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, then once more there are D_7, 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 $k$ .
But one can also order them in columns corresponding to the number of moved elements $i=n-k$ (sequence A098825 in the OEIS).
Each entry in the table below on the left can be factored into two terms given in the table below on the right: the product of a binomial coefficient (given first in red) and a subfactorial (given second in blue).
In this order each column corresponds to one subfactorial: $~~~T(n,i)={\binom {n}{i}}~\cdot ~!i$

$i$ $n$	0	1	2	3	4	5	6	7	8
0	1
1	1	0
2	1	0	1
3	1	0	3	2
4	1	0	6	8	9
5	1	0	10	20	45	44
6	1	0	15	40	135	264	265
7	1	0	21	70	315	924	1855	1854
8	1	0	28	112	630	2464	7420	14832	14833

$i$ $n$	0	1	2	3	4	5	6	7	8
0	1⋅1
1	1⋅1	1⋅0
2	1⋅1	2⋅0	1⋅1
3	1⋅1	3⋅0	3⋅1	1⋅2
4	1⋅1	4⋅0	6⋅1	4⋅2	1⋅9
5	1⋅1	5⋅0	10⋅1	10⋅2	5⋅9	1⋅44
6	1⋅1	6⋅0	15⋅1	20⋅2	15⋅9	6⋅44	1⋅265
7	1⋅1	7⋅0	21⋅1	35⋅2	35⋅9	21⋅44	7⋅265	1⋅1854
8	1⋅1	8⋅0	28⋅1	56⋅2	70⋅9	56⋅44	28⋅265	8⋅1854	1⋅14833

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 − 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 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.[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

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