Evidence for parking conjectures

TL;DR

This paper proves that for certain infinite families of irreducible real reflection groups, the strongest conjecture related to parking functions and Coxeter-Catalan theory holds true in a generic sense, advancing understanding in algebraic combinatorics.

Contribution

The paper generalizes parking function conjectures to Fuss-Catalan levels and proves the strongest case for infinite families ABCDI, linking to Coxeter-Catalan theory.

Findings

Proves the strongest parking conjecture for infinite families ABCDI.

Establishes generic truth of conjectures in Coxeter-Catalan theory.

Extends parking function models to Fuss-Catalan generalizations.

Abstract

Let $W$ be an irreducible real reflection group. Armstrong, Reiner, and the author presented a model for parking functions attached to W and made three increasingly strong conjectures about these objects. The author generalized these objects and conjectures to the Fuss-Catalan level of generality. Even the weakest of these conjectures would imply a collection of facts in Coxeter-Catalan theory which are at present understood only in a case-by-case fashion. We prove that when $W$ belongs to any the infinite families ABCDI, the strongest of these conjectures is generically true.