Probability distribution of number of fixed points of permutations

Definition
Let $$n$$ be a natural number. Consider the uniform distribution on the symmetric group of degree $$n$$. The probability distribution of number of fixed points is the probability distribution on the set $$\{ 0,1,2,\dots,n \}$$ where the probability associated with $$r$$ is the probability that a given permutation has $$r$$ fixed points.

The probability distribution is given explicitly by:

$$\! \operatorname{Pr}(n,r) = \frac{D_{n-r}}{r!(n - r)!}$$

where $$D_{n-r}$$ is the derangement number, i.e., the number of permutations on a set of size $$n - r$$ with no fixed point.

The mean (expected value) of this distribution is one.

Particular cases
The first table lists number of permutations. The probabilities are obtained by dividing these numbers by $$n!$$. The column headers are for number of fixed points.

The second table gives the probabilities:

Mathematica implementation
We need the Combinatorica package, that can be loaded using:

<< Combinatorica`

To determine the probability that a permutation on $$n$$ letters has $$r$$ fixed points, do:

Binomial[n, r] * NumberOfDerangements[n - r]/Factorial[n]

To output the whole list for a given $$n$$, do:

(Binomial[n, #] * NumberOfDerangements[n - #]/Factorial[n]) & /@ Range[0, n]

This can be further mapped for small values of $$n$$; for instance:

f[n_] := (Binomial[n, #] * NumberOfDerangements[n - #]/Factorial[n]) & /@ Range[0, n]

f /@ Range[7]