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 (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 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
The usual way (table above) to show the rencontres numbers is in columns corresponding to the number of fixed points . | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
|
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 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.
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
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
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.
See also
- Oberwolfach problem, a different mathematical problem involving the arrangement of diners at tables
- Problème des ménages, a similar problem involving partial derangements
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.
- v
- t
- e
univariate
with finite support | |
---|---|
with infinite support |
univariate
univariate
continuous- discrete |
---|
(joint)
- Discrete:
- Ewens
- Multinomial
- Continuous:
- Dirichlet
- Multivariate Laplace
- Multivariate normal
- Multivariate stable
- Multivariate t
- Normal-gamma
- Matrix-valued:
- LKJ
- Matrix normal
- Matrix t
- Matrix gamma
- Wishart
- Univariate (circular) directional
- Circular uniform
- Univariate von Mises
- Wrapped normal
- Wrapped Cauchy
- Wrapped exponential
- Wrapped asymmetric Laplace
- Wrapped Lévy
- Bivariate (spherical)
- Kent
- Bivariate (toroidal)
- Bivariate von Mises
- Multivariate
- von Mises–Fisher
- Bingham
and singular
- Degenerate
- Dirac delta function
- Singular
- Cantor
Category
Commons