AI 中文总结
研究将停车函数推广到停车完成，解决了Adeniran等人提出的关于停车完成数量与Pitman-Stanley多面体体积联系的开放问题，通过证明定理说明停车完成集是$\mathbf{x}$-停车函数集。
AI 中文摘要
停车函数对应于$n$辆汽车依次进入单行道停车的偏好情况，每辆车停在大于或等于其偏好的第一个可用车位且所有车都成功停车。我们将停车函数推广到停车完成：给定一些车已停在一组按序列$\mathbf{t}$索引的车位上，再考虑一个偏好列表$\mathbf{c}$，其中$\mathbf{t}$的长度加上$\mathbf{c}$的长度等于$n$。若所有车都能停车，则称$\mathbf{c}$是一个停车完成。Adeniran等人（2020）提出一个开放问题，通过对小$n$值的显式计算来建立停车完成数量与Pitman-Stanley多面体体积之间的联系。本文通过证明一个定理解决了这个开放问题，该定理表明停车完成集就是$\mathbf{x}$-停车函数集。
英文摘要
Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. We generalize parking functions to parking completions: Here, we are given that some cars have already parked in a set of spots, which are indexed in a sequence $\mathbf{t}$. We then consider a preference list $\mathbf{c}$, where length of $\mathbf{t}$ + length of $\mathbf{c}$ = $n$. If all cars can park, we say that $\mathbf{c}$ is a parking completion. Adeniran et al. (2020) state an open problem which proposes a connection between the number of parking completions to the volumes of Pitman-Stanley polytopes by explicit computation on small values of $n$. In this paper, we provide a solution to this open problem by proving a theorem which explains that the set of parking completions is the set of $\mathbf{x}$-parking functions.
Comments5 pages + 1 page references/appendices
