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Identifier

Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001821: Signed permutations ⟶ ℤ

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] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,3,2] => [1,2,3] => [1,2,3] => 0
[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] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[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] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[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] => [1,2,3,4] => [1,2,3,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[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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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6$

$F_{4} = 18 + 6\ q$

Description

The sorting index of a signed permutation.
A signed permutation

$\sigma = [\sigma(1),\ldots,\sigma(n)]$ can be sorted

$[1,\ldots,n]$ by signed transpositions in the following way:
First move

$\pm n$ to its position and swap the sign if needed, then

$\pm (n-1), \pm (n-2)$ and so on.
For example for

$[2,-4,5,-1,-3]$ we have the swaps

$[2,-4,5,-1,-3] \rightarrow [2,-4,-3,-1,5] \rightarrow [2,1,-3,4,5] \rightarrow [2,1,3,4,5] \rightarrow [1,2,3,4,5]$
given by the signed transpositions

$(3,5), (-2,4), (-3,3), (1,2)$ .
If

$(i_1,j_1),\ldots,(i_n,j_n)$ is the decomposition of

$\sigma$ obtained this way (including trivial transpositions) then the sorting index of

$\sigma$ is defined as

$\operatorname{sor}_B(\sigma) = \sum_{k=1}^{n-1} j_k - i_k - \chi(i_k < 0),$
where

$\chi(i_k < 0)$ is 1 if

$i_k$ is negative and 0 otherwise.
For

$\sigma = [2,-4,5,-1,-3]$ we have

$\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.$

Map

to signed permutation

Description

The signed permutation with all signs positive.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.