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Identifier
Values
=>
[1,2]=>0 [2,1]=>0 [1,2,3]=>0 [1,3,2]=>1 [2,1,3]=>0 [2,3,1]=>1 [3,1,2]=>0 [3,2,1]=>1 [1,2,3,4]=>0 [1,2,4,3]=>0 [1,3,2,4]=>1 [1,3,4,2]=>1 [1,4,2,3]=>2 [1,4,3,2]=>2 [2,1,3,4]=>0 [2,1,4,3]=>0 [2,3,1,4]=>1 [2,3,4,1]=>1 [2,4,1,3]=>2 [2,4,3,1]=>2 [3,1,2,4]=>0 [3,1,4,2]=>0 [3,2,1,4]=>1 [3,2,4,1]=>1 [3,4,1,2]=>2 [3,4,2,1]=>2 [4,1,2,3]=>0 [4,1,3,2]=>0 [4,2,1,3]=>1 [4,2,3,1]=>1 [4,3,1,2]=>2 [4,3,2,1]=>2 [1,2,3,4,5]=>0 [1,2,3,5,4]=>0 [1,2,4,3,5]=>0 [1,2,4,5,3]=>0 [1,2,5,3,4]=>0 [1,2,5,4,3]=>0 [1,3,2,4,5]=>1 [1,3,2,5,4]=>1 [1,3,4,2,5]=>1 [1,3,4,5,2]=>1 [1,3,5,2,4]=>1 [1,3,5,4,2]=>1 [1,4,2,3,5]=>2 [1,4,2,5,3]=>2 [1,4,3,2,5]=>2 [1,4,3,5,2]=>2 [1,4,5,2,3]=>2 [1,4,5,3,2]=>2 [1,5,2,3,4]=>3 [1,5,2,4,3]=>3 [1,5,3,2,4]=>3 [1,5,3,4,2]=>3 [1,5,4,2,3]=>3 [1,5,4,3,2]=>3 [2,1,3,4,5]=>0 [2,1,3,5,4]=>0 [2,1,4,3,5]=>0 [2,1,4,5,3]=>0 [2,1,5,3,4]=>0 [2,1,5,4,3]=>0 [2,3,1,4,5]=>1 [2,3,1,5,4]=>1 [2,3,4,1,5]=>1 [2,3,4,5,1]=>1 [2,3,5,1,4]=>1 [2,3,5,4,1]=>1 [2,4,1,3,5]=>2 [2,4,1,5,3]=>2 [2,4,3,1,5]=>2 [2,4,3,5,1]=>2 [2,4,5,1,3]=>2 [2,4,5,3,1]=>2 [2,5,1,3,4]=>3 [2,5,1,4,3]=>3 [2,5,3,1,4]=>3 [2,5,3,4,1]=>3 [2,5,4,1,3]=>3 [2,5,4,3,1]=>3 [3,1,2,4,5]=>0 [3,1,2,5,4]=>0 [3,1,4,2,5]=>0 [3,1,4,5,2]=>0 [3,1,5,2,4]=>0 [3,1,5,4,2]=>0 [3,2,1,4,5]=>1 [3,2,1,5,4]=>1 [3,2,4,1,5]=>1 [3,2,4,5,1]=>1 [3,2,5,1,4]=>1 [3,2,5,4,1]=>1 [3,4,1,2,5]=>2 [3,4,1,5,2]=>2 [3,4,2,1,5]=>2 [3,4,2,5,1]=>2 [3,4,5,1,2]=>2 [3,4,5,2,1]=>2 [3,5,1,2,4]=>3 [3,5,1,4,2]=>3 [3,5,2,1,4]=>3 [3,5,2,4,1]=>3 [3,5,4,1,2]=>3 [3,5,4,2,1]=>3 [4,1,2,3,5]=>0 [4,1,2,5,3]=>0 [4,1,3,2,5]=>0 [4,1,3,5,2]=>0 [4,1,5,2,3]=>0 [4,1,5,3,2]=>0 [4,2,1,3,5]=>1 [4,2,1,5,3]=>1 [4,2,3,1,5]=>1 [4,2,3,5,1]=>1 [4,2,5,1,3]=>1 [4,2,5,3,1]=>1 [4,3,1,2,5]=>2 [4,3,1,5,2]=>2 [4,3,2,1,5]=>2 [4,3,2,5,1]=>2 [4,3,5,1,2]=>2 [4,3,5,2,1]=>2 [4,5,1,2,3]=>3 [4,5,1,3,2]=>3 [4,5,2,1,3]=>3 [4,5,2,3,1]=>3 [4,5,3,1,2]=>3 [4,5,3,2,1]=>3 [5,1,2,3,4]=>0 [5,1,2,4,3]=>0 [5,1,3,2,4]=>0 [5,1,3,4,2]=>0 [5,1,4,2,3]=>0 [5,1,4,3,2]=>0 [5,2,1,3,4]=>1 [5,2,1,4,3]=>1 [5,2,3,1,4]=>1 [5,2,3,4,1]=>1 [5,2,4,1,3]=>1 [5,2,4,3,1]=>1 [5,3,1,2,4]=>2 [5,3,1,4,2]=>2 [5,3,2,1,4]=>2 [5,3,2,4,1]=>2 [5,3,4,1,2]=>2 [5,3,4,2,1]=>2 [5,4,1,2,3]=>3 [5,4,1,3,2]=>3 [5,4,2,1,3]=>3 [5,4,2,3,1]=>3 [5,4,3,1,2]=>3 [5,4,3,2,1]=>3
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
Code
def statistic(pi):
    k=2
    n=len(pi)
    return(sum(1 for i in [k+1 .. n] if pi(k)>pi(i)))
Created
Jun 25, 2020 at 09:59 by Kathrin Meier
Updated
Jun 25, 2020 at 10:52 by Christian Stump