Partial parking functions

Rui Duarte,António Guedes De Oliveira

doi:10.1016/j.disc.2018.10.032

Rui Duarte, António Guedes De Oliveira

Open Access

https://doi.org/10.1016/j.disc.2018.10.032

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Journal: Discrete Mathematics	Publication Date: Nov 15, 2018
Citations: 12	License type: publisher-specific-oa

Affiliation: University of Aveiro, University of Porto

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.

Highlights

In this paper, we characterise the Pak-Stanley labels of the regions of the recently introduced family of the arrangements of hyperplanes “between Shi and Ish”.In other words, for n ∈ N = {1, 2, . . . } there is a labelling of the regions of the n-dimensional Shi arrangement by the n-dimensional parking functions, and the labelling in this case is a bijection
Remember that the parking functions can be characterised as a = (a1, . . . , an) ∈ [n]n such that there is a permutation σ ∈ Sn with aσ(i) ≤ i, for every i ∈ [n]
By labelling under the same rules the regions of the n-dimensional Ish arrangement, we obtain a new bijection between these regions and the so-called Ish-parking functions [5] which can be characterised as a = (a1, . . . , an) ∈ [n]n such that there is a permutation σ ∈ Sn with aσ(i) ≤ i, for every i ∈ [a1] ; σ(i + 1) < σ(i), for every i ∈ [a1 − 1]

Summary

Introduction

We characterise the Pak-Stanley labels of the regions of the recently introduced family of the arrangements of hyperplanes “between Shi and Ish” (cf. [6]). } there is a labelling (due to Pak and Stanley [13]) of the regions of the n-dimensional Shi arrangement (that is, the connected components of the complement in Rn of the union of the hyperplanes of the arrangement) by the n-dimensional parking functions, and the labelling in this case is a bijection. An) ∈ [n]n such that there is a permutation σ ∈ Sn with aσ(i) ≤ i for every i ∈ [a1] and for every i ∈ [k, n] such that σ(i) ≥ k ; σ(i + 1) < σ(i) for every i ∈ [a1 − 1] such that σ(i) < k We call these sets of labels partial parking functions and note that they all have the same number of elements, viz.

Preliminaries

Graphical parking functions

The DFS-Burning Algorithm

Main Theorem