Talmudic Lattice Path Counting

Consider all planar walks, with positive unit steps (1, 0) and (0, 1), from the origin (0, 0) to a given point (a, b). Let L be the line joining the beginning to the end, i.e., the line b x a y = 0. Call the region below L "downtown," and the region above L "uptown," the line L being the border-line between downtown and uptown. Each such walk has a + b 1 points, not counting the endpoints. For i = 0 , . . . , a + b 1, let W~ be the set of walks with "exactly" i points downtown and "exactly" a + b 1 i points uptown. H o w do we treat those walks that have some points o n L ? If there are i points downtown and j points on L, then each of the sets IV,,., W~+1,... , W/+ i has equal claim to this walk. If it is possible to