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:

$\mathrm{P}\mathrm{r}(n,r)=\frac{D_{n-r}}{r!(n-r)!}$

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. Further information: expected number of fixed points of permutation equals one

Particular cases

Number of permutations

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.

$n$	0	1	2	3	4	5	6
1	0	1
2	1	0	1
3	2	3	0	1
4	9	8	6	0	1
5	44	45	20	10	0	1
6	265	264	135	40	15	0	1