Distribution of maximum frequency of uniformly distributed integers If I roll an M sided dice N times, there will be at least one number that occurs most frequently. What's the distribution of that maximum frequency in terms of M and N? (its pmf and name if it has one)
e.g. say I roll a 6 sided dice 4 times and get [6, 2, 1, 1] the maximum frequency is 2, if I get [5, 3, 5, 3] then it is also 2.
 A: We roll an $m$-sided die $n$ times. Let $X_i$ denote the number of times that side $i$ of the die appears, where $i \in \{1, \dots, m\}$.  Then, the joint pmf of $(X_1, X_2,\dots, X_m)$ is $\text{Multinomial}(n,\frac1m)$ i.e.:
$$P\left(X_1=x_1,\ldots ,X_m=x_m\right) \; = \; \frac{n! }{ x_1! \cdots  x_m!}  \; \frac{1}{m^n} \quad \text{ subject to: } \quad 
\sum _{i=1}^m x_i=n$$
The OP's problem is to find the pmf of: $\quad Y = \max \big(X_i \big)$
which can be termed the maximum order statistic of a Multinomial. 
Calculation
I am not aware that the pmf of $Y$ has a closed-form  ... nor a name. However, exact solutions to the pmf can be calculated for given values of $n$ and $m$. There are some papers that propose methods ... using iterated stochastic matrices or iterated probability calculations, but seem to involve a lot of work.
But there seems to be an easier way. Here is some code, using  Mathematica,  that reduces the above problem to a one-liner:
MaxPMF[n_, m_] := ParallelTable[ 
  PDF[OrderDistribution[ MultinomialDistribution[n, Table[1/m, m]], m], i], {i,n}]

For example, with  $n=4$ throws and $m=6$ sides, the exact pmf of the maximum is then obtained with:
MaxPMF[4, 6]

which returns:

$\left\{\frac{5}{18},\frac{5}{8},\frac{5}{54},\frac{1}{216}\right\}$

... denoting the pmf of $Y$, corresponding to $Y =$ 1, 2, 3 and 4 i.e.

A calculation such as:  MaxPMF[20, 10] takes about $\frac13$ of a second on my computer.
I have checked these results with exact alternative methods (evaluating every possible composition of $n$ objects into $m$ places, and then mapping the Multinomial probabilities over every possible combination)- and get a perfect match - and checked with Monte Carlo too.
The same approach will work just as well, of course, even if the probability vector is not identical.
Alternative
There is a paper by:


*

*Corrado (2010), "The exact distribution of the maximum, minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies,  Statistics and Computing, p.349-359


which can be downloaded here: https://www.jstor.org/stable/2347220
... that finds the solution as a product of stochastic matrices.
