Even and Odd Pairs of Lattice Paths with Multiple Intersections

LetMr,sn,kbe the number of ordered pairs of paths in the plane, with unit stepsEorN, that intersectktimes in which the first path ends at the point (r,n−r) and the second path ends at the point (s,n−s). Let[formula]and[formula]We study the numbersMr,sn,k,Nkn,r=Mr,rn,k,NE(n,k,p), andNO(n,k,p), prove several simple relations among them, and derive a simpler formula forMr,sn,kthan appears in [1].