Identifier
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [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] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [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 first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.