We analyze the asymptotic behavior of the average maximal number of balls
in a bin obtained by throwing uniformly at random
r balls without
replacement into
n bins,
T times. Writing the expected maximum
as
a recent preprint
of Behrouzi-Far and Zeilberger asks for an explicit expression for
Cn,r in terms of
n,
r and
![$\pi$](abs/img2.gif)
.
In this short note, we find an
expression for
Cn,r in terms of
n,
r and the expected maximum of
n independent standard Gaussians. This provides asymptotics for large
n as well as closed forms for small
n--e.g.,
![$C_{4,2} = \frac{3}{2
\pi^{3/2}} \arccos(-1/3)$](abs/img3.gif)
--and shows that computing a closed form for
Cn,r is precisely as hard as the difficult question of finding the
expected maximum of
n independent standard Gaussians.
Received August 19 2019; revised versions received October 17 2019; October 23 2019.
Published in Journal of Integer Sequences,
December 30 2019.