Rencontres numbers

In combinatorics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements. (Rencontre is French for encounter. By some accounts, the problem is named after a solitaire game.)  For n ≥ 0 and 0 ≤ k ≤ n, the rencontres number Dn, k is the number of permutations of { 1, ..., n } 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 D7, 2 = 924 ways this could happen. Another often cited example is that of a dance school with 7 couples, where, after tea-break the participants are told to randomly find a partner to continue, then once more there are D7, 2 = 924 possibilities that 2 previous couples meet again by chance.

Numerical values
Here is the beginning of this array :

Formulas
The numbers in the k = 0 column enumerate derangements. Thus
 * $$D_{0,0} = 1, \!$$
 * $$D_{1,0} = 0, \!$$
 * $$D_{n+2,0} = (n + 1)(D_{n+1,0} + D_{n,0}) \!$$

for non-negative n. It turns out that


 * $$D_{n,0} = \left\lceil\frac{n!}{e}\right\rfloor,$$

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 $$k\geq 0$$, we have


 * $$D_{n,k} = {n \choose k} \cdot D_{n-k,0}.$$

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 &minus; k points.

The numbers $D_{n,0}/(n!)$ are generated by the power series $e^{&minus;z}/(1 &minus; z)$; accordingly, an explicit formula for Dn, m can be derived as follows:
 * $$ D_{n, m}

= \frac{n!}{m!} [z^{n-m}] \frac{e^{-z}}{1-z} = \frac{n!}{m!} \sum_{k=0}^{n-m} \frac{(-1)^k}{k!}.$$

This immediately implies that


 * $$ D_{n, m} = {n \choose m} D_{n-m, 0} \; \; \mbox{ and } \; \;

\frac{D_{n, m}}{n!} \approx \frac{e^{-1}}{m!}$$

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 { 1, ..., n }, 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 { 1, ..., n }. The probability that the number of fixed points is k is


 * $${D_{n,k} \over n!}.$$

For n ≥ 1, the expected number of fixed points is 1 (a fact that follows from linearity of expectation).

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


 * $$\lim_{n\to\infty} {D_{n,k} \over n!} = {e^{-1} \over k!}. $$

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.