## 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.
- 2
^{n}, triangular numbers, binomial coefficients, multinomial coefficients
- the partition number p(n)
- Stirling numbers of the 1st and 2nd kind
- (n+1)
^{n-1}, Catalan, Narayana, and Kirkman numbers
- tableaux numbers

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.

- Things we count
- What is a finite reflection group?
- Taxonomy of reflection groups
- Back to the Twelvefold Way
- Transitive actions and CSPs
- Multinomials, flags, and parabolic subgroups
- Fake degrees
- The Catalan and parking function family
- Bibliography
- Exercises

**Suggested reading**

- S. Fomin and N. Reading,
"Root Systems and
Generalized Associahedra",

from IAS/Park City Summer Math Institute on Geometric Combinatorics, 2004.
- D. Armstrong,
"Generalized noncrossing partitions and combinatorics of Coxeter groups",
Mem. Amer. Math. Soc. 202 (2009), no. 949.
- A. Björner and F. Brenti, "The combinatorics of Coxeter groups",

Springer-Verlag, Graduate Texts in Mathematics 227.
- J.E. Humphreys "Reflection groups and Coxeter groups",

Cambridge Studies in Advanced Mathematics 29.
- R. Kane, "Reflection groups and invariant theory",
CMS Books in Math. 5, Springer-Verlag.
- Chapter 1 of
R.P. Stanley, "Enumerative Combinatorics, Vol. 1" (Chapter 1),

Cambridge Studies in Advanced Mathematics 29.