OpenAlex ·
Points of maximal traffic on a grid with obstruction
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.01562
- Original
- Read it at the publisher
- Topics
Abstract
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 $n\ge 496$.
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