发表机构
RECITS Laboratory, USTHB（USTHB RECITS实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出圆形s-选择停车模型，利用旋转对称性证明空车位等分布，得到首个闭式乘积公式，并验证其细化性质。
AI 中文摘要
我们研究了 $s$-选择停车模型的圆形变体：$n$ 辆车停放在排列在圆上的 $m=n+1$ 个车位上，每辆车携带一个锚点以及 $s-1$ 个顺时针增量，这些增量至少相隔 $d$，且返回间隙至少为 $d$；一辆车按顺序尝试其选择，然后从最后一次选择开始顺时针扫描。在圆上，每辆车都能停车，且恰好有一个车位保持为空。利用 Pollak 证明计数 $(n+1)^{n-1}$ 的精神中的旋转对称性，我们证明了空车位恰好是等分布的，这产生了使任何指定车位为空的偏好数量的闭式公式 $m^{n-1}\inom{m-sd+s-1}{s-1}^{n}$。这似乎是多选择停车领域中第一个闭式乘积公式。我们进一步证明了一个细化：在具有指定增量的每一类偏好中，空车位仍然是恰好等分布的，这解释了公式的乘积结构，并且对于 $s=2$、$d=1$，解释了经典计数 $(n+1)^{n-1}$ 作为 $(n+1)^{n-1}n^{n}$ 的一个因子出现。可接受的元组通过 Kaplansky 关于圆形选择的引理进行枚举，所有结果均通过穷举计算机枚举验证。
英文摘要
We study a circular variant of the $s$-choice parking model: $n$ cars park on $m=n+1$ spots arranged on a circle, each car carrying an anchor and $s-1$ clockwise increments at least $d$ apart with return gap at least $d$; a car tries its choices in order and then sweeps clockwise from its last choice. On the circle every car parks and exactly one spot remains empty. Exploiting rotational symmetry in the spirit of Pollak's proof of the count $(n+1)^{n-1}$, we prove that the empty spot is exactly equidistributed, which yields the closed formula $m^{n-1}\binom{m-sd+s-1}{s-1}^{n}$ for the number of preferences leaving any prescribed spot empty. This appears to be the first closed product formula in the multi-choice parking landscape. We further prove a refinement: within every class of preferences with prescribed increments, the empty spot is still exactly equidistributed, which explains the product structure of the formula and, for $s=2$, $d=1$, the appearance of the classical count $(n+1)^{n-1}$ as a factor of $(n+1)^{n-1}n^{n}$. The admissible tuples are enumerated through Kaplansky's lemma on circular selections, and all results are verified by exhaustive computer enumeration.
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