Function Repository Resource:

Derangements

Give all the ways the numbers 1, 2, …, n can be scrambled completely

Contributed by: Wolfram Staff

ResourceFunction["Derangements"][n]

gives all lists that are rearrangements of {1,2,…,n} such that no element appears in its original position.

Examples

In the following two lists, no element appears in its original place:

In[1]:=

Out[1]=

Here is the next case, condensed:

In[2]:=

Out[2]=

The number of lists grows with n!/e:

In[3]:=

Out[3]=

In fact the number of derangements is given by the built-in subfactorial function:

In[4]:=

Out[4]=