Combinatorics Through Guided Discovery, 2004a

172 D. Hints to Selected Problems
48.b. InProblem 47 you saw that we had to make ten choices of north or east,
choosing north four times.
48.c. This problem is actually a bit tricky. What happens to the answer if i > m or
j > n? Remember that paths go up or to the right.
49.a. Where can you go from (0,0) in one step? In two steps? In any of these cases,
what can you say about the sum of the coordinates of a point you can get to?
Can you find any other relationship between the x- and y-coordinates of a
point you can get to? For example, can you get to the point (1, 3)?
49.c. How many choices do you have to make in order to choose a path?
50.c. In each part, each such sequence corresponds to a path that can’t cross over
(but may touch) a certain line.
51.b. Given a path from (0, 0) to (n, n) which touches or crosses the line y = x +1,
how can you modify the part of the path from (0, 0) to the first touch of
y = x +1so that the modified path starts instead at (−1, 1)? The trick is to
do this in a systematic way that will give you your bijection.
51.c. A path either touches the line y = x +1or it doesn’t. This partitions the set
of paths into two blocks.
52.b. Look back at the definition of a Dyck Path and a Catalan Path.
52.c. What makes this part difficult is understanding how we are partitioning the
paths. As an example, B 0 is the set of all paths that have no upsteps following
the last absolute minimum. Can such a path have downsteps after the last
absolute minimum? (The description we gave of B 0 is not succinct enough to
be the answer to the second question of this part.) As another example B 1 is
the set of all paths that have exactly one upstep and perhaps some downsteps
after the last absolute minimum. Is it possible, though, for a path in B 1 to have
any downsteps after the last absolute minimum? A path in B 2 has exactly two
upsteps after its last absolute minimum. If is possible to have one downstep
after the last absolute minimum, but it has to be in a special place. What
place is that? Now to figure out how many parts our partition has, we need
to know the maximum number of upsteps a path can have following its last
absolute minimum. What is this maximum? It might help to draw some
pictures with n =5or 6. In particular, is it possible that all upsteps occur
after the last absolute minimum?
52.e. Using d fordownandu for up, we could have uudduuddudud as our Catalan
path. Suppose that i =5. The fifth upstep is the u in position 9. Thus
F = uudduudd, U = u, and B = dud. Now BUF is duduuudduudd. This
is a Dyck path that begins by going below the x-axis. The d’s in positions 1
and 3 take the path to the y-coordinate −1. Then the y coordinate climbs to 2,