Rencontres numbers

In combinatorial mathematics, 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 D_n, 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 D_7, 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, and there are D_7, 2 = 924 possibilities once more, now, that 2 previous couples meet again just by chance.

Numerical values

Here is the beginning of this array (sequence A008290 in OEIS):

$_n\!\!\diagdown\!\!^k$	0	1	2	3	4	5	6	7
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

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

$D_{n,0} = \left[{n! \over e}\right]$

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, 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 − k points.

The numbers D_n,0/(n!) are generated by the power series e^−z/(1 − z); accordingly, an explicit formula for D_n, 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 is the whole 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.^[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.

References

1. Jim Pitman, "Some Probabilistic Aspects of Set Partitions", American Mathematical Monthly, volume 104, number 3, March 1997, pages 201–209.

• Riordan, John, An Introduction to Combinatorial Analysis, New York, Wiley, 1958, pages 57, 58, and 65.
• Weisstein, Eric W., "Partial Derangements", MathWorld.