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] ...

• Type: Journal
• Year: 1998
• Where: COMBINATORICS
• Authors: Heinrich Niederhausen
• Keywords: COMBINATORICS 1998, Positive Integers, Symmetric Backwards, Symmetric Backwards Recursion