- FindStat

Identifier

St001556: Permutations ⟶ ℤ

Values

=>
                            
                            
                            
                            

                        [1,2]=>0
[2,1]=>0
[1,2,3]=>0
[1,3,2]=>0
[2,1,3]=>0
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>0
[1,2,3,4]=>0
[1,2,4,3]=>1
[1,3,2,4]=>0
[1,3,4,2]=>1
[1,4,2,3]=>0
[1,4,3,2]=>1
[2,1,3,4]=>0
[2,1,4,3]=>1
[2,3,1,4]=>0
[2,3,4,1]=>1
[2,4,1,3]=>0
[2,4,3,1]=>1
[3,1,2,4]=>0
[3,1,4,2]=>1
[3,2,1,4]=>0
[3,2,4,1]=>1
[3,4,1,2]=>0
[3,4,2,1]=>1
[4,1,2,3]=>0
[4,1,3,2]=>1
[4,2,1,3]=>0
[4,2,3,1]=>1
[4,3,1,2]=>0
[4,3,2,1]=>1
[1,2,3,4,5]=>0
[1,2,3,5,4]=>0
[1,2,4,3,5]=>1
[1,2,4,5,3]=>1
[1,2,5,3,4]=>2
[1,2,5,4,3]=>2
[1,3,2,4,5]=>0
[1,3,2,5,4]=>0
[1,3,4,2,5]=>1
[1,3,4,5,2]=>1
[1,3,5,2,4]=>2
[1,3,5,4,2]=>2
[1,4,2,3,5]=>0
[1,4,2,5,3]=>0
[1,4,3,2,5]=>1
[1,4,3,5,2]=>1
[1,4,5,2,3]=>2
[1,4,5,3,2]=>2
[1,5,2,3,4]=>0
[1,5,2,4,3]=>0
[1,5,3,2,4]=>1
[1,5,3,4,2]=>1
[1,5,4,2,3]=>2
[1,5,4,3,2]=>2
[2,1,3,4,5]=>0
[2,1,3,5,4]=>0
[2,1,4,3,5]=>1
[2,1,4,5,3]=>1
[2,1,5,3,4]=>2
[2,1,5,4,3]=>2
[2,3,1,4,5]=>0
[2,3,1,5,4]=>0
[2,3,4,1,5]=>1
[2,3,4,5,1]=>1
[2,3,5,1,4]=>2
[2,3,5,4,1]=>2
[2,4,1,3,5]=>0
[2,4,1,5,3]=>0
[2,4,3,1,5]=>1
[2,4,3,5,1]=>1
[2,4,5,1,3]=>2
[2,4,5,3,1]=>2
[2,5,1,3,4]=>0
[2,5,1,4,3]=>0
[2,5,3,1,4]=>1
[2,5,3,4,1]=>1
[2,5,4,1,3]=>2
[2,5,4,3,1]=>2
[3,1,2,4,5]=>0
[3,1,2,5,4]=>0
[3,1,4,2,5]=>1
[3,1,4,5,2]=>1
[3,1,5,2,4]=>2
[3,1,5,4,2]=>2
[3,2,1,4,5]=>0
[3,2,1,5,4]=>0
[3,2,4,1,5]=>1
[3,2,4,5,1]=>1
[3,2,5,1,4]=>2
[3,2,5,4,1]=>2
[3,4,1,2,5]=>0
[3,4,1,5,2]=>0
[3,4,2,1,5]=>1
[3,4,2,5,1]=>1
[3,4,5,1,2]=>2
[3,4,5,2,1]=>2
[3,5,1,2,4]=>0
[3,5,1,4,2]=>0
[3,5,2,1,4]=>1
[3,5,2,4,1]=>1
[3,5,4,1,2]=>2
[3,5,4,2,1]=>2
[4,1,2,3,5]=>0
[4,1,2,5,3]=>0
[4,1,3,2,5]=>1
[4,1,3,5,2]=>1
[4,1,5,2,3]=>2
[4,1,5,3,2]=>2
[4,2,1,3,5]=>0
[4,2,1,5,3]=>0
[4,2,3,1,5]=>1
[4,2,3,5,1]=>1
[4,2,5,1,3]=>2
[4,2,5,3,1]=>2
[4,3,1,2,5]=>0
[4,3,1,5,2]=>0
[4,3,2,1,5]=>1
[4,3,2,5,1]=>1
[4,3,5,1,2]=>2
[4,3,5,2,1]=>2
[4,5,1,2,3]=>0
[4,5,1,3,2]=>0
[4,5,2,1,3]=>1
[4,5,2,3,1]=>1
[4,5,3,1,2]=>2
[4,5,3,2,1]=>2
[5,1,2,3,4]=>0
[5,1,2,4,3]=>0
[5,1,3,2,4]=>1
[5,1,3,4,2]=>1
[5,1,4,2,3]=>2
[5,1,4,3,2]=>2
[5,2,1,3,4]=>0
[5,2,1,4,3]=>0
[5,2,3,1,4]=>1
[5,2,3,4,1]=>1
[5,2,4,1,3]=>2
[5,2,4,3,1]=>2
[5,3,1,2,4]=>0
[5,3,1,4,2]=>0
[5,3,2,1,4]=>1
[5,3,2,4,1]=>1
[5,3,4,1,2]=>2
[5,3,4,2,1]=>2
[5,4,1,2,3]=>0
[5,4,1,3,2]=>0
[5,4,2,1,3]=>1
[5,4,2,3,1]=>1
[5,4,3,1,2]=>2
[5,4,3,2,1]=>2
                    

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Description

The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the second entry is St001557The number of inversions of the second entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.

Code

def statistic(pi):
    k=3
    n=len(pi)
    return(sum(1 for i in [k+1 .. n] if pi(k)>pi(i)))

Created

Jun 25, 2020 at 10:01 by Kathrin Meier

Updated

Jun 25, 2020 at 10:52 by Christian Stump