Identifier
Values
=>
[1]=>[1]=>0 [1,2]=>[1,2]=>0 [2,1]=>[2,1]=>1 [1,2,3]=>[1,2,3]=>0 [1,3,2]=>[1,3,2]=>1 [2,1,3]=>[2,1,3]=>1 [2,3,1]=>[2,3,1]=>2 [3,1,2]=>[3,1,2]=>3 [3,2,1]=>[3,2,1]=>2 [1,2,3,4]=>[1,2,3,4]=>0 [1,2,4,3]=>[1,2,4,3]=>1 [1,3,2,4]=>[1,3,2,4]=>1 [1,3,4,2]=>[1,3,4,2]=>2 [1,4,2,3]=>[1,4,2,3]=>3 [1,4,3,2]=>[1,4,3,2]=>2 [2,1,3,4]=>[2,1,3,4]=>1 [2,1,4,3]=>[2,1,4,3]=>2 [2,3,1,4]=>[2,3,1,4]=>2 [2,3,4,1]=>[2,3,4,1]=>3 [2,4,1,3]=>[2,4,1,3]=>4 [2,4,3,1]=>[2,4,3,1]=>3 [3,1,2,4]=>[3,1,2,4]=>3 [3,1,4,2]=>[3,1,4,2]=>4 [3,2,1,4]=>[3,2,1,4]=>2 [3,2,4,1]=>[3,2,4,1]=>3 [3,4,1,2]=>[3,4,1,2]=>4 [3,4,2,1]=>[3,4,2,1]=>5 [4,1,2,3]=>[4,1,2,3]=>6 [4,1,3,2]=>[4,1,3,2]=>4 [4,2,1,3]=>[4,2,1,3]=>5 [4,2,3,1]=>[4,2,3,1]=>3 [4,3,1,2]=>[4,3,1,2]=>5 [4,3,2,1]=>[4,3,2,1]=>4
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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.