##### Department of Mathematics,

University of California San Diego

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### Final Defense

## Emily Sergel

#### UCSD

## The Combinatorics of nabla pn and connections to the Rational Shuffle Conjecture

##### Abstract:

The symmetric function operator, nabla, introduced by Bergeron and Garsia (1999), has many astounding combinatorial properties. The (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005) relates nabla en to parking functions. The rational Compositional Shuffle Conjecture of the author, Bergeron, Garsia, and Xin (2015) relates a whole family of operators (closely linked to nabla) to rational parking functions. In (2007), Loehr and Warrington conjectured a relationship between nabla pn and preference functions. We prove this conjecture and provide another combinatorial interpretation in terms of parking functions. This new formula reveals a connection between nabla pn and an operator appearing in the rational Compositional Shuffle Conjecture at $t = 1/q$.

Advisor: Adriano Garsia

### May 17, 2016

### 11:30 AM

### AP&M 7321

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