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Statistic identifier: St001556

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Collection: Permutations

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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 [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

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References: 

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

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Statistic 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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Created: Jun 25, 2020 at 10:01 by Kathrin Meier

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Last Updated: Jun 25, 2020 at 10:52 by Christian Stump