Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [3,1,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,4,2] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,4,3] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,1,4] => [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,3,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,3,1,2] => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
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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
to signed permutation
Description
The signed permutation with all signs positive.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.