Mitsuki Hanada, John Lentfer, Andrés R. Vindas-Meléndez
{"title":"Generalized Parking Function Polytopes","authors":"Mitsuki Hanada, John Lentfer, Andrés R. Vindas-Meléndez","doi":"10.1007/s00026-023-00671-1","DOIUrl":null,"url":null,"abstract":"<div><p>A classical parking function of length <i>n</i> is a list of positive integers <span>\\((a_1, a_2, \\ldots , a_n)\\)</span> whose nondecreasing rearrangement <span>\\(b_1 \\le b_2 \\le \\cdots \\le b_n\\)</span> satisfies <span>\\(b_i \\le i\\)</span>. The convex hull of all parking functions of length <i>n</i> is an <i>n</i>-dimensional polytope in <span>\\({\\mathbb {R}}^n\\)</span>, which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of <span>\\({\\textbf{x}}\\)</span>-parking functions for <span>\\({\\textbf{x}}=(a,b,\\dots ,b)\\)</span>, which we refer to as <span>\\({\\textbf{x}}\\)</span>-parking function polytopes. We explore connections between these <span>\\({\\textbf{x}}\\)</span>-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of <span>\\({\\textbf{x}}\\)</span>-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 2","pages":"575 - 613"},"PeriodicalIF":0.6000,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-023-00671-1.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00026-023-00671-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
A classical parking function of length n is a list of positive integers \((a_1, a_2, \ldots , a_n)\) whose nondecreasing rearrangement \(b_1 \le b_2 \le \cdots \le b_n\) satisfies \(b_i \le i\). The convex hull of all parking functions of length n is an n-dimensional polytope in \({\mathbb {R}}^n\), which we refer to as the classical parking function polytope. Its geometric properties have been explored in Amanbayeva and Wang (Enumer Combin Appl 2(2):Paper No. S2R10, 10, 2022) in response to a question posed by Stanley (Amer Math Mon 127(6):563–571, 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of \({\textbf{x}}\)-parking functions for \({\textbf{x}}=(a,b,\dots ,b)\), which we refer to as \({\textbf{x}}\)-parking function polytopes. We explore connections between these \({\textbf{x}}\)-parking function polytopes, the Pitman–Stanley polytope, and the partial permutahedra of Heuer and Striker (SIAM J Discrete Math 36(4):2863–2888, 2022). In particular, we establish a closed-form expression for the volume of \({\textbf{x}}\)-parking function polytopes. This allows us to answer a conjecture of Behrend et al. (2022) and also obtain a new closed-form expression for the volume of the convex hull of classical parking functions as a corollary.
期刊介绍:
Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board.
The scope of Annals of Combinatorics is covered by the following three tracks:
Algebraic Combinatorics:
Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices
Analytic and Algorithmic Combinatorics:
Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms
Graphs and Matroids:
Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches