Proof of a Conjecture on the Slit Plane Problem

الأصل · EN

Let aᵢ,ⱼ(n) denote the number of walks in n steps from (0,0) to (i,j), with steps (± 1,0) and (0,± 1), never touching a point (-k,0) with k≥ 0 after the starting point. and Schaeffer conjectured a closed form for the number a₋ᵢ,ᵢ(2n) when i≥ 1. In this paper, we prove their conjecture, and give a formula for a₋ᵢ,ᵢ(2n) for i≤ -1.