Rencontres numbers

In combinatorics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set with specified numbers of fixed points: in other words, partial derangements. For n ≥ 0 and 0 ≤ k ≤ n, the rencontres number D_{n, k} is the number of permutations of that have exactly k fixed points.
For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are D_{7, 2} = 924 ways this could happen. Another often cited example is that of a dance school with 7 opposite-sex couples, where, after tea-break the participants are told to randomly find an opposite-sex partner to continue, then once more there are D_{7, 2} = 924 possibilities that exactly 2 previous couples meet again by chance.

Numerical values

Here is the beginning of this array :

	0	1	2	3	4	5	6	7	8
0	1
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
7	1854	1855	924	315	70	21	0	1
8	14833	14832	7420	2464	630	112	28	0	1

Formulas

The numbers in the k = 0 column enumerate derangements. Thus
for non-negative n. It turns out that
where the ratio is rounded up for even n and rounded down for odd n. For n ≥ 1, this gives the nearest integer.
More generally, for any, we have
The proof is easy after one knows how to enumerate derangements: choose the k fixed points out of n; then choose the derangement of the other n − k points.
The numbers are generated by the power series ; accordingly,
an explicit formula for D_{n, m} can be derived as follows:
This immediately implies that
for n large, m fixed.

Probability distribution

The sum of the entries in each row for the table in "Numerical Values" is the total number of permutations of, and is therefore n!. If one divides all the entries in the nth row by n!, one gets the probability distribution of the number of fixed points of a uniformly distributed random permutation of. The probability that the number of fixed points is k is
For n ≥ 1, the expected number of fixed points is 1.
More generally, for i ≤ n, the ith moment of this probability distribution is the ith moment of the Poisson distribution with expected value 1. For i > n, the ith moment is smaller than that of that Poisson distribution. Specifically, for i ≤ n, the ith moment is the ith Bell number, i.e. the number of partitions of a set of size i.

Limiting probability distribution

As the size of the permuted set grows, we get
This is just the probability that a Poisson-distributed random variable with expected value 1 is equal to k. In other words, as n grows, the probability distribution of the number of fixed points of a random permutation of a set of size n approaches the Poisson distribution with expected value 1.