Identifier
Values
[1] => [1,0,1,0] => [2,1] => [1,2] => 0
[2] => [1,1,0,0,1,0] => [3,1,2] => [1,3,2] => 0
[1,1] => [1,0,1,1,0,0] => [2,3,1] => [1,2,3] => 0
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [1,4,3,2] => 1
[2,1] => [1,0,1,0,1,0] => [2,1,3] => [1,2,3] => 0
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [1,2,3,4] => 0
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [1,3,2,4] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [1,3,2,4] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [1,2,3,4] => 0
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [1,4,3,2,5] => 1
[3,2] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [1,3,4,2] => 1
[3,1,1] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [1,2,3,4] => 0
[2,2,1] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [1,2,4,3] => 1
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [1,2,3,4,5] => 0
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [1,4,5,3,2] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [1,3,2,4,5] => 0
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [1,4,2,5,3] => 0
[3,2,1] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [1,2,3,4] => 0
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [1,2,3,4,5] => 0
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [1,3,5,2,4] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [1,2,3,5,4] => 0
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [1,4,2,3,5] => 0
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [1,3,2,4,5] => 0
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [1,2,3,4,5] => 0
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [1,3,2,5,4] => 0
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [1,3,2,4,5] => 0
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [1,2,3,4,5] => 0
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [1,2,4,3,5] => 1
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [1,3,5,4,2] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [1,3,4,5,2] => 1
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [1,2,3,4,5] => 0
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [1,3,2,4,5] => 0
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [1,2,5,4,3] => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [1,2,4,5,3] => 1
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [1,3,4,2,5] => 1
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [1,2,3,5,4] => 0
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [1,2,3,4,5] => 0
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [1,2,4,3,5] => 1
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [1,2,3,4,5] => 0
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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.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.