Lattice paths between diagonal boundaries

Abstract

A bivariate symmetric backwards recursion is of the form d[m, n] = w 0(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-s k, 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.