Points of maximal traffic on a grid with obstruction

Abstract

For n∈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≥ 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≤ n≤ 375, and intermittently up to n=495. We conjecture that the anomaly disappears for n≥ 496.

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BibTeX RIS

BibTeX

@misc{gil2026points,
  title = {{Points of maximal traffic on a grid with obstruction}},
  author = {Juan Gil and Zhenni Liang and Ayodeji Odetola and Michael Weiner},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01562},
  eprint = {2609.01562},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01562},
}

RIS

TY  - UNPB
TI  - Points of maximal traffic on a grid with obstruction
AU  - Juan Gil
AU  - Zhenni Liang
AU  - Ayodeji Odetola
AU  - Michael Weiner
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.01562
UR  - https://arxiv.org/abs/2609.01562
AB  - For n∈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≥ 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≤ n≤ 375, and intermittently up to n=495. We conjecture that the anomaly disappears for n≥ 496.
ER  -

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