Identifier
Values
[1] => [1,0,1,0] => [2,1] => 0
[2] => [1,1,0,0,1,0] => [3,1,2] => 0
[1,1] => [1,0,1,1,0,0] => [2,3,1] => 0
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 0
[2,1] => [1,0,1,0,1,0] => [3,2,1] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 0
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 1
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 0
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 0
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 0
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 1
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 1
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 2
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 1
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 2
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].