Partial Parking Functions

TL;DR

This paper characterizes the labels of regions in a family of hyperplane arrangements that smoothly transition between the Shi and Ish arrangements, providing new insights into their combinatorial structure.

Contribution

It introduces a characterization of Pak-Stanley labels for a continuum of hyperplane arrangements bridging Shi and Ish arrangements.

Findings

Identifies the structure of labels across the interpolating arrangements

Provides a unified framework connecting Shi and Ish arrangements

Enhances understanding of hyperplane arrangement combinatorics

Abstract

We characterise the Pak-Stanley labels of the regions of a family of hyperplane arrangements that interpolate between the Shi arrangement and the Ish arrangement.