• 1 Department of Mathematics [Univ California San Diego]

In a 2010 paper Haglund, Morse, and Zabrocki studied the family of polynomials $\nabla C_{p1}\dots C_{pk}1$ , where $p=(p_1,\ldots,p_k)$ is a composition, $\nabla$ is the Bergeron-Garsia Macdonald operator and the $C_\alpha$ are certain slightly modified Hall-Littlewood vertex operators. They conjecture that these polynomials enumerate a composition indexed family of parking functions by area, dinv and an appropriate quasi-symmetric function. This refinement of the nearly decade old ``Shuffle Conjecture,'' when combined with properties of the Hall-Littlewood operators can be shown to imply the existence of certain bijections between these families of parking functions. In previous work to appear in her PhD thesis, the author has shown that the existence of these bijections follows from some relatively simple properties of a certain family of polynomials in one variable x with coefficients in $\mathbb{N}[q]$. In this paper we introduce those polynomials, explain their connection to the conjecture of Haglund, Morse, and Zabrocki, and explore some of their surprising properties, both proven and conjectured.