# 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):

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

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, 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 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:

This immediately implies that

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 *n*th 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

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 *i*th moment of this probability distribution is the *i*th moment of the Poisson distribution with expected value 1.^{[1]} For *i* > *n*, the *i*th moment is smaller than that of that Poisson distribution. Specifically, for *i* ≤ *n*, the *i*th moment is the *i*th 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.

## References

- ↑ 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*.

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