edit this statistic or download as text // json
Identifier
Values
[+,+] => 0
[-,+] => 1
[+,-] => 2
[-,-] => 3
[2,1] => 1
[+,+,+] => 0
[-,+,+] => 1
[+,-,+] => 2
[+,+,-] => 3
[-,-,+] => 3
[-,+,-] => 4
[+,-,-] => 5
[-,-,-] => 6
[+,3,2] => 2
[-,3,2] => 3
[2,1,+] => 1
[2,1,-] => 4
[2,3,1] => 1
[3,1,2] => 3
[3,+,1] => 1
[3,-,1] => 3
[+,+,+,+] => 0
[-,+,+,+] => 1
[+,-,+,+] => 2
[+,+,-,+] => 3
[+,+,+,-] => 4
[-,-,+,+] => 3
[-,+,-,+] => 4
[-,+,+,-] => 5
[+,-,-,+] => 5
[+,-,+,-] => 6
[+,+,-,-] => 7
[-,-,-,+] => 6
[-,-,+,-] => 7
[-,+,-,-] => 8
[+,-,-,-] => 9
[-,-,-,-] => 10
[+,+,4,3] => 3
[-,+,4,3] => 4
[+,-,4,3] => 5
[-,-,4,3] => 6
[+,3,2,+] => 2
[-,3,2,+] => 3
[+,3,2,-] => 6
[-,3,2,-] => 7
[+,3,4,2] => 2
[-,3,4,2] => 3
[+,4,2,3] => 5
[-,4,2,3] => 6
[+,4,+,2] => 2
[-,4,+,2] => 3
[+,4,-,2] => 5
[-,4,-,2] => 6
[2,1,+,+] => 1
[2,1,-,+] => 4
[2,1,+,-] => 5
[2,1,-,-] => 8
[2,1,4,3] => 4
[2,3,1,+] => 1
[2,3,1,-] => 5
[2,3,4,1] => 1
[2,4,1,3] => 4
[2,4,+,1] => 1
[2,4,-,1] => 4
[3,1,2,+] => 3
[3,1,2,-] => 7
[3,1,4,2] => 3
[3,+,1,+] => 1
[3,-,1,+] => 3
[3,+,1,-] => 5
[3,-,1,-] => 7
[3,+,4,1] => 1
[3,-,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 3
[4,1,2,3] => 6
[4,1,+,2] => 3
[4,1,-,2] => 6
[4,+,1,3] => 4
[4,-,1,3] => 6
[4,+,+,1] => 1
[4,-,+,1] => 3
[4,+,-,1] => 4
[4,-,-,1] => 6
[4,3,1,2] => 3
[4,3,2,1] => 3
[+,+,+,+,+] => 0
[-,+,+,+,+] => 1
[+,-,+,+,+] => 2
[+,+,-,+,+] => 3
[+,+,+,-,+] => 4
[+,+,+,+,-] => 5
[-,-,+,+,+] => 3
[-,+,-,+,+] => 4
[-,+,+,-,+] => 5
[-,+,+,+,-] => 6
[+,-,-,+,+] => 5
[+,-,+,-,+] => 6
[+,-,+,+,-] => 7
[+,+,-,-,+] => 7
[+,+,-,+,-] => 8
>>> Load all 412 entries. <<<
[+,+,+,-,-] => 9
[-,-,-,+,+] => 6
[-,-,+,-,+] => 7
[-,-,+,+,-] => 8
[-,+,-,-,+] => 8
[-,+,-,+,-] => 9
[-,+,+,-,-] => 10
[+,-,-,-,+] => 9
[+,-,-,+,-] => 10
[+,-,+,-,-] => 11
[+,+,-,-,-] => 12
[-,-,-,-,+] => 10
[-,-,-,+,-] => 11
[-,-,+,-,-] => 12
[-,+,-,-,-] => 13
[+,-,-,-,-] => 14
[-,-,-,-,-] => 15
[+,+,+,5,4] => 4
[-,+,+,5,4] => 5
[+,-,+,5,4] => 6
[+,+,-,5,4] => 7
[-,-,+,5,4] => 7
[-,+,-,5,4] => 8
[+,-,-,5,4] => 9
[-,-,-,5,4] => 10
[+,+,4,3,+] => 3
[-,+,4,3,+] => 4
[+,-,4,3,+] => 5
[+,+,4,3,-] => 8
[-,-,4,3,+] => 6
[-,+,4,3,-] => 9
[+,-,4,3,-] => 10
[-,-,4,3,-] => 11
[+,+,4,5,3] => 3
[-,+,4,5,3] => 4
[+,-,4,5,3] => 5
[-,-,4,5,3] => 6
[+,+,5,3,4] => 7
[-,+,5,3,4] => 8
[+,-,5,3,4] => 9
[-,-,5,3,4] => 10
[+,+,5,+,3] => 3
[-,+,5,+,3] => 4
[+,-,5,+,3] => 5
[+,+,5,-,3] => 7
[-,-,5,+,3] => 6
[-,+,5,-,3] => 8
[+,-,5,-,3] => 9
[-,-,5,-,3] => 10
[+,3,2,+,+] => 2
[-,3,2,+,+] => 3
[+,3,2,-,+] => 6
[+,3,2,+,-] => 7
[-,3,2,-,+] => 7
[-,3,2,+,-] => 8
[+,3,2,-,-] => 11
[-,3,2,-,-] => 12
[+,3,2,5,4] => 6
[-,3,2,5,4] => 7
[+,3,4,2,+] => 2
[-,3,4,2,+] => 3
[+,3,4,2,-] => 7
[-,3,4,2,-] => 8
[+,3,4,5,2] => 2
[-,3,4,5,2] => 3
[+,3,5,2,4] => 6
[-,3,5,2,4] => 7
[+,3,5,+,2] => 2
[-,3,5,+,2] => 3
[+,3,5,-,2] => 6
[-,3,5,-,2] => 7
[+,4,2,3,+] => 5
[-,4,2,3,+] => 6
[+,4,2,3,-] => 10
[-,4,2,3,-] => 11
[+,4,2,5,3] => 5
[-,4,2,5,3] => 6
[+,4,+,2,+] => 2
[-,4,+,2,+] => 3
[+,4,-,2,+] => 5
[+,4,+,2,-] => 7
[-,4,-,2,+] => 6
[-,4,+,2,-] => 8
[+,4,-,2,-] => 10
[-,4,-,2,-] => 11
[+,4,+,5,2] => 2
[-,4,+,5,2] => 3
[+,4,-,5,2] => 5
[-,4,-,5,2] => 6
[+,4,5,2,3] => 5
[-,4,5,2,3] => 6
[+,4,5,3,2] => 5
[-,4,5,3,2] => 6
[+,5,2,3,4] => 9
[-,5,2,3,4] => 10
[+,5,2,+,3] => 5
[-,5,2,+,3] => 6
[+,5,2,-,3] => 9
[-,5,2,-,3] => 10
[+,5,+,2,4] => 6
[-,5,+,2,4] => 7
[+,5,-,2,4] => 9
[-,5,-,2,4] => 10
[+,5,+,+,2] => 2
[-,5,+,+,2] => 3
[+,5,-,+,2] => 5
[+,5,+,-,2] => 6
[-,5,-,+,2] => 6
[-,5,+,-,2] => 7
[+,5,-,-,2] => 9
[-,5,-,-,2] => 10
[+,5,4,2,3] => 5
[-,5,4,2,3] => 6
[+,5,4,3,2] => 5
[-,5,4,3,2] => 6
[2,1,+,+,+] => 1
[2,1,-,+,+] => 4
[2,1,+,-,+] => 5
[2,1,+,+,-] => 6
[2,1,-,-,+] => 8
[2,1,-,+,-] => 9
[2,1,+,-,-] => 10
[2,1,-,-,-] => 13
[2,1,+,5,4] => 5
[2,1,-,5,4] => 8
[2,1,4,3,+] => 4
[2,1,4,3,-] => 9
[2,1,4,5,3] => 4
[2,1,5,3,4] => 8
[2,1,5,+,3] => 4
[2,1,5,-,3] => 8
[2,3,1,+,+] => 1
[2,3,1,-,+] => 5
[2,3,1,+,-] => 6
[2,3,1,-,-] => 10
[2,3,1,5,4] => 5
[2,3,4,1,+] => 1
[2,3,4,1,-] => 6
[2,3,4,5,1] => 1
[2,3,5,1,4] => 5
[2,3,5,+,1] => 1
[2,3,5,-,1] => 5
[2,4,1,3,+] => 4
[2,4,1,3,-] => 9
[2,4,1,5,3] => 4
[2,4,+,1,+] => 1
[2,4,-,1,+] => 4
[2,4,+,1,-] => 6
[2,4,-,1,-] => 9
[2,4,+,5,1] => 1
[2,4,-,5,1] => 4
[2,4,5,1,3] => 4
[2,4,5,3,1] => 4
[2,5,1,3,4] => 8
[2,5,1,+,3] => 4
[2,5,1,-,3] => 8
[2,5,+,1,4] => 5
[2,5,-,1,4] => 8
[2,5,+,+,1] => 1
[2,5,-,+,1] => 4
[2,5,+,-,1] => 5
[2,5,-,-,1] => 8
[2,5,4,1,3] => 4
[2,5,4,3,1] => 4
[3,1,2,+,+] => 3
[3,1,2,-,+] => 7
[3,1,2,+,-] => 8
[3,1,2,-,-] => 12
[3,1,2,5,4] => 7
[3,1,4,2,+] => 3
[3,1,4,2,-] => 8
[3,1,4,5,2] => 3
[3,1,5,2,4] => 7
[3,1,5,+,2] => 3
[3,1,5,-,2] => 7
[3,+,1,+,+] => 1
[3,-,1,+,+] => 3
[3,+,1,-,+] => 5
[3,+,1,+,-] => 6
[3,-,1,-,+] => 7
[3,-,1,+,-] => 8
[3,+,1,-,-] => 10
[3,-,1,-,-] => 12
[3,+,1,5,4] => 5
[3,-,1,5,4] => 7
[3,+,4,1,+] => 1
[3,-,4,1,+] => 3
[3,+,4,1,-] => 6
[3,-,4,1,-] => 8
[3,+,4,5,1] => 1
[3,-,4,5,1] => 3
[3,+,5,1,4] => 5
[3,-,5,1,4] => 7
[3,+,5,+,1] => 1
[3,-,5,+,1] => 3
[3,+,5,-,1] => 5
[3,-,5,-,1] => 7
[3,4,1,2,+] => 3
[3,4,1,2,-] => 8
[3,4,1,5,2] => 3
[3,4,2,1,+] => 3
[3,4,2,1,-] => 8
[3,4,2,5,1] => 3
[3,4,5,1,2] => 3
[3,4,5,2,1] => 3
[3,5,1,2,4] => 7
[3,5,1,+,2] => 3
[3,5,1,-,2] => 7
[3,5,2,1,4] => 7
[3,5,2,+,1] => 3
[3,5,2,-,1] => 7
[3,5,4,1,2] => 3
[3,5,4,2,1] => 3
[4,1,2,3,+] => 6
[4,1,2,3,-] => 11
[4,1,2,5,3] => 6
[4,1,+,2,+] => 3
[4,1,-,2,+] => 6
[4,1,+,2,-] => 8
[4,1,-,2,-] => 11
[4,1,+,5,2] => 3
[4,1,-,5,2] => 6
[4,1,5,2,3] => 6
[4,1,5,3,2] => 6
[4,+,1,3,+] => 4
[4,-,1,3,+] => 6
[4,+,1,3,-] => 9
[4,-,1,3,-] => 11
[4,+,1,5,3] => 4
[4,-,1,5,3] => 6
[4,+,+,1,+] => 1
[4,-,+,1,+] => 3
[4,+,-,1,+] => 4
[4,+,+,1,-] => 6
[4,-,-,1,+] => 6
[4,-,+,1,-] => 8
[4,+,-,1,-] => 9
[4,-,-,1,-] => 11
[4,+,+,5,1] => 1
[4,-,+,5,1] => 3
[4,+,-,5,1] => 4
[4,-,-,5,1] => 6
[4,+,5,1,3] => 4
[4,-,5,1,3] => 6
[4,+,5,3,1] => 4
[4,-,5,3,1] => 6
[4,3,1,2,+] => 3
[4,3,1,2,-] => 8
[4,3,1,5,2] => 3
[4,3,2,1,+] => 3
[4,3,2,1,-] => 8
[4,3,2,5,1] => 3
[4,3,5,1,2] => 3
[4,3,5,2,1] => 3
[4,5,1,2,3] => 6
[4,5,1,3,2] => 6
[4,5,2,1,3] => 6
[4,5,2,3,1] => 6
[4,5,+,1,2] => 3
[4,5,-,1,2] => 6
[4,5,+,2,1] => 3
[4,5,-,2,1] => 6
[5,1,2,3,4] => 10
[5,1,2,+,3] => 6
[5,1,2,-,3] => 10
[5,1,+,2,4] => 7
[5,1,-,2,4] => 10
[5,1,+,+,2] => 3
[5,1,-,+,2] => 6
[5,1,+,-,2] => 7
[5,1,-,-,2] => 10
[5,1,4,2,3] => 6
[5,1,4,3,2] => 6
[5,+,1,3,4] => 8
[5,-,1,3,4] => 10
[5,+,1,+,3] => 4
[5,-,1,+,3] => 6
[5,+,1,-,3] => 8
[5,-,1,-,3] => 10
[5,+,+,1,4] => 5
[5,-,+,1,4] => 7
[5,+,-,1,4] => 8
[5,-,-,1,4] => 10
[5,+,+,+,1] => 1
[5,-,+,+,1] => 3
[5,+,-,+,1] => 4
[5,+,+,-,1] => 5
[5,-,-,+,1] => 6
[5,-,+,-,1] => 7
[5,+,-,-,1] => 8
[5,-,-,-,1] => 10
[5,+,4,1,3] => 4
[5,-,4,1,3] => 6
[5,+,4,3,1] => 4
[5,-,4,3,1] => 6
[5,3,1,2,4] => 7
[5,3,1,+,2] => 3
[5,3,1,-,2] => 7
[5,3,2,1,4] => 7
[5,3,2,+,1] => 3
[5,3,2,-,1] => 7
[5,3,4,1,2] => 3
[5,3,4,2,1] => 3
[5,4,1,2,3] => 6
[5,4,1,3,2] => 6
[5,4,2,1,3] => 6
[5,4,2,3,1] => 6
[5,4,+,1,2] => 3
[5,4,-,1,2] => 6
[5,4,+,2,1] => 3
[5,4,-,2,1] => 6
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 sum of the indices of the first term of the associated 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
    for i in range(0,len(tau[0])):
        sum = sum + tau[0][i]
    return sum
Created
May 14, 2020 at 20:46 by Danny Luecke
Updated
May 14, 2020 at 22:05 by Danny Luecke