<p>A natural question here is to ask: for any given k, what is the maximum number of monotonic paths avoiding a set S of k points (where the maximum is over all possible sets S)? </p>

<p>This is actually an open problem raised in a paper of Johnson, Leader and Russell: <a href="http://arxiv.org/pdf/1309.4643.pdf" rel="nofollow">http://arxiv.org/pdf/1309.4643.pdf</a></p>
Number of monotonic paths in a rectangular grid avoiding certain points
<p>In a rectangular grid of size m*n, the number of paths from (0,0) to (m,n) (without backtracking) is (m+n)!/(m!*n!). Now if there are certain points in the grid which we want to avoid, how can we calculate the number of paths avoiding those points?</p>