  We consider three directed walkers on the square lattice, which move
simultaneously at each tick of a clock and never cross. Their trajectories form
a non-crossing configuration of walks. This configuration is said to be
osculating if the walkers never share an edge, and vicious (or:
non-intersecting) if they never meet. We give a closed form expression for the
generating function of osculating configurations starting from prescribed
points. This generating function turns out to be algebraic. We also relate the
enumeration of osculating configurations with prescribed starting and ending
points to the (better understood) enumeration of non-intersecting
configurations. Our method is based on a step by step decomposition of
osculating configurations, and on the solution of the functional equation
provided by this decomposition.
