Talmudic lattice path counting (Q1336450)

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. For \(i= 0,1,\dots, a+ b-1\), let \(W_ i\) be the set of walks with ``exactly'' \(i\) points above and ``exactly'' \(a+ b+ 1- i\) points below \(L\). The authors show that it is possible to distribute the points on \(L\) in such a way that the sets \(W_ i\) are equinumerous.