An Arithmetic Triangle Arising From a One-Way Street Grid

Summary Pascal’s triangle arises by counting the number of shortest paths from (0, 0) to (n, k) on a square street grid. The length of the shortest path is the Manhattan distance from (0, 0) to (n, k). We consider the case of a square street grid of one-way streets, with successive parallel streets being oppositely directed. We investigate the associated distance function q (which is only a quasi-metric, since q(A, B) may differ from q(B, A)) and the arithmetic triangle obtained by counting shortest routes on the one-way grid from (0, 0) to (n, k).