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Identifier

St001821: Signed permutations ⟶ ℤ

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

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

References

[1] Petersen, T. K. The sorting index arXiv:1007.1207

Code

def statistic(pi):
    T = []
    n = pi.parent().rank()
    w = list(pi)
    iw = [abs(i) for i in w]
    
    for i in range(n):
        ind = iw.index(n - i)
        
        if ind + 1 < iw[ind]:
            T.append([sgn(w[ind]) * (ind+1), n-i])
            w[ind] = sgn(w[ind]) * w[n-i-1]
            w[n-i-1] = n-i
        
        if ind + 1 == iw[ind] and w[ind] < ind + 1:
            T.append([-(ind+1),ind+1])
            w[ind] = -w[ind]
        
        iw = [abs(x) for x in w]

    sort = 0
    for t in T:
        if t[0] > 0:
            sort += (t[1] - t[0])
        else:
            sort += (t[1] - t[0] - 1)
    
    return sort

Created

Jul 21, 2022 at 15:03 by Dennis Jahn

Updated

Jul 21, 2022 at 15:03 by Dennis Jahn