Lattice Paths Between Diagonal Boundaries

A bivariate symmetric backwards recursion is of the form d[m,n] = w0(d[m− 1, n]+d[m,n−1])+ω1(d[m−r1, n−s1]+d[m−s1, n−r1])+· · ·+ωk(d[m−rk, n−sk] +d[m− sk, n− rk]) where ω0, . . . ωk are weights, r1, . . . rk and s1, . . . sk are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x+ u < y < x− l. With a solution we mean a formula that expresses d[m,n] as a sum of differences of recursions without the band restriction. Depending on the application, the boundary conditions can take different forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x ≥ 0, and d[x− l, x] = 0 for all x ≥ l (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), d[x + u, x] = 0 for all x ≥ 0, and d[x− l+ 1, x] = d[x− l+ 1, x− 1] for x ≥ l (ordinary lattice paths with weighted left turns), and d[y, y − u + 1] = d[y − 1, y − u + 1] for all y ≥ u and d[x − l + 1, x] = d[x − l + 1, x − 1] for x ≥ l. The first theorem is a general form of what is commonly known as repeated application of the Reflection Principle. The second and third theorem are new; we apply them to lattice paths which in addition to the usual North and East steps also make two hook moves, East-North-North and North-East-East. Hook moves differ from knight moves (covered by the first theorem) by being blocked by any piece of the barrier they encounter along their three part move. Submitted: September 9, 1997; Accepted: June 15, 1998 AMS Subject Classification: 0A15