Even and Odd Pairs of Lattice Paths with Multiple Intersections

Abstract

Let M n, k r, s be the number of ordered pairs of 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). Let

On Pairs of Lattice Paths with a Given Number of Intersections

“…Moreover, we may drop the condition that endpoints do not coincide (as is the case in Theorem 1): Remember that we are dealing with ordered pairs of paths, i.e., every pair ( p, q) of paths p, q, where both paths start at the origin and end in the same point P (and where p{q), is considered different from the``reversed'' pair (q, p). Here, the common endpoint counts as point of intersection (contrary to the definition in [1,2]; pointed out by Sulanke [6]): Corollary 1. Let 0 t v and u w 0; let P=(t, u), Q=(v, w) be two points in the integer lattice Z_Z (P=Q is possible).…”

“…5]. The definition for M n, k r, s in [1,2] differs slightly from the one given above insofar as intersections at the endpoint (if r=s) are also not counted: This difference is irrelevant for the following theorem, but will simplify notation of the subsequent corollary.…”

“…We now consider the number of ordered pairs of lattice paths (as defined in Corollary 1) for the special case P=Q=(x, y): Adapting the notation from [1,2] (remember that the intersection at P=Q is not counted there), we denote this number by N x+ y, x k&1 . First, it is clear that if k is maximal, i.e., k=x+ y, then the number of pairs of lattice paths is simply ( x+ y x ) (since the paths must be equal in this case).…”

“…A first formula for M n, k r, s was given in [1,Thm. 2]; the formula we want to prove here was given in [2,Thm. 5].…”