##### Department of Mathematics,

University of California San Diego

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### Combinatorics Seminar

## Adriano Garsia

#### UCSD

## Some new Symmetric Functions Operators and Parking Functions - Part 2

##### Abstract:

The main result presented in this talk is a plethystic formula for the specialization at $t = 1/q$ of the $Q_{u,v}$ operators studied in [math arKiv:1405.0316]. This discovery yields elementary and direct derivations of several identities relating these operators at $t =1/q$ to the Rational Compositional Shuffle Conjecture of [math arKiv: 1404.4616]. In particular we are able to give a direct derivation of a simple formula for the symmetric polynomial $$Q_{km,kn}1|_{t=1/q} \ \mbox{(for all $m,n$ co-prime and $k \geq 1 $.)}$$ We also give an elementary proof that this polynomial is Schur positive. Moreover, by combining our main result with the Rational Compositional Shuffle Conjecture, we obtain a completely elementary derivation of the identity expressing this polynomial in terms of Parking functions in the $km \times km$ rectangle.

### December 17, 2014

### 9:00 AM

### AP&M 7218

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