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