##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## Drew Armstrong

#### University of Miami

## Rational Catalan Combinatorics

##### Abstract:

In recent years the sequence of integers Cat(n)=(n choose 2)/(n+1) called ``Catalan numbers'' has been extended to a family of integers Cat(a,b)=(a+b choose a)/(a+b) that is parametrized by rational numbers a/b. These numbers originally showed up as the number of lattice paths in an aXb rectangle that stay above the diagonal. In 2002, Jaclyn Anderson gave a bijection between these paths and so-called (a,b)-core partitions. These are integer partitions in which no cell has hook length divisible by a or b. This result unlocked many new ideas in the area between combinatorics and representation theory. On the one hand, there have been many combinatorial conjectures and slightly fewer proofs. On the other hand, it seems that the numbers Cat(a,b) ultimately come from the representation theory of rational Cherednik algebras. The existence of a symmetric (q,t)- graded version of the numbers Cat(a,b) suggests that there should be a ``rational'' generalization of Mark Haiman's results on the Hilbert scheme of points in $C^2$.

Host: Brendon Rhoades

### March 16, 2017

### 4:00 PM

### AP&M 6402

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