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A permutation involution with

1↔2

,

3↔4

:

In[]:=

[◼]	PermutationInvolutionQ

@{2,1,4,3}

Out[]=

True

The permutation

{2,3,1}

is not an involution because

1↦2↦3≠1

:

In[]:=

[◼]	PermutationInvolutionQ

@{2,3,1}

Out[]=

False

These are the 10 involutive permutation on four elements, shown compactly:

In[]:=

Row/@SelectPermutations@Range@4,

[◼]	PermutationInvolutionQ



Out[]=

{1234,1243,1324,1432,2134,2143,3214,3412,4231,4321}