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.

Here is the beginning of this array (sequence in the OEIS):

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 ${\displaystyle k\geq 0}$, 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.

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