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arXiv · 1306.6015

Koroljuk's formula for counting lattice paths revisited

James J. Y. Zhao
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Abstract

Koroljuk gave a summation formula for counting the number of lattice paths from $(0,0)$ to $(m,n)$ with $(1,0), (0,1)$-steps in the plane that stay strictly above the line $y=k(x-d)$, where $k$ and $d$ are positive integers. In this paper we obtain an explicit formula for the number of lattice paths from $(a,b)$ to $(m,n)$ above the diagonal $y=kx-r$, where $r$ is a rational number. Our result slightly generalizes Koroljuk's formula, while the former can be essentially derived from the latter. However, our proof uses a recurrence with respect to the starting points, and hereby presents a new approach to Koroljuk's formula.

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James J. Y. Zhao. 2013-06-25. Koroljuk's formula for counting lattice paths revisited. https://arxiv.org/abs/1306.6015

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