Counting Pairs of Lattice Paths by Intersections

Consider an r × (n − r) plane lattice rectangle, and walks that begin at the origin (south-west corner), proceed with unit steps in either of the directions east or north, and terminate at the north-east corner of the rectangle. For each integer k we ask for N k , the number of ordered pairs of these walks that intersect in exactly k points. The number of points in the intersection of two such walks is defined as the cardinality of the intersection of their two sets of vertices, excluding the initial and terminal vertices. The figure below shows a pair of such walks where r = 9, n = 17, and k = 5.