发表机构
Penn State Altoona; University of North Texas（宾夕法尼亚州立大学阿洛纳分校; 北德克萨斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨存在障碍物B的n×n网格上，避开B的格路的最大通行点位置，发现n≥9时最大值集中在近端点的10个点，而B在反对角线x+y=n上时会出现异常迁移，推测n≥496时该异常消失。
AI 中文摘要
对于自然数n，我们考虑仅使用单位北向和东向步长从(0,0)到(n,n)的格路集合。给定一个需要避开的点B，我们询问在以(0,0)和(n,n)为角的网格上，不同于端点的哪个点A会被最多的避开B的格路经过？我们证明，对于n≥9，无论B的位置如何，最大值都出现在靠近网格两个端点的10个特定点之一。然而，这种稳定性掩盖了一个有趣的异常现象：当障碍物B位于反对角线x+y=n上时，最大通行点会从近角点(1,1)和(n-1,n-1)迁移到可能的最大化点集合中的边界点。这种迁移在8≤n≤375时都会发生，在n=495时仍间歇性出现。我们推测，该异常现象在n≥496时消失。
英文摘要
For $n\in\mathbb{N}$, we consider the set of lattice paths from $(0,0)$ to $(n,n)$ using only unit north and east steps. Given a point $B$ to be avoided, we ask: at which point $A$ on the grid with corners $(0,0)$ and $(n,n)$, different from the endpoints, does the largest number of $B$-avoiding lattice paths pass through? We show that for $n\ge 9$, regardless of the location of $B$, the maximum is attained at one of ten specific points clustered near the two endpoints of the grid. This stability, however, conceals an interesting anomaly. When the obstruction $B$ lies on the antidiagonal $x+y=n$, the points of maximal traffic migrate from the near-corner points $(1,1)$ and $(n-1,n-1)$ to boundary points in the set of possible maximizers. The migration occurs for every $8\le n\le 375$, and intermittently up to $n=495$. We conjecture that the anomaly disappears for $n\ge 496$.
Comments19 pages, 7 figures
