Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.7 |

University of Illinois at Chicago

Department of Mathematics, Statistics and Computer Science

Chicago, IL 60607-7045

USA

**Abstract:**

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*C*_{n,r} in terms of *n*,*r* and .
In this short note, we find an
expression for *C*_{n,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.,
--and shows that computing a closed form for
*C*_{n,r} is precisely as hard as the difficult question of finding the
expected maximum of *n* independent standard Gaussians.

a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for

Received August 19 2019; revised versions received October 17 2019; October 23 2019.
Published in *Journal of Integer Sequences*,
December 30 2019.

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