A Proof the Functional Equation Conjecture

TL;DR

This paper proves a key functional equation related to the Compositional Shuffle Conjecture, advancing the Garsia-Hicks approach towards its full proof, and discusses future conjectures for completing this proof.

Contribution

It provides a proof of a crucial functional equation that was a major obstacle in proving the Compositional Shuffle Conjecture.

Findings

Proof of the functional equation for a Catalan family of polynomials

Removes a key obstacle in the Garsia-Hicks approach

Formulates new conjectures for future work

Abstract

In the early 2000's the first and second named authors worked for a period of six years in an attempt of proving the Compositional Shuffle Conjecture [1]. Their approach was based on the discovery that all the Combinatorial properties predicted by the Compositional Shuffle Conjecture remain valid for each family of Parking Functions with prescribed diagonal cars. The validity of this property was reduced to the proof of a functional equation satisfied by a Catalan family of univariate polynomials. The main result in this paper is a proof of this functional equation. The Compositional Shuffle Conjecture was proved in 2015 by Eric Carlsson and Anton Mellit [3]. Our proof of the Functional Equation removes one of the main obstacles in the completion of the Garsia-Hicks approach to the proof of the Compositional Shuffle Conjecture. At the end of this writing we formulate a few further conjectures including what remains to be proved to complete this approach.