Generalized Parking Function Polytopes

IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED Annals of Combinatorics Pub Date : 2023-11-06 DOI:10.1007/s00026-023-00671-1
Mitsuki Hanada, John Lentfer, Andrés R. Vindas-Meléndez
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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.

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广义停车函数多面体
长度为 n 的经典驻留函数是一个正整数列表((a_1, a_2, \ldots , a_n)\),它的非递减重排(b_1 \le b_2 \le \cdots \le b_n\)满足(b_i \le i\ )。长度为 n 的所有驻车函数的凸环是 \({\mathbb {R}}^n\) 中的一个 n 维多形,我们称之为经典驻车函数多形。Amanbayeva 和 Wang(Enumer Combin Appl 2(2):论文编号 S2R10, 10, 2022)针对 Stanley(Amer Math Mon 127(6):563-571, 2020)提出的问题探讨了它的几何性质。我们通过研究\({textbf{x}}=(a,b,\dots ,b)\)的\({textbf{x}}\)-泊车函数的凸壳的几何性质来推广这个多边形家族,我们称之为\({textbf{x}}\)-泊车函数多边形。我们探讨了这些({\textbf{x}}\)-泊车函数多面体、皮特曼-斯坦利多面体以及豪尔和斯特里克的部分包络多面体之间的联系(SIAM J Discrete Math 36(4):2863-2888, 2022)。特别是,我们建立了一个闭式表达式,用于表示停泊函数多面体的体积。这使我们能够回答贝伦德等人(2022)的猜想,并作为推论得到经典停车函数凸壳体积的新闭式表达式。
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来源期刊
Annals of Combinatorics
Annals of Combinatorics 数学-应用数学
CiteScore
1.00
自引率
0.00%
发文量
56
审稿时长
>12 weeks
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
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