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Identifier
Values
[+,+] => 0
[-,+] => 2
[+,-] => 4
[-,-] => 6
[2,1] => 3
[+,+,+] => 0
[-,+,+] => 3
[+,-,+] => 6
[+,+,-] => 9
[-,-,+] => 9
[-,+,-] => 12
[+,-,-] => 15
[-,-,-] => 18
[+,3,2] => 7
[-,3,2] => 10
[2,1,+] => 4
[2,1,-] => 13
[2,3,1] => 6
[3,1,2] => 12
[3,+,1] => 7
[3,-,1] => 13
[+,+,+,+] => 0
[-,+,+,+] => 4
[+,-,+,+] => 8
[+,+,-,+] => 12
[+,+,+,-] => 16
[-,-,+,+] => 12
[-,+,-,+] => 16
[-,+,+,-] => 20
[+,-,-,+] => 20
[+,-,+,-] => 24
[+,+,-,-] => 28
[-,-,-,+] => 24
[-,-,+,-] => 28
[-,+,-,-] => 32
[+,-,-,-] => 36
[-,-,-,-] => 40
[+,+,4,3] => 13
[-,+,4,3] => 17
[+,-,4,3] => 21
[-,-,4,3] => 25
[+,3,2,+] => 9
[-,3,2,+] => 13
[+,3,2,-] => 25
[-,3,2,-] => 29
[+,3,4,2] => 11
[-,3,4,2] => 15
[+,4,2,3] => 23
[-,4,2,3] => 27
[+,4,+,2] => 12
[-,4,+,2] => 16
[+,4,-,2] => 24
[-,4,-,2] => 28
[2,1,+,+] => 5
[2,1,-,+] => 17
[2,1,+,-] => 21
[2,1,-,-] => 33
[2,1,4,3] => 18
[2,3,1,+] => 7
[2,3,1,-] => 23
[2,3,4,1] => 10
[2,4,1,3] => 21
[2,4,+,1] => 11
[2,4,-,1] => 23
[3,1,2,+] => 15
[3,1,2,-] => 31
[3,1,4,2] => 17
[3,+,1,+] => 8
[3,-,1,+] => 16
[3,+,1,-] => 24
[3,-,1,-] => 32
[3,+,4,1] => 11
[3,-,4,1] => 19
[3,4,1,2] => 20
[3,4,2,1] => 21
[4,1,2,3] => 30
[4,1,+,2] => 19
[4,1,-,2] => 31
[4,+,1,3] => 23
[4,-,1,3] => 31
[4,+,+,1] => 13
[4,-,+,1] => 21
[4,+,-,1] => 25
[4,-,-,1] => 33
[4,3,1,2] => 21
[4,3,2,1] => 22
[+,+,+,+,+] => 0
[-,+,+,+,+] => 5
[+,-,+,+,+] => 10
[+,+,-,+,+] => 15
[+,+,+,-,+] => 20
[+,+,+,+,-] => 25
[-,-,+,+,+] => 15
[-,+,-,+,+] => 20
[-,+,+,-,+] => 25
[-,+,+,+,-] => 30
[+,-,-,+,+] => 25
[+,-,+,-,+] => 30
[+,-,+,+,-] => 35
[+,+,-,-,+] => 35
[+,+,-,+,-] => 40
>>> Load all 412 entries. <<<
[+,+,+,-,-] => 45
[-,-,-,+,+] => 30
[-,-,+,-,+] => 35
[-,-,+,+,-] => 40
[-,+,-,-,+] => 40
[-,+,-,+,-] => 45
[-,+,+,-,-] => 50
[+,-,-,-,+] => 45
[+,-,-,+,-] => 50
[+,-,+,-,-] => 55
[+,+,-,-,-] => 60
[-,-,-,-,+] => 50
[-,-,-,+,-] => 55
[-,-,+,-,-] => 60
[-,+,-,-,-] => 65
[+,-,-,-,-] => 70
[-,-,-,-,-] => 75
[+,+,+,5,4] => 21
[-,+,+,5,4] => 26
[+,-,+,5,4] => 31
[+,+,-,5,4] => 36
[-,-,+,5,4] => 36
[-,+,-,5,4] => 41
[+,-,-,5,4] => 46
[-,-,-,5,4] => 51
[+,+,4,3,+] => 16
[-,+,4,3,+] => 21
[+,-,4,3,+] => 26
[+,+,4,3,-] => 41
[-,-,4,3,+] => 31
[-,+,4,3,-] => 46
[+,-,4,3,-] => 51
[-,-,4,3,-] => 56
[+,+,4,5,3] => 18
[-,+,4,5,3] => 23
[+,-,4,5,3] => 28
[-,-,4,5,3] => 33
[+,+,5,3,4] => 38
[-,+,5,3,4] => 43
[+,-,5,3,4] => 48
[-,-,5,3,4] => 53
[+,+,5,+,3] => 19
[-,+,5,+,3] => 24
[+,-,5,+,3] => 29
[+,+,5,-,3] => 39
[-,-,5,+,3] => 34
[-,+,5,-,3] => 44
[+,-,5,-,3] => 49
[-,-,5,-,3] => 54
[+,3,2,+,+] => 11
[-,3,2,+,+] => 16
[+,3,2,-,+] => 31
[+,3,2,+,-] => 36
[-,3,2,-,+] => 36
[-,3,2,+,-] => 41
[+,3,2,-,-] => 56
[-,3,2,-,-] => 61
[+,3,2,5,4] => 32
[-,3,2,5,4] => 37
[+,3,4,2,+] => 13
[-,3,4,2,+] => 18
[+,3,4,2,-] => 38
[-,3,4,2,-] => 43
[+,3,4,5,2] => 16
[-,3,4,5,2] => 21
[+,3,5,2,4] => 35
[-,3,5,2,4] => 40
[+,3,5,+,2] => 17
[-,3,5,+,2] => 22
[+,3,5,-,2] => 37
[-,3,5,-,2] => 42
[+,4,2,3,+] => 28
[-,4,2,3,+] => 33
[+,4,2,3,-] => 53
[-,4,2,3,-] => 58
[+,4,2,5,3] => 30
[-,4,2,5,3] => 35
[+,4,+,2,+] => 14
[-,4,+,2,+] => 19
[+,4,-,2,+] => 29
[+,4,+,2,-] => 39
[-,4,-,2,+] => 34
[-,4,+,2,-] => 44
[+,4,-,2,-] => 54
[-,4,-,2,-] => 59
[+,4,+,5,2] => 17
[-,4,+,5,2] => 22
[+,4,-,5,2] => 32
[-,4,-,5,2] => 37
[+,4,5,2,3] => 33
[-,4,5,2,3] => 38
[+,4,5,3,2] => 34
[-,4,5,3,2] => 39
[+,5,2,3,4] => 51
[-,5,2,3,4] => 56
[+,5,2,+,3] => 32
[-,5,2,+,3] => 37
[+,5,2,-,3] => 52
[-,5,2,-,3] => 57
[+,5,+,2,4] => 37
[-,5,+,2,4] => 42
[+,5,-,2,4] => 52
[-,5,-,2,4] => 57
[+,5,+,+,2] => 19
[-,5,+,+,2] => 24
[+,5,-,+,2] => 34
[+,5,+,-,2] => 39
[-,5,-,+,2] => 39
[-,5,+,-,2] => 44
[+,5,-,-,2] => 54
[-,5,-,-,2] => 59
[+,5,4,2,3] => 34
[-,5,4,2,3] => 39
[+,5,4,3,2] => 35
[-,5,4,3,2] => 40
[2,1,+,+,+] => 6
[2,1,-,+,+] => 21
[2,1,+,-,+] => 26
[2,1,+,+,-] => 31
[2,1,-,-,+] => 41
[2,1,-,+,-] => 46
[2,1,+,-,-] => 51
[2,1,-,-,-] => 66
[2,1,+,5,4] => 27
[2,1,-,5,4] => 42
[2,1,4,3,+] => 22
[2,1,4,3,-] => 47
[2,1,4,5,3] => 24
[2,1,5,3,4] => 44
[2,1,5,+,3] => 25
[2,1,5,-,3] => 45
[2,3,1,+,+] => 8
[2,3,1,-,+] => 28
[2,3,1,+,-] => 33
[2,3,1,-,-] => 53
[2,3,1,5,4] => 29
[2,3,4,1,+] => 11
[2,3,4,1,-] => 36
[2,3,4,5,1] => 15
[2,3,5,1,4] => 33
[2,3,5,+,1] => 16
[2,3,5,-,1] => 36
[2,4,1,3,+] => 25
[2,4,1,3,-] => 50
[2,4,1,5,3] => 27
[2,4,+,1,+] => 12
[2,4,-,1,+] => 27
[2,4,+,1,-] => 37
[2,4,-,1,-] => 52
[2,4,+,5,1] => 16
[2,4,-,5,1] => 31
[2,4,5,1,3] => 31
[2,4,5,3,1] => 33
[2,5,1,3,4] => 48
[2,5,1,+,3] => 29
[2,5,1,-,3] => 49
[2,5,+,1,4] => 35
[2,5,-,1,4] => 50
[2,5,+,+,1] => 18
[2,5,-,+,1] => 33
[2,5,+,-,1] => 38
[2,5,-,-,1] => 53
[2,5,4,1,3] => 32
[2,5,4,3,1] => 34
[3,1,2,+,+] => 18
[3,1,2,-,+] => 38
[3,1,2,+,-] => 43
[3,1,2,-,-] => 63
[3,1,2,5,4] => 39
[3,1,4,2,+] => 20
[3,1,4,2,-] => 45
[3,1,4,5,2] => 23
[3,1,5,2,4] => 42
[3,1,5,+,2] => 24
[3,1,5,-,2] => 44
[3,+,1,+,+] => 9
[3,-,1,+,+] => 19
[3,+,1,-,+] => 29
[3,+,1,+,-] => 34
[3,-,1,-,+] => 39
[3,-,1,+,-] => 44
[3,+,1,-,-] => 54
[3,-,1,-,-] => 64
[3,+,1,5,4] => 30
[3,-,1,5,4] => 40
[3,+,4,1,+] => 12
[3,-,4,1,+] => 22
[3,+,4,1,-] => 37
[3,-,4,1,-] => 47
[3,+,4,5,1] => 16
[3,-,4,5,1] => 26
[3,+,5,1,4] => 34
[3,-,5,1,4] => 44
[3,+,5,+,1] => 17
[3,-,5,+,1] => 27
[3,+,5,-,1] => 37
[3,-,5,-,1] => 47
[3,4,1,2,+] => 23
[3,4,1,2,-] => 48
[3,4,1,5,2] => 26
[3,4,2,1,+] => 24
[3,4,2,1,-] => 49
[3,4,2,5,1] => 28
[3,4,5,1,2] => 30
[3,4,5,2,1] => 31
[3,5,1,2,4] => 46
[3,5,1,+,2] => 28
[3,5,1,-,2] => 48
[3,5,2,1,4] => 47
[3,5,2,+,1] => 30
[3,5,2,-,1] => 50
[3,5,4,1,2] => 31
[3,5,4,2,1] => 32
[4,1,2,3,+] => 36
[4,1,2,3,-] => 61
[4,1,2,5,3] => 38
[4,1,+,2,+] => 22
[4,1,-,2,+] => 37
[4,1,+,2,-] => 47
[4,1,-,2,-] => 62
[4,1,+,5,2] => 25
[4,1,-,5,2] => 40
[4,1,5,2,3] => 41
[4,1,5,3,2] => 42
[4,+,1,3,+] => 27
[4,-,1,3,+] => 37
[4,+,1,3,-] => 52
[4,-,1,3,-] => 62
[4,+,1,5,3] => 29
[4,-,1,5,3] => 39
[4,+,+,1,+] => 14
[4,-,+,1,+] => 24
[4,+,-,1,+] => 29
[4,+,+,1,-] => 39
[4,-,-,1,+] => 39
[4,-,+,1,-] => 49
[4,+,-,1,-] => 54
[4,-,-,1,-] => 64
[4,+,+,5,1] => 18
[4,-,+,5,1] => 28
[4,+,-,5,1] => 33
[4,-,-,5,1] => 43
[4,+,5,1,3] => 33
[4,-,5,1,3] => 43
[4,+,5,3,1] => 35
[4,-,5,3,1] => 45
[4,3,1,2,+] => 24
[4,3,1,2,-] => 49
[4,3,1,5,2] => 27
[4,3,2,1,+] => 25
[4,3,2,1,-] => 50
[4,3,2,5,1] => 29
[4,3,5,1,2] => 31
[4,3,5,2,1] => 32
[4,5,1,2,3] => 45
[4,5,1,3,2] => 46
[4,5,2,1,3] => 46
[4,5,2,3,1] => 48
[4,5,+,1,2] => 33
[4,5,-,1,2] => 48
[4,5,+,2,1] => 34
[4,5,-,2,1] => 49
[5,1,2,3,4] => 60
[5,1,2,+,3] => 41
[5,1,2,-,3] => 61
[5,1,+,2,4] => 46
[5,1,-,2,4] => 61
[5,1,+,+,2] => 28
[5,1,-,+,2] => 43
[5,1,+,-,2] => 48
[5,1,-,-,2] => 63
[5,1,4,2,3] => 43
[5,1,4,3,2] => 44
[5,+,1,3,4] => 51
[5,-,1,3,4] => 61
[5,+,1,+,3] => 32
[5,-,1,+,3] => 42
[5,+,1,-,3] => 52
[5,-,1,-,3] => 62
[5,+,+,1,4] => 38
[5,-,+,1,4] => 48
[5,+,-,1,4] => 53
[5,-,-,1,4] => 63
[5,+,+,+,1] => 21
[5,-,+,+,1] => 31
[5,+,-,+,1] => 36
[5,+,+,-,1] => 41
[5,-,-,+,1] => 46
[5,-,+,-,1] => 51
[5,+,-,-,1] => 56
[5,-,-,-,1] => 66
[5,+,4,1,3] => 35
[5,-,4,1,3] => 45
[5,+,4,3,1] => 37
[5,-,4,3,1] => 47
[5,3,1,2,4] => 48
[5,3,1,+,2] => 30
[5,3,1,-,2] => 50
[5,3,2,1,4] => 49
[5,3,2,+,1] => 32
[5,3,2,-,1] => 52
[5,3,4,1,2] => 33
[5,3,4,2,1] => 34
[5,4,1,2,3] => 46
[5,4,1,3,2] => 47
[5,4,2,1,3] => 47
[5,4,2,3,1] => 49
[5,4,+,1,2] => 34
[5,4,-,1,2] => 49
[5,4,+,2,1] => 35
[5,4,-,2,1] => 50
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Description
The sum of all indices from every element of the Grassmann necklace.
Here, we use Postnikov's map (p.59) from decorated permutations to Grassmann necklaces.
References
[1] A. Postnikov, Total positivity, Grassmannians, and networks. 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks arXiv:math/0609764
Code
def dectoneck(pi):
    tau=list(pi)
    n=len(tau)
    perm=[]
    neck=[ [] for _ in range(n) ]
    
    for j in range(0,n):
        if tau[j]<0:
            for k in range(0,n):
                neck[k].append(abs(tau[j]))
        perm.append(abs(tau[j]))

    perminv=Permutation(perm).inverse()
   
    for el in range(1,n+1):
        adjust_index=[]
        adjust_perminv=[]
        for m in range(0,n):
            if el>(m+1):
                adjust_index.append(m+1+n)
            else:
                adjust_index.append(m+1)

            if el>perminv[m]:
                adjust_perminv.append(perminv[m]+n)
            else:
                adjust_perminv.append(perminv[m])
            
        for x in range(0,n):
            if adjust_index[x] < adjust_perminv[x]:
                neck[el-1].append(x+1)
   
    for y in range(0,n):
        neck[y].sort()
    
    return neck

def statistic(pi):
    tau=dectoneck(pi)
    sum=0
    k=len(tau[0])
    for i in range(0,len(tau)):
        for j in range(0,k):
            sum = sum + tau[i][j]
    return sum
Created
May 14, 2020 at 20:41 by Danny Luecke
Updated
May 14, 2020 at 20:41 by Danny Luecke