Identifier
Values
[] => [] => [1] => [1] => 0
[[]] => [1,0] => [2,1] => [2,1] => 1
[[],[]] => [1,0,1,0] => [3,1,2] => [3,1,2] => 1
[[[]]] => [1,1,0,0] => [2,3,1] => [2,3,1] => 1
[[],[],[]] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 2
[[],[[]]] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 3
[[[]],[]] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 1
[[[],[]]] => [1,1,0,1,0,0] => [4,3,1,2] => [4,3,1,2] => 3
[[[[]]]] => [1,1,1,0,0,0] => [2,3,4,1] => [2,3,4,1] => 2
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Description
The Stasinski-Voll length of a signed permutation.
The Stasinski-Voll length of a signed permutation $\sigma$ is
$$ L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\}, $$
where $n$ is the size of $\sigma$.
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
to signed permutation
Description
The signed permutation with all signs positive.