Summer School
on Algebraic and Enumerative Combinatorics 2012
S. Miguel de Seide, Portugal

Lectures by Vic Reiner

Topic: Reflection group counting and q-counting

Certain families of numbers, such as in G.-C. Rota's "Twelve-fold way", appear repeatedly as solution to enumeration problems, and in other locations, e.g. Much effort has gone into understanding how to view these numbers as coming from the symmetric groups, or the finite reflection/Weyl groups/Weyl group of type A, and generalizing them to all finite reflection groups.

This viewpoint not only illuminates connections between them, and other areas of mathematics, but also on how to define useful q-analogues of these numbers. We hope to illustrate this here.

  1. Things we count
  2. What is a finite reflection group?
  3. Taxonomy of reflection groups
  4. Back to the Twelvefold Way
  5. Transitive actions and CSPs
  6. Multinomials, flags, and parabolic subgroups
  7. Fake degrees
  8. The Catalan and parking function family
  9. Bibliography
  10. Exercises

Suggested reading