Suppose you roll a dice 100 times, How many times would you expect the most common number to show up.


I.e. roll a dice 100 times and document the frequency of each value, then repeat this process infinitely many times and take the mean of the highest frequency from each trial.


Is there a way to derive a formula or approach to calculate such a value? thanks.



Dice roll expectation question
According to the multinomial probability mass function formula, the expected maximum frequency in $n$ rolls of a fair die is
$$e\_n=\frac1{6^n}\sum\frac{n!}{x\_1!\cdots x\_6!}\,\max(x\_1,\dots,x\_6),$$
where the sum is taken over all $n$-tuples $(x\_1,\dots,x\_6)$ of nonnegative integers such that $x\_1+\dots+x\_6=n$. There seems to be no reason for the existence of a simpler expression for $e\_n$.


Mathematica computes
$$e\_{30}=\frac{3063261583291047469655}{383808888404050968576}$$
in about 7 sec, and
$$e\_{40}=\frac{936567872552422596737147305735}{92829823186414819915547541504}$$
in about 33 sec.
It will likely take too long to compute $e\_{100}$.


