Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 PermutationInvolutionQ

Source Notebook

Test whether a permutation equals its inverse

Contributed by: Wolfram Staff

ResourceFunction["PermutationInvolutionQ"][p]

tests whether a permutation p equals its inverse.

Details

In general, a function f is an involution if f(f(x))=x for all x in the domain of f. For a permutation p, that is equivalent to p=p-1. A permutation is easily seen to be involutive if and only if it is comprised entirely of cycles of lengths 1 or 2.

Examples

Basic Examples (2) 

A permutation involution with 1↔2, 3↔4:

In[1]:=
ResourceFunction["PermutationInvolutionQ"][{2, 1, 4, 3}]
Out[1]=

The permutation {2,3,1} is not an involution because 1↦2↦3≠1:

In[2]:=
ResourceFunction["PermutationInvolutionQ"][{2, 3, 1}]
Out[2]=

Scope (1) 

Test if a permutation in disjoint cyclic format is an involution:

In[3]:=
ResourceFunction["PermutationInvolutionQ"][Cycles[{{1, 3}, {4, 5}}]]
Out[3]=

Applications (1) 

These are the 10 involutive permutation on 4 elements, shown compactly:

In[4]:=
Row /@ Select[Permutations[Range[4]], ResourceFunction[ "PermutationInvolutionQ"]]
Out[4]=

Publisher

George Beck

Version History

  • 1.0.1 – 07 February 2022
  • 1.0.0 – 08 July 2019

License Information