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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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
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.