If f is an arbitrary surjection from N onto M, then we can think of f as partitioning N into m different groups, each group of inputs representing the same output point in M. The Stirling Numbers of the second kind count how many ways to partition an N element set into m groups. But this undercounts it, because any permutation of those m groups defines a different surjection but gets counted the same. There are m! such permutations, so our total number of surjections is
m! S(n,m)
To look at the maximum values, define a sequence S_n = n - M_n where M_n is the m that attains maximum value for a given n - in other words, S_n is the "distance from the right edge" for the maximum value. Computer-generated tables suggest that this function is constant for 3-4 values of n before increasing by 1. If this is true, then the m coordinate that maximizes m! S(n,m) is bounded by n - ceil(n/3) - 1 and n - floor(n/4) + 1.
I have no proof of the above, but it gives you a conjecture to work with in the meantime.