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 .