A Parking Function Bijection supporting the Haglund–Morse–Zabrocki Conjectures

The shuffle conjecture expresses a relationship between parking functions, diagonal harmonics, and the Bergeron–Garsia ∇ operator. Recent conjectures by Haglund, Morse, and Zabrocki about a family of modified Hall–Littlewood operators sharpen the shuffle conjecture and suggest a variety of combinatorial properties of parking functions. In particular, these conjectures combined with previously established commutativity laws of the Hall–Littlewood operators suggest the existence of certain bijections relating parking functions with different diagonal compositions. In this paper, we formulate a conjecture which yields an algorithm for the construction of these bijections, prove a special case, and give some applications.