Today we will continue our discussion on path counting in random walks.
The number of paths from (m,x) to (n,y) that do not touch level z < min(x,y) is given by :
(n-m) Choose 1/2 (n-m+y-x) - (n-m) Choose (1/2(n-m-y-x)-z)
We prove it this way:
Here the relative positions of x and y with respect to z plays any role as compared to the equation we derived in the earlier post where we found the number of paths for level z = 0 or the horizontal axis.  So we can take the relative distance the other way and state that
the number of paths between (m,x) to (n,y) that do  not touch level z =
the number of paths between (m, x+z) to (n, y+z) that do not touch level 0 or horizontal axis.
which is equal to
(n-m) Choose 1/2(n-m+ y + z -x -z ) - (n-m) Choose 1/2 (n - m - y -z -x -z )
resulting in
(n-m) Choose 1/2 (n-m+y-x) - (n-m) Choose (1/2(n-m-y-x)-z)
Another corollary is that the number of paths from (0,0) to (n,y) where y >=0 , that remain >= 0 is given by
n Choose (n+y)/2  - n Choose ((n+y)/2 + 1)
We prove it this way:
The target number of paths is the same as the number of paths that do not touch level -1 and are above it
which is the same as 
the number of paths between (0,1) and (n, y+1) that remain > 0
which we can write as
n Choose 1/2(n+y) - n Choose 1/2(n-y-1-1)
=n Choose 1/2(n+y) - n Choose (1/2(n-y) - 1)
Now we use the equation we derived at the beginning of this post for paths that do not touch level z < min(x,y) = -1

and rewrite the above as
n Choose 1/2(n+y) - n Choose (1/2 (n-y-1-1) - (-1))