Abstract
The shuffle conjecture expresses a relationship between parking functions, diagonal harmonics, and the Bergeron-Garsia $\nabla$ operator. Recent conjectures about a family of modified Hall-Littlewood operators made by Haglund, Morse, and Zabrocki sharpen the shuffle conjecture and suggest a variety of combinatorial properties of parking functions. In particular, their 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.
Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
