论文标题
广义停车功能多面体
Generalized parking function polytopes
论文作者
论文摘要
长度为$ n $的经典停车功能是正整数$（a_1，a_2，\ ldots，a_n）$的列表，其非额定重新安排$ b_1 \ leq b_2 \ leq b_2 \ leq \ leq \ cdots \ cdots \ leq b_n $满意。所有长度$ n $的所有停车功能的凸壳是$ \ Mathbb {r}^n $中的$ n $二维polytope，我们称之为经典的停车功能polytope。它的几何特性已在（Amanbayeva和Wang 2022）中探讨了（Stanley 2020）中提出的问题。我们通过研究$ \ Mathbf {x} $的凸壳的几何特性来概括这个多面体 - $ \ Mathbf {x} =（a，b，b，\ dots，b）$的停车功能，我们称为$ \ Mathbf {x} $ - {x} $ - 停车功能函数。我们探索这些$ \ Mathbf {x} $ - 停车功能polytopes，Pitman-Stanley Polytope和（Heuer and Striker 2022）的部分定居者之间的连接。特别是，我们为$ \ mathbf {x} $ - 停车功能polytopes的音量建立了封闭形式的表达式。这使我们能够回答（Behrend等，2022）的猜想，并为经典停车功能作为推论的凸壳的体积获得新的闭合形式表达式。
A classical parking function of length $n$ is a list of positive integers $(a_1, a_2, \ldots, a_n)$ whose nondecreasing rearrangement $b_1 \leq b_2 \leq \cdots \leq b_n$ satisfies $b_i \leq i$. 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 2022) in response to a question posed in (Stanley 2020). We generalize this family of polytopes by studying the geometric properties of the convex hull of $\mathbf{x}$-parking functions for $\mathbf{x}=(a,b,\dots,b)$, which we refer to as $\mathbf{x}$-parking function polytopes. We explore connections between these $\mathbf{x}$-parking function polytopes, the Pitman-Stanley polytope, and the partial permutahedra of (Heuer and Striker 2022). In particular, we establish a closed-form expression for the volume of $\mathbf{x}$-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.
