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Function Repository Resource:

ToInversionVector

Source Notebook

Get the inversion vector of a permutation written as a list

Contributed by: Wolfram Staff

ResourceFunction["ToInversionVector"][p]

gives the inversion vector of the permutation list p.

Details and Options

The inversion vector of a permutation of length n lists the number of times k is preceded by an entry greater than k, where k runs from 1 to n.

Examples

Basic Examples

This permutation has three numbers greater than 1 before 1, two numbers greater than 2 before 2 and so on:

In[1]:=
p1 = {4, 3, 2, 1};

Therefore, this is its inversion vector:

In[2]:=
ResourceFunction["ToInversionVector"]@p1
Out[2]=

Here is another example:

In[3]:=
p2 = PermutationList@RandomPermutation@12
Out[3]=
In[4]:=
ResourceFunction["ToInversionVector"]@p2
Out[4]=

Scope

Here are all six permutation of {1,2,3}:

In[5]:=
s3 = Permutations@{1, 2, 3}
Out[5]=

Here are the corresponding inversion vectors:

In[6]:=
ivs = ResourceFunction["ToInversionVector"] /@ s3
Out[6]=

In the factorial base, ivs represents the numbers from 0 to 3!-1:

In[7]:=
FromDigits[#, MixedRadix[{3, 2, 1}]] & /@ ivs
Out[7]=

Resource History

See Also

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