
 Catwalks, Sandsteps and Pascal Pyramids 
  Their number is the same as the total number of paths already counted, except that y is replaced by y+2, i.e., r and s are each decreased by 1. 
  For a given path, say C, of such a random walk the total number of paths is found which, after each step, do not lie to the left of the corresponding point of C, and which touch C in a prescribed way in exactly r of the last s left steps. 
  The number of ``good paths'' from p to q is the total number of paths from p to q minus the number of ``bad paths'' from p to q (those which touch the line). 
 www.cs.uwaterloo.ca /journals/JIS/VOL3/GUY/catwalks.html (7889 words) 
