Lattice Paths Between Diagonal Boundaries

Abstract

A bivariate symmetric backwards recursion is of the form $d[m,n]=w_{0}(d[m-1,n]+d[m,n-1])+\omega_{1}(d[m-r_{1},n-s_{1}]+d[m-s_{1},n-r_{1}])+\dots+\omega_{k}(d[m-r_{k},n-s_{k}]+d[m-s_{k},n-r_{k}])$ where $\omega_{0},\dots\omega_{k}$ are weights, $r_{1},\dots r_{k}$ and $s_{1},\dots s_{k}$ 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 $x\geq0$, and $d[x-l,x]=0$ for all $x\geq l$ (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), $d[x+u,x]=0$ for all $x\geq0$, and $d[x-l+1,x]=d[x-l+1,x-1]$ for $x\geq l$ (ordinary lattice paths with weighted left turns), and $d[y,y-u+1]=d[y-1,y-u+1]$ for all $y\geq u$ and $d[x-l+1,x]=d[x-l+1,x-1]$ for $x\geq l$. 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.