To sum up the derivation for those looking for crib notes; model the problem as choosing k items u.a.r. from [1..N]. Compute P(max = i), and from this compute the expectation of the max. After simpliciation this is given in terms of k and N, and hence we have an estimator for N in terms of k and max.

According to Wikipedia the estimator is (n+1)/ns-1 not (n+1)/n(s+1) as the author stated.