Even and Odd Pairs of Lattice Paths with Multiple Intersections

TL;DR

This paper studies lattice paths with multiple intersections, deriving simplified formulas for counting paths with specific endpoint conditions and analyzing even and odd sum cases of endpoint coordinates.

Contribution

It introduces a new, simpler formula for counting intersecting lattice paths and analyzes the sum of such paths based on the parity of endpoint coordinate sums.

Findings

Derived a simpler formula for M(n,k,r,s)

Analyzed even and odd cases of r+s separately

Provided explicit counts for paths with multiple intersections

Abstract

Let M(n,k,r,s) be the number of ordered paths in the plane, with unit steps E or N, that intersect k times in which the first path ends at the point (r,n-r) and the second path ends at the point (s,n-s). Our main object of study in this paper is the sum of the numbers M(n,k,r,s) over r and s where r+s is fixed. We consider even and odd values of r+s separately, and we derive a simpler formula for M(n,k,r,s) than previously appeared in the literature.