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Derangement Diagrams

number of points n
3
derangement
1
A derangement is a permutation that leaves no element in its original position. For example, (1234) shifts every element over (cyclically), so it is a derangement, but (124) leaves 3 fixed in place, so it is not a derangement. The number of derangements on a set of
n
elements is called the subfactorial of
n
(with notation
!n
), given by the formula
!n=n!
1
0!
-
1
1!
+
1
2!
-
1
3!
+
1
4!
++
n
(-1)
n!
, which is highly reminiscent of
1
e
=
1
2!
-
1
3!
+
1
4!
-
1
5!
+
. The sequence of subfactorials is
!n=0,1,2,9,44,265,1854,
, for
n=1,2,
.
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