  Suppose $k$ balls are dropped into $n$ boxes independently with uniform
probability, where $n, k$ are large with ratio approximately equal to some
positive real $\lambda$. The maximum box count has a counterintuitive behavior:
first of all, with high probability it takes at most two values $m_n$ or
$m_n+1$, where $m_n$ is roughly $\frac{\ln n}{\ln \ln n}$. Moreover, it
oscillates between these two values with an unusual periodicity. In order to
prove this statement and various generalizations, it is first shown that for
$X_1,...,X_n$ independent and identically distributed discrete random variables
with common distribution $F$, under mild conditions, the limiting distribution
of their maximum oscillates in three possible families, depending on the tail
of the distribution. The result stated at the beginning follows from the
equivalence of ensemble for the order statistics in various allocations
problems, obtained via conditioning limit theory. Results about the number of
ties for the maximum, as well as applications, are also provided.
