Identifier
Values
{{1}} => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [2,3,1] => 2
{{1,2},{3}} => [2,1,3] => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [2,3,4,1] => 3
{{1,2,3},{4}} => [2,3,1,4] => [2,3,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [2,4,3,1] => 3
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}} => [3,2,4,1] => [3,2,4,1] => 3
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => 4
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => 2
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => 4
{{1},{2,3,4}} => [1,3,4,2] => [1,3,4,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => 1
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
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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 permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to signed permutation
Description
The signed permutation with all signs positive.