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Bijectivity of a generalized Pak-Stanley labeling

Olivier Bernardi, Neha Goregaokar

Abstract

The Pak-Stanley labeling is a bijection between the regions of the $m$-Shi arrangement and the $m$-parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class $\mathcal{C}$ of arrangements for which this labeling is bijective. The class $\mathcal{C}$ includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in $\mathcal{C}$ are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.

Bijectivity of a generalized Pak-Stanley labeling

Abstract

The Pak-Stanley labeling is a bijection between the regions of the -Shi arrangement and the -parking functions. Mazin generalized this labeling to every deformation of the braid arrangement and proved that this labeling is always surjective onto a set of directed multigraph parking functions. We provide a right inverse to the generalized Pak-Stanley labeling, and identify a class of arrangements for which this labeling is bijective. The class includes the multi-Shi arrangements and the multi-Catalan arrangements. We also show that the arrangements in are the only transitive arrangements for which the generalized Pak-Stanley labeling is bijective.

Paper Structure

This paper contains 7 sections, 14 theorems, 22 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Parking functions and the generalized Pak-Stanley labeling
  4. Sketches and Bernardi labeling
  5. Right inverse to the generalized Pak-Stanley labeling
  6. Bijectivity of the generalized Pak-Stanley labeling for $(\bm, \varepsilon)$-arrangements
  7. Bijectivity of the generalized Pak-Stanley labeling for transitive arrangements

Key Result

Theorem 2.3

For any tuple $\textbf{S}=(S_{i,j})$, the GPS labeling $\lambda : R(\mathcal{A}_{\textbf{S}}) \rightarrow \text{Park}_{\textbf{S}}$ is surjective.

Figures (6)

  • Figure 1: An $(\bm, \varepsilon)$-arrangement with its generalized Pak-Stanley labeling. Here we have $\bm = (1,0,3)$, $\varepsilon_{2,3} = 1$, and $\varepsilon_{i,j} = 0$ for all $(i,j)\neq (2,3)$.
  • Figure 2: The directed multigraph $D_{\textbf{S}}$ for $S_{1,2} = \{0\}$, $S_{1,3} = S_{2,3} = \{0,1\}$.
  • Figure 3: The generalized Pak-Stanley labeling, and Bernardi labeling for $\mathcal{A}_{\textbf{S}}$ with $S_{1,2} = \{0\}$, $S_{1,3} = S_{2,3} = \{0,1\}$. The region $R_0$ is shaded. The associated directed multigraph $D_{\textbf{S}}$ is shown in Fig. \ref{['fig:multigraph']}.
  • Figure 4: Computing $\Psi(p)$ for the parking function $p=(1,0,1)$ for two different $\textbf{S}$.
  • Figure 5: Two arrangements for which either $(X)$ or $(Y)$ does not hold and the GPS labeling is not injective.
  • ...and 1 more figures

Theorems & Definitions (37)

  • Remark 1.2
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: MR3721647
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: Bernardi
  • Definition 2.7
  • Definition 2.8
  • ...and 27 more