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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St001821
-----------------------------------------------------------------------------
Collection: Signed permutations
-----------------------------------------------------------------------------
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
$$
\operatorname{sor}_B(\sigma) = (5-3) + (4-(-2)-1) + (3-(-3)-1) + (2-1) = 13.
$$
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References: [1] Petersen, T. K. The sorting index [[arXiv:1007.1207]]
-----------------------------------------------------------------------------
Code:
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 3
[-1,2] => 1
[-1,-2] => 4
[2,1] => 1
[2,-1] => 2
[-2,1] => 3
[-2,-1] => 2
[1,2,3] => 0
[1,2,-3] => 5
[1,-2,3] => 3
[1,-2,-3] => 8
[-1,2,3] => 1
[-1,2,-3] => 6
[-1,-2,3] => 4
[-1,-2,-3] => 9
[1,3,2] => 1
[1,3,-2] => 4
[1,-3,2] => 7
[1,-3,-2] => 4
[-1,3,2] => 2
[-1,3,-2] => 5
[-1,-3,2] => 8
[-1,-3,-2] => 5
[2,1,3] => 1
[2,1,-3] => 6
[2,-1,3] => 2
[2,-1,-3] => 7
[-2,1,3] => 3
[-2,1,-3] => 8
[-2,-1,3] => 2
[-2,-1,-3] => 7
[2,3,1] => 2
[2,3,-1] => 3
[2,-3,1] => 6
[2,-3,-1] => 5
[-2,3,1] => 4
[-2,3,-1] => 3
[-2,-3,1] => 6
[-2,-3,-1] => 7
[3,1,2] => 3
[3,1,-2] => 5
[3,-1,2] => 4
[3,-1,-2] => 4
[-3,1,2] => 6
[-3,1,-2] => 4
[-3,-1,2] => 5
[-3,-1,-2] => 5
[3,2,1] => 2
[3,2,-1] => 3
[3,-2,1] => 5
[3,-2,-1] => 6
[-3,2,1] => 4
[-3,2,-1] => 3
[-3,-2,1] => 7
[-3,-2,-1] => 6
[1,2,3,4] => 0
[1,2,3,-4] => 7
[1,2,-3,4] => 5
[1,2,-3,-4] => 12
[1,-2,3,4] => 3
[1,-2,3,-4] => 10
[1,-2,-3,4] => 8
[1,-2,-3,-4] => 15
[-1,2,3,4] => 1
[-1,2,3,-4] => 8
[-1,2,-3,4] => 6
[-1,2,-3,-4] => 13
[-1,-2,3,4] => 4
[-1,-2,3,-4] => 11
[-1,-2,-3,4] => 9
[-1,-2,-3,-4] => 16
[1,2,4,3] => 1
[1,2,4,-3] => 6
[1,2,-4,3] => 11
[1,2,-4,-3] => 6
[1,-2,4,3] => 4
[1,-2,4,-3] => 9
[1,-2,-4,3] => 14
[1,-2,-4,-3] => 9
[-1,2,4,3] => 2
[-1,2,4,-3] => 7
[-1,2,-4,3] => 12
[-1,2,-4,-3] => 7
[-1,-2,4,3] => 5
[-1,-2,4,-3] => 10
[-1,-2,-4,3] => 15
[-1,-2,-4,-3] => 10
[1,3,2,4] => 1
[1,3,2,-4] => 8
[1,3,-2,4] => 4
[1,3,-2,-4] => 11
[1,-3,2,4] => 7
[1,-3,2,-4] => 14
[1,-3,-2,4] => 4
[1,-3,-2,-4] => 11
[-1,3,2,4] => 2
[-1,3,2,-4] => 9
[-1,3,-2,4] => 5
[-1,3,-2,-4] => 12
[-1,-3,2,4] => 8
[-1,-3,2,-4] => 15
[-1,-3,-2,4] => 5
[-1,-3,-2,-4] => 12
[1,3,4,2] => 2
[1,3,4,-2] => 5
[1,3,-4,2] => 10
[1,3,-4,-2] => 7
[1,-3,4,2] => 8
[1,-3,4,-2] => 5
[1,-3,-4,2] => 10
[1,-3,-4,-2] => 13
[-1,3,4,2] => 3
[-1,3,4,-2] => 6
[-1,3,-4,2] => 11
[-1,3,-4,-2] => 8
[-1,-3,4,2] => 9
[-1,-3,4,-2] => 6
[-1,-3,-4,2] => 11
[-1,-3,-4,-2] => 14
[1,4,2,3] => 3
[1,4,2,-3] => 9
[1,4,-2,3] => 6
[1,4,-2,-3] => 6
[1,-4,2,3] => 12
[1,-4,2,-3] => 6
[1,-4,-2,3] => 9
[1,-4,-2,-3] => 9
[-1,4,2,3] => 4
[-1,4,2,-3] => 10
[-1,4,-2,3] => 7
[-1,4,-2,-3] => 7
[-1,-4,2,3] => 13
[-1,-4,2,-3] => 7
[-1,-4,-2,3] => 10
[-1,-4,-2,-3] => 10
[1,4,3,2] => 2
[1,4,3,-2] => 5
[1,4,-3,2] => 7
[1,4,-3,-2] => 10
[1,-4,3,2] => 8
[1,-4,3,-2] => 5
[1,-4,-3,2] => 13
[1,-4,-3,-2] => 10
[-1,4,3,2] => 3
[-1,4,3,-2] => 6
[-1,4,-3,2] => 8
[-1,4,-3,-2] => 11
[-1,-4,3,2] => 9
[-1,-4,3,-2] => 6
[-1,-4,-3,2] => 14
[-1,-4,-3,-2] => 11
[2,1,3,4] => 1
[2,1,3,-4] => 8
[2,1,-3,4] => 6
[2,1,-3,-4] => 13
[2,-1,3,4] => 2
[2,-1,3,-4] => 9
[2,-1,-3,4] => 7
[2,-1,-3,-4] => 14
[-2,1,3,4] => 3
[-2,1,3,-4] => 10
[-2,1,-3,4] => 8
[-2,1,-3,-4] => 15
[-2,-1,3,4] => 2
[-2,-1,3,-4] => 9
[-2,-1,-3,4] => 7
[-2,-1,-3,-4] => 14
[2,1,4,3] => 2
[2,1,4,-3] => 7
[2,1,-4,3] => 12
[2,1,-4,-3] => 7
[2,-1,4,3] => 3
[2,-1,4,-3] => 8
[2,-1,-4,3] => 13
[2,-1,-4,-3] => 8
[-2,1,4,3] => 4
[-2,1,4,-3] => 9
[-2,1,-4,3] => 14
[-2,1,-4,-3] => 9
[-2,-1,4,3] => 3
[-2,-1,4,-3] => 8
[-2,-1,-4,3] => 13
[-2,-1,-4,-3] => 8
[2,3,1,4] => 2
[2,3,1,-4] => 9
[2,3,-1,4] => 3
[2,3,-1,-4] => 10
[2,-3,1,4] => 6
[2,-3,1,-4] => 13
[2,-3,-1,4] => 5
[2,-3,-1,-4] => 12
[-2,3,1,4] => 4
[-2,3,1,-4] => 11
[-2,3,-1,4] => 3
[-2,3,-1,-4] => 10
[-2,-3,1,4] => 6
[-2,-3,1,-4] => 13
[-2,-3,-1,4] => 7
[-2,-3,-1,-4] => 14
[2,3,4,1] => 3
[2,3,4,-1] => 4
[2,3,-4,1] => 9
[2,3,-4,-1] => 8
[2,-3,4,1] => 7
[2,-3,4,-1] => 6
[2,-3,-4,1] => 11
[2,-3,-4,-1] => 12
[-2,3,4,1] => 5
[-2,3,4,-1] => 4
[-2,3,-4,1] => 9
[-2,3,-4,-1] => 10
[-2,-3,4,1] => 7
[-2,-3,4,-1] => 8
[-2,-3,-4,1] => 13
[-2,-3,-4,-1] => 12
[2,4,1,3] => 4
[2,4,1,-3] => 8
[2,4,-1,3] => 5
[2,4,-1,-3] => 7
[2,-4,1,3] => 11
[2,-4,1,-3] => 7
[2,-4,-1,3] => 10
[2,-4,-1,-3] => 8
[-2,4,1,3] => 6
[-2,4,1,-3] => 8
[-2,4,-1,3] => 5
[-2,4,-1,-3] => 9
[-2,-4,1,3] => 11
[-2,-4,1,-3] => 9
[-2,-4,-1,3] => 12
[-2,-4,-1,-3] => 8
[2,4,3,1] => 3
[2,4,3,-1] => 4
[2,4,-3,1] => 8
[2,4,-3,-1] => 9
[2,-4,3,1] => 7
[2,-4,3,-1] => 6
[2,-4,-3,1] => 12
[2,-4,-3,-1] => 11
[-2,4,3,1] => 5
[-2,4,3,-1] => 4
[-2,4,-3,1] => 10
[-2,4,-3,-1] => 9
[-2,-4,3,1] => 7
[-2,-4,3,-1] => 8
[-2,-4,-3,1] => 12
[-2,-4,-3,-1] => 13
[3,1,2,4] => 3
[3,1,2,-4] => 10
[3,1,-2,4] => 5
[3,1,-2,-4] => 12
[3,-1,2,4] => 4
[3,-1,2,-4] => 11
[3,-1,-2,4] => 4
[3,-1,-2,-4] => 11
[-3,1,2,4] => 6
[-3,1,2,-4] => 13
[-3,1,-2,4] => 4
[-3,1,-2,-4] => 11
[-3,-1,2,4] => 5
[-3,-1,2,-4] => 12
[-3,-1,-2,4] => 5
[-3,-1,-2,-4] => 12
[3,1,4,2] => 4
[3,1,4,-2] => 6
[3,1,-4,2] => 11
[3,1,-4,-2] => 9
[3,-1,4,2] => 5
[3,-1,4,-2] => 5
[3,-1,-4,2] => 10
[3,-1,-4,-2] => 10
[-3,1,4,2] => 7
[-3,1,4,-2] => 5
[-3,1,-4,2] => 10
[-3,1,-4,-2] => 12
[-3,-1,4,2] => 6
[-3,-1,4,-2] => 6
[-3,-1,-4,2] => 11
[-3,-1,-4,-2] => 11
[3,2,1,4] => 2
[3,2,1,-4] => 9
[3,2,-1,4] => 3
[3,2,-1,-4] => 10
[3,-2,1,4] => 5
[3,-2,1,-4] => 12
[3,-2,-1,4] => 6
[3,-2,-1,-4] => 13
[-3,2,1,4] => 4
[-3,2,1,-4] => 11
[-3,2,-1,4] => 3
[-3,2,-1,-4] => 10
[-3,-2,1,4] => 7
[-3,-2,1,-4] => 14
[-3,-2,-1,4] => 6
[-3,-2,-1,-4] => 13
[3,2,4,1] => 3
[3,2,4,-1] => 4
[3,2,-4,1] => 9
[3,2,-4,-1] => 8
[3,-2,4,1] => 6
[3,-2,4,-1] => 7
[3,-2,-4,1] => 12
[3,-2,-4,-1] => 11
[-3,2,4,1] => 5
[-3,2,4,-1] => 4
[-3,2,-4,1] => 9
[-3,2,-4,-1] => 10
[-3,-2,4,1] => 8
[-3,-2,4,-1] => 7
[-3,-2,-4,1] => 12
[-3,-2,-4,-1] => 13
[3,4,1,2] => 4
[3,4,1,-2] => 7
[3,4,-1,2] => 5
[3,4,-1,-2] => 8
[3,-4,1,2] => 10
[3,-4,1,-2] => 7
[3,-4,-1,2] => 11
[3,-4,-1,-2] => 8
[-3,4,1,2] => 6
[-3,4,1,-2] => 9
[-3,4,-1,2] => 5
[-3,4,-1,-2] => 8
[-3,-4,1,2] => 12
[-3,-4,1,-2] => 9
[-3,-4,-1,2] => 11
[-3,-4,-1,-2] => 8
[3,4,2,1] => 5
[3,4,2,-1] => 6
[3,4,-2,1] => 7
[3,4,-2,-1] => 6
[3,-4,2,1] => 9
[3,-4,2,-1] => 8
[3,-4,-2,1] => 9
[3,-4,-2,-1] => 10
[-3,4,2,1] => 8
[-3,4,2,-1] => 7
[-3,4,-2,1] => 6
[-3,4,-2,-1] => 7
[-3,-4,2,1] => 10
[-3,-4,2,-1] => 11
[-3,-4,-2,1] => 10
[-3,-4,-2,-1] => 9
[4,1,2,3] => 6
[4,1,2,-3] => 9
[4,1,-2,3] => 8
[4,1,-2,-3] => 7
[4,-1,2,3] => 7
[4,-1,2,-3] => 8
[4,-1,-2,3] => 7
[4,-1,-2,-3] => 8
[-4,1,2,3] => 10
[-4,1,2,-3] => 7
[-4,1,-2,3] => 8
[-4,1,-2,-3] => 9
[-4,-1,2,3] => 9
[-4,-1,2,-3] => 8
[-4,-1,-2,3] => 9
[-4,-1,-2,-3] => 8
[4,1,3,2] => 4
[4,1,3,-2] => 6
[4,1,-3,2] => 9
[4,1,-3,-2] => 11
[4,-1,3,2] => 5
[4,-1,3,-2] => 5
[4,-1,-3,2] => 10
[4,-1,-3,-2] => 10
[-4,1,3,2] => 7
[-4,1,3,-2] => 5
[-4,1,-3,2] => 12
[-4,1,-3,-2] => 10
[-4,-1,3,2] => 6
[-4,-1,3,-2] => 6
[-4,-1,-3,2] => 11
[-4,-1,-3,-2] => 11
[4,2,1,3] => 5
[4,2,1,-3] => 7
[4,2,-1,3] => 6
[4,2,-1,-3] => 6
[4,-2,1,3] => 8
[4,-2,1,-3] => 10
[4,-2,-1,3] => 9
[4,-2,-1,-3] => 9
[-4,2,1,3] => 8
[-4,2,1,-3] => 6
[-4,2,-1,3] => 7
[-4,2,-1,-3] => 7
[-4,-2,1,3] => 11
[-4,-2,1,-3] => 9
[-4,-2,-1,3] => 10
[-4,-2,-1,-3] => 10
[4,2,3,1] => 3
[4,2,3,-1] => 4
[4,2,-3,1] => 8
[4,2,-3,-1] => 9
[4,-2,3,1] => 6
[4,-2,3,-1] => 7
[4,-2,-3,1] => 11
[4,-2,-3,-1] => 12
[-4,2,3,1] => 5
[-4,2,3,-1] => 4
[-4,2,-3,1] => 10
[-4,2,-3,-1] => 9
[-4,-2,3,1] => 8
[-4,-2,3,-1] => 7
[-4,-2,-3,1] => 13
[-4,-2,-3,-1] => 12
[4,3,1,2] => 5
[4,3,1,-2] => 7
[4,3,-1,2] => 6
[4,3,-1,-2] => 6
[4,-3,1,2] => 9
[4,-3,1,-2] => 9
[4,-3,-1,2] => 8
[4,-3,-1,-2] => 10
[-4,3,1,2] => 8
[-4,3,1,-2] => 6
[-4,3,-1,2] => 7
[-4,3,-1,-2] => 7
[-4,-3,1,2] => 10
[-4,-3,1,-2] => 10
[-4,-3,-1,2] => 11
[-4,-3,-1,-2] => 9
[4,3,2,1] => 4
[4,3,2,-1] => 5
[4,3,-2,1] => 7
[4,3,-2,-1] => 8
[4,-3,2,1] => 10
[4,-3,2,-1] => 11
[4,-3,-2,1] => 7
[4,-3,-2,-1] => 8
[-4,3,2,1] => 6
[-4,3,2,-1] => 5
[-4,3,-2,1] => 9
[-4,3,-2,-1] => 8
[-4,-3,2,1] => 12
[-4,-3,2,-1] => 11
[-4,-3,-2,1] => 9
[-4,-3,-2,-1] => 8
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Created: Jul 21, 2022 at 15:03 by Dennis Jahn
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Last Updated: Jul 21, 2022 at 15:03 by Dennis Jahn