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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.
Here is the beginning of this array (sequence A008290 in the OEIS):
k n | 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 |
ordered by number of moved elements | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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The usual way (table above) to show the rencontres numbers is in columns corresponding to the number of fixed points . | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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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 Dn,0/(n!) are generated by the power series e−z/(1 − z); accordingly, an explicit formula for Dn, m can be derived as follows:
This immediately implies that
for n large, m fixed.
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
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.
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.
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