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

TL;DR

This paper introduces a conjecture and an algorithm for constructing bijections between parking functions with different diagonal compositions, advancing understanding of the shuffle conjecture and related combinatorial structures.

Contribution

It formulates a new conjecture, provides an algorithm for bijection construction, proves a special case, and explores applications in parking function theory.

Findings

Proposed a conjecture for parking function bijections.

Developed an algorithm to construct these bijections.

Proved a special case of the conjecture.

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.