Asymptotic distribution of multiplicities when selecting $k$ integers from $[1,n]$ When selecting $k$ integers from $[1,n]$ uniformly and with replacement, what does the distribution of multiplicities look like? For example, on average how many numbers will be selected exactly $m$ times? And what is the standard deviation?
Instead of an exact answer, I would like to consider where $k$ is a fraction of $n$, and discussing the asymptotics of the results as $n$ increases. So if we write $k=cn$ for some constant $0<c$, then I shall denote the distribution $f(c,m)$ as the expected fraction of the values selected exactly $m$ times. I am also interested in the standard deviation from this expected value.
I've tried some numerical tests with $k=n=10000000$, and then at least up to m=6 the distribution appears to follow:
$$ f(1,m) = \frac{1}{e\ m!}$$
And the fluctuations from these averages appear quite small.  I'm really hoping there is a nice closed form solution for these distributions.
 A: The number of values that occur exactly $m$ times can be written as the sum of indicator random variables.
Letting $U_i:={\bf 1}_{[\text{value } i \text{ occurs }m\text{ times}]}$ we have 
$N_m=\sum_{i=1}^n U_i$. Using the binomial distribution,
 we calculate $$\mathbb{E}(U_i)=\mathbb{P}(\text{value } i \text{ occurs }m\text{ times})
={k\choose m}\left({1\over n}\right)^m\left(1-{1\over n}\right)^{k-m},\tag1$$
so that $$\mathbb{E}(N_m)=n{k\choose m}\left({1\over n}\right)^m\left(1-{1\over n}\right)^{k-m}.$$ 
If you set $k=cn$ and let $n\to\infty$ in (1) you find that $\mathbb{E}(N_m/n)\to {c^m\over m!}\, e^{-c}.$

The random variables $U_i$ are exchangeable so that 
$$\mbox{Var}(N_m)=\sum_{i,j}\mbox{Cov}(U_i,U_j)=n\mbox{Var}(U_1)+n(n-1)\mbox{Cov}(U_1,U_2),$$ and thus
$$n\mbox{Var}(N_m/n)=\mbox{Var}(U_1)+(n-1)\mbox{Cov}(U_1,U_2).\tag2$$
The limit of the right hand side of (2) as $n\to\infty$ is the coefficient of $n^{-1}$ in the asymptotic expansion of $\mbox{Var}(N_m/n).$ 
The first term is easy. Defining $p={c^m\over m!}\, e^{-c}$, we have
$$\mbox{Var}(U_1)=\mathbb{E}(U_1)-\mathbb{E}(U_1)^2\to p-p^2. $$
The second term is a bit more complicated. 
Using the multinomial distribution we have 
\begin{eqnarray*}
\mbox{Cov}(U_1,U_2)&=&\mathbb{E}(U_1U_2)-\mathbb{E}(U_1)\mathbb{E}(U_2)\\[8pt]
&=&{k\choose m, m, k-2m}\left({1\over n}\right)^m\left({1\over n}\right)^m\left(1-{1\over n}\right)^{k-2m}-\left[{k\choose m}\left({1\over n}\right)^m\left(1-{1\over n}\right)^{k-m}\right]^2.
\end{eqnarray*}
Multiplying by $n-1$ and taking the limit gives 
$$(n-1)\mbox{Cov}(U_1,U_2)\to p^2(m^2c+2m-c),$$
so that the desired coefficient is $p+p^2(m^2c+2m-c-1).$
