UC San Diego

The "Shuffle Conjecture" states that the bigraded Frobeneus characteristic of the space of diagonal harmonics (equal to the operator nabla applied to the nth elementary symmetric function) can be computed as the weighted sum of combinatorial objects called parking functions. In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials given by nabla applied to composition-indexed, rescaled Hall-Littlewood polynomials. They conjecture that these polynomials enumerate a composition indexed family of parking functions weighted by the same statistics. This refinement of the nearly decade old "Shuffle Conjecture" when combined with properties of the Hall-Littlewood polynomials implies the existence of certain bijections between these families of parking functions. The existence of these bijections then follows from some relatively simple properties of a certain recursively constructed family of polynomials. This work introduces those polynomials, explains their connection to the conjecture of Haglund, Morse, and Zabrocki, and explores some of their surprising properties, both proven and conjectured. The result is an intriguing new approach to the Shuffle Conjecture and a deeper understanding of some classical parking function statistics