Identifier
Values
[[]] => [1,0] => [2,1] => [2,1] => 0
[[],[]] => [1,0,1,0] => [3,1,2] => [1,3,2] => 0
[[[]]] => [1,1,0,0] => [2,3,1] => [3,1,2] => 0
[[],[],[]] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 0
[[],[[]]] => [1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 1
[[[]],[]] => [1,1,0,0,1,0] => [2,4,1,3] => [1,3,4,2] => 0
[[[],[]]] => [1,1,0,1,0,0] => [4,3,1,2] => [1,4,3,2] => 0
[[[[]]]] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,4,5,1,3] => 2
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,4,5,2] => 0
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [2,1,5,4,3] => 0
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,2,4] => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,2,5,4] => 0
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 1
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [1,4,2,5,3] => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,3,4,5,2] => 1
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [1,2,5,4,3] => 0
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,5,2,1,3] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [1,4,5,3,2] => 1
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [1,5,2,4,3] => 1
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 1
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Description
The number of strict 3-descents.
A strict 3-descent of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.