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.
Another often cited example is that of a dance school with 7 opposite-sex couples, where, after tea-break the participants are told to randomly find an opposite-sex partner to continue, then once more there are D7, 2 = 924 possibilities that exactly 2 previous couples meet again by chance.
Here is the beginning of this array (sequence A008290 in the OEIS): The usual way (table above) to show the rencontres numbers is in columns corresponding to the number of fixed points
But one can also order them in columns corresponding to the number of moved elements
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!.
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.