A partial derangement is a permutation of distinct items with a specified number of fixed points, generalizing
a derangement, which has none (Evans et al. 2002,
p. 385).
Let
count the permutations of distinct items with exactly items in their original positions, where is a nonnegative integer
and .
These counts include the cases in which all or none of the items remain in place.
Then
(1)
(2)
(3)
where
is a binomial coefficient and is the subfactorial.
The second identity assumes . The counts are called rencontres numbers. In particular, is the subfactorial,
and the name rencontres numbers is also used for this one-parameter sequence.
The following table lists for , 1, ..., 8, with running from 0 to in each row:
count the permutations of 
distinct items with exactly 
items in their original positions, where 
is a nonnegative integer
and 
.
These counts include the cases in which all or none of the items remain in place.
Then
is a binomial coefficient and 
is the subfactorial.
The second identity assumes 
. The counts 
are called rencontres numbers. In particular, 
is the subfactorial,
and the name rencontres numbers is also used for this one-parameter sequence.
for 
, 1, ..., 8, with 
running from 0 to 
in each row:
