Parking functions and Łukasiewicz paths

Thomas Selig; Haoyue Zhu

Parking functions and Łukasiewicz paths

Research output: Contribution to journal › Article › peer-review

Abstract

We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and Łukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the corresponding Łukasiewicz path, and the total displacement to the area of the path. We also study this bijection restricted to two specific families of parking-functions: unit-interval parking functions, and prime parking functions.

Original language	English
Pages (from-to)	77-84
Journal	Discrete Mathematics Letters
Volume	14
Publication status	Published - 6 Nov 2024

Keywords

Parking functions
Łukasiewicz paths
Catalan numbers
Displacement statistic
Bijection

Access to Document

https://www.dmlett.com/archive/v14/DML24_v14_pp77-84.pdfLicence: CC BY

Towards a combinatorial theory of sandpile models
Selig, T.
1/01/23 → 31/12/25
Project: Internal Research Project
Stochastic variants of the Abelian sandpile model
Selig, T.
1/01/22 → 31/12/24
Project: Governmental Research Project

Cite this

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title = "Parking functions and {\L}ukasiewicz paths",

abstract = "We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and {\L}ukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the corresponding {\L}ukasiewicz path, and the total displacement to the area of the path. We also study this bijection restricted to two specific families of parking-functions: unit-interval parking functions, and prime parking functions.",

keywords = "Parking functions, {\L}ukasiewicz paths, Catalan numbers, Displacement statistic, Bijection",

author = "Thomas Selig and Haoyue Zhu",

year = "2024",

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journal = "Discrete Mathematics Letters",

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AU - Selig, Thomas

AU - Zhu, Haoyue

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N2 - We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and Łukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the corresponding Łukasiewicz path, and the total displacement to the area of the path. We also study this bijection restricted to two specific families of parking-functions: unit-interval parking functions, and prime parking functions.

AB - We present a bijection between two well-known objects in the ubiquitous Catalan family: non-decreasing parking functions and Łukasiewicz paths. This bijection maps the maximal displacement of a parking function to the height of the corresponding Łukasiewicz path, and the total displacement to the area of the path. We also study this bijection restricted to two specific families of parking-functions: unit-interval parking functions, and prime parking functions.

KW - Parking functions

KW - Łukasiewicz paths

KW - Catalan numbers

KW - Displacement statistic

KW - Bijection

M3 - Article

SN - 2664-2557

VL - 14

SP - 77

EP - 84

JO - Discrete Mathematics Letters

JF - Discrete Mathematics Letters

ER -

Parking functions and Łukasiewicz paths

Abstract

Keywords

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Projects

Towards a combinatorial theory of sandpile models

Stochastic variants of the Abelian sandpile model

Cite this