Rational Catalan combinatorics

This web page contains material for the workshop Rational Catalan combinatorics.

Contributions from the workshop participants are available in dvi, postscript or pdf.

Experimental material

AIM is developing some new web-based tools for research mathematics. This workshop is the first time we are trying this, and all comments are welcome.
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Reading list

The organizers have prepared a reading list:

1. Dusa McDuff "What is symplectic geometry?". The first few pages give some helpful intuition.


2. Mark Haiman "Conjectures on the quotient ring by diagonal invariants". This paper in some sense launched the subject. See Section 7.4.


3. Mark Haiman's ICM notes on Cherednik algebras, CDM notes on the Hilbert scheme of points in C^2 and PCMI notes on q- and q,t-enumeration.



4. Michel Broue gave three expository lectures on "Complex reflection groups and their associated braid groups and Hecke algebras" at the Isaac Newton Institute program on Algebraic Lie Theory, Spring 2009. There are other useful videos here as well.


5. Vic Reiner gave a lecture series on "Reflection group counting and q-counting" at the 2012 Summer School on Algebraic and Enumerative Combinatorics, S. Miguel de Seide, Portugal.


6. Pavel Etingof "Lecture notes on Cherednik algebras" and "Lectures on Calogero-Moser systems". The Calogero-Moser notes explain the physical motivation behind the subject.



7. Iain Gordon and Stephen Griffeth "Catalan numbers for complex reflection groups". This paper studies Catalan numbers in their fullest generality. See equation (4).


8. Drew Armstrong talks called "Rational Catalan Combinatorics 1" and "Rational Catalan Combinatorics 2". These expository talks describe classical (type A) Catalan combinatorics at the "rational" level of generality.



7. Armstrong-Reiner-Rhoades "Parking spaces" and Armstrong-Stump-Thomas "A uniform bijection between nonnesting and noncrossing partitions". These papers discuss nonnesting and noncrossing Catalan phenomena. The first extends NN and NC phenomena to parking functions and makes a powerful conjecture relating them. See the Main Conjecture, Section 2.6.



8. Drew Armstrong "Generalized noncrossing partitions and combinatorics of Coxeter groups". Chapter 5 of this memoir is an exposition of Catalan combinatorics at the "Fuss" level of generality.


9. Francois Bergeron "Combinatorics of r-Dyck paths, r-Parking functions and the r-Tamari lattices" and "Multivariate diagonal coinvariant spaces for complex reflection groups". This (multivariate) level of generality is slightly outside the workshop goals, but it is an important direction for the future.



10. Kraft and Procesi's "Classical Invariant Theory: A Primer". Here "classical" does not mean "old," but related to the classical matrix groups. This reference is helpful when reading the Haiman and Etingof papers.


11. Eugene Gorsky and Mikhail Mazin "Compactified Jacobians and q,t-Catalan numbers I and II". This is an interesting new direction, closely related to several aspects of the workshop.

12. Sergey Fomin and Nathan Reading "Root systems and generalized associahedra". A very nice set of notes from PCMI 2004, describing clusters/associahedra and Tamari/Cambrian lattices. (Lots of good pictures.) The cluster side of Catalan combinatorics is slightly under-represented at this workshop, but we want to keep it in mind.


A list of registered participants is available.

Questions or comments to workshops@aimath.org