受挫司机的停车问题
Parking with Frustrated Drivers
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中文总结 AI 辅助
本文研究受挫停车函数的计数性质，证明长度为n的此类函数数量为(2n-1)!!，并通过与高度标记Dyck路径关联，给出幸运车辆/停车位分布的相关计数公式。
中文摘要 AI 辅助
设想有 $n$ 辆车沿着一条单行道排列，路上有 $n$ 个停车位。每辆车都载着一群朋友，其中包括一位不情愿的司机。每辆车都有一个首选停车位，车辆依次进入。车辆驶向首选停车位，如果该位为空则停在那里。如果不为空，后排的一位朋友会喊道：“嘿！你本该开快点！”司机对此感到沮丧，继续沿路行驶，直到找到最后一个空位（如果存在的话）并停在那里。我们说一个首选停车位序列 $(a_1,a_2,\dots, a_n)$ 是一个受挫停车函数，如果所有车辆都能在此规则下停车。在本文中，我们研究受挫停车函数的计数性质。特别地，我们证明长度为 $n$ 的受挫停车函数的数量为 $(2n-1)!!$。这是通过将受挫停车函数与高度标记的 Dyck 路径相关联来实现的。利用这种关联，我们能够更好地理解受挫停车函数中幸运车辆和幸运停车位的集合。我们证明长度为 $n$ 的受挫停车函数中，前 $k$ 辆车（或前 $k$ 个停车位）是幸运的数量由 $k!S(n,k)$ 给出，其中 $S(n,k)$ 是第二类斯特林数。这进而意味着，一旦一辆车（或一个停车位）不幸，剩余车辆（或停车位）都不幸的受挫停车函数的数量由第 $n$ 个 Fubini 数计数。我们还证明了具有 $k$ 辆幸运车辆（或幸运停车位）的受挫停车函数的数量由二阶欧拉数给出。
英文摘要
Imagine there are $n$ cars lined up along a one-way street containing $n$ spots. Each car contains a group of friends, including a reluctant driver. Each car has a preferred spot and cars enter one by one. The cars drive to their preferred spot and if it is empty park there. If it is not empty, a friend in the back yells out ``Hey! You should have driven faster!". Frustrated by this, the driver continues down the road until they find the last unoccupied spot (if one exists) and parks there. We say a sequence $(a_1,a_2,\dots, a_n)$ of preferred spots is a frustrated parking function if all cars can park under this rule. In this paper, we study the enumerative properties of frustrated parking functions. In particular, we show that the number of frustrated parking functions of length $n$ is $(2n-1)!!$. This is done by associating frustrated parking functions with height labeled Dyck paths. Using this association, we are then able to better understand the sets of lucky cars and lucky spots for frustrated parking functions. We show that the frustrated parking functions of length $n$ where the first $k$ cars (or first $k$ spots) are lucky is given by $k!S(n,k)$ where $S(n,k)$ is the Stirling number of the second kind. This in turn implies that the number of frustrated parking functions where once a car (or spot) is unlucky, the remaining cars (or spots) are unlucky is counted by the $n^{th}$ Fubini number. We also show that the number of frustrated parking functions with $k$ lucky cars (or spots) is given by the second order Eulerian number.
发表机构
- Loyola Marymount University（洛约拉马利蒙特大学）
- University of California Davis（加州大学戴维斯分校）
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