[Paper Review] Note on the bijectivity of the Pak-Stanley labelling

Rui Duarte, António Guedes de Oliveira|arXiv (Cornell University)|Jan 10, 2015

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper provides a self-contained, direct proof of the bijectivity of the Pak-Stanley labeling, which maps regions of the Shi arrangement in $$\mathbb{R}^n$$ to parking functions of size $n$. The proof establishes a reversible correspondence via a novel s-parking construction that maps parking functions back to valid pairs of words and intervals, confirming the labeling's injectivity and surjectivity.

ABSTRACT

This article has the sole purpose of presenting a simple, self-contained and direct proof of the fact that the Pak-Stanley labeling is a bijection. The construction behind the proof is subsumed in a forthcoming paper [R. Duarte and A. Guedes de Oliveira, The braid and the Shi arrangements and the Pak-Stanley labeling, Eur. J. Combinatorics, in press.], but an actual self-contained proof is not explicitly included in that paper.

Motivation & Objective

• To provide a self-contained, direct proof of the bijectivity of the Pak-Stanley labeling from regions of the Shi arrangement to parking functions.
• To clarify the structural correspondence between combinatorial objects—regions, valid pairs, and parking functions—via an explicit inverse construction.
• To resolve a gap in prior literature by offering a complete proof of bijectivity not explicitly included in the original Pak-Stanley work.
• To define and analyze the s-parking operation as a key mechanism for reversing the labeling process.

Proposed method

• The labeling is defined via a region-specific permutation $w$ and a set $\mathfrak{I}$ of maximal intervals $[o_i, c_i]$ satisfying $w_{o_i} > w_{c_i}$ and $0 < x_{w_{c_i}} - x_{w_{o_i}} < 1$.
• A valid pair $(w, \mathfrak{I})$ is constructed from any point in a region, and this pair uniquely identifies the region.
• The s-parking operation $S(f)$ is defined to reconstruct the word $w$ from a parking function $f$, using the inverse of the contraction map $\widehat{w}$.
• The inverse map $\varphi_A$ is constructed via $S(f)$, showing that $S(\widehat{w}) = w$ and $\widehat{S(f)} = f$, proving bijectivity.
• The proof relies on the fact that $\lambda(R) = f$ if and only if $S(f)$ yields the unique word $w$ and interval set $\mathfrak{I}$ corresponding to region $R$.
• The construction is shown to be independent of the representative point $x$ in the region, ensuring well-definedness.

Experimental results

Research questions

• RQ1 Is the Pak-Stanley labeling a bijection between regions of the Shi arrangement and parking functions of size $n$?
• RQ2 Can a direct, self-contained proof of this bijectivity be constructed without relying on external results?
• RQ3 What is the precise inverse construction that recovers the region's defining word and interval set from a given parking function?
• RQ4 How does the s-parking operation $S(f)$ reconstruct the original word $w$ from a parking function $f$?
• RQ5 What structural properties of the labeling ensure injectivity and surjectivity?

Key findings

• The Pak-Stanley labeling is a well-defined bijection from the set of regions of the Shi arrangement $\mathcal{S}_n$ to the set of parking functions of size $n$.
• The inverse of the labeling is explicitly constructed via the s-parking operation $S(f)$, which recovers the word $w$ from a parking function $f$.
• The contraction map $\widehat{w}$ satisfies $\widehat{S(f)} = f$, proving that $S$ is a left inverse of $\widehat{w}$.
• The map $S$ is also a right inverse, as $S(\widehat{w}) = w$, confirming that $S$ and $\widehat{w}$ are mutual inverses.
• The labeling is independent of the choice of representative point in a region, ensuring consistency and well-definedness.
• The proof establishes that every valid pair $(w, \mathfrak{I})$ corresponds to exactly one region, and vice versa, confirming the bijective correspondence.