##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Angela Hicks

#### UCSD

## Parking Function Bijection Suggested by the Haglund-Morse-Zabrocki Conjecture

##### Abstract:

In recent work Jim Haglund, Jennifer Morse and Mike Zabrocki introduce a new statistic on Parking Functions, the ``diagonal composition,'' which gives the lengths of the intervals between successive diagonal hits of the Dyck path. They conjectured that the nabla operator, when applied to certain modified Hall-Littlewood functions indexed by compositions, yields the weighted sum of the corresponding Parking Functions by area, dinv, and Gessel quasisymmetric function. This conjecture then gives a sharpening of the ``shuffle conjecture'' and suggests several combinatorial conjectures about the parking functions. In particular, we discuss a bijective map on the parking functions implied by the commutativity properties of the modified Hall-Littlewood polynomials that appear in their conjecture.

### November 23, 2010

### 3:00 PM

### AP&M 7321

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