论文标题
概率停车功能
Probabilistic Parking Functions
论文作者
论文摘要
我们通过引入随机性和新的停车协议来考虑经典停车功能的概念，这是受论文``停车功能：选择自己的冒险''中的工作的启发。在我们的结果中，我们证明，从长度$ n $偏好向量获得停车功能的可能性独立于概率参数$ p $。我们还探索了优先矢量的属性，因为它是一个停车功能，并讨论了概率参数$ p $的效果。特别感兴趣的是$ p = 1/2 $，我们在某些停车统计中表现出急剧过渡。我们还提出了停车协议的几个有趣的组合后果。特别是，我们为OEI A220884中描述的阵列提供了一种组合解释，作为与占领停车位有关的特定属性的预期数量序列，该序列解决了2020年提出的Novelli和Thibon的开放问题（Arxiv：Arxiv：1209.5959）。最后，我们将结果与组合学中的其他加权现象联系起来，并为研究提供了进一步的方向。
We consider the notion of classical parking functions by introducing randomness and a new parking protocol, as inspired by the work presented in the paper ``Parking Functions: Choose your own adventure,'' (arXiv:2001.04817) by Carlson, Christensen, Harris, Jones, and Rodríguez. Among our results, we prove that the probability of obtaining a parking function, from a length $n$ preference vector, is independent of the probabilistic parameter $p$. We also explore the properties of a preference vector given that it is a parking function and discuss the effect of the probabilistic parameter $p$. Of special interest is when $p=1/2$, where we demonstrate a sharp transition in some parking statistics. We also present several interesting combinatorial consequences of the parking protocol. In particular, we provide a combinatorial interpretation for the array described in OEIS A220884 as the expected number of preference sequences with a particular property related to occupied parking spots, which solves an open problem of Novelli and Thibon posed in 2020 (arXiv:1209.5959). Lastly, we connect our results to other weighted phenomena in combinatorics and provide further directions for research.
