S. Mahajan

A Survey of the Riordan Group by Louis Shapiro

Wikipedia:

Blogs: John Baez:This week’s finds.

Gil Kalai: Combinatorics and more.

GRADING POLICY: The following percentages will be used in grading:

50% Homework

30% Tests (2 Tests at 15% each.)

20% Final Exam/Project

90% guarantees an A

80% guarantees a B

70% guarantees a C

60% guarantees a D

Homework problems should be attempted individually at first. After that, research and collaboration are permitted as long as you actually cite any published sources and credit any persons who helped you.

Course Outline with dates:

•Jan. 9. Day one.

•Jan 15: Last day to add.

•Jan. 16: No class on MLK day.

• Jan. 23: Last day to drop.

• Feb.21: No class on Pres. day.

• ~Feb.? Test 1.

• Feb26: Last day to w/draw w/out signature.

• Mar. 12-17: Spring break.

• ~April ?: Test 2.

•Apr. 28: Last day.

• Exam TBA

Tentative Topic outline:

  1. Posets
  1. Orders, finite topology
  2. Lattices
  3. Examples: Tamari, Weak lattice of permutations, Boolean lattice
  1. Geometric Combinatorics
  1. Polytopes: convex hulls, half-spaces
  2. Hasse diagrams
  3. Skeletons
  4. Associahedra
  5. Minkowski sums
  6. Euler’s formula and Platonic solids
  1. Species
  1. Definitions and examples.
  2. Categories and functors.
  3. Species and generating functions
  4. Examples
  5. Operations on species (+, . , o)
  6. Transforms, Riordan group.
  7. Operads
  1. Algebraic Combinatorics

A. Algebras,Graphs and Trees: planar trees, grafting, splitting

B. Coalgebras

C. Bialgebras

  1. Polytopes again.
  2. Moebius inversion, Algebras.
  1. Tutte Polynomial
  1. Recursive calculation
  2. Interpretation
  3. Jones polynomial
  1. Computer Science, Chemistry and Biology
  1. P vs NP
  2. Benzenoids and polyhexes
  3. Phylogenetic trees and DNA .