# Derangement Diagrams

Derangement Diagrams

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 elements is called the subfactorial of (with notation ), given by the formula , which is highly reminiscent of =-+-+⋯. The sequence of subfactorials is , for .

n

n

!n

!n=n!-+-++⋯+

1

0!

1

1!

1

2!

1

3!

1

4!

n

(-1)

n!

1

e

1

2!

1

3!

1

4!

1

5!

!n=0,1,2,9,44,265,1854,…

n=1,2,…