Koroljuk's formula for counting lattice paths revisited

TL;DR

This paper generalizes Koroljuk's formula for counting lattice paths above certain lines, providing an explicit formula for paths from arbitrary points and introducing a new recurrence-based proof approach.

Contribution

It extends Koroljuk's original formula to paths from arbitrary starting points above lines with rational slopes, using a novel recurrence method.

Findings

Derived an explicit formula for lattice paths from (a,b) to (m,n) above y=kx-r.

Introduced a new recurrence-based proof approach.

Generalized Koroljuk's original formula.

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.