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Identifier
Values
[+,+] => 0
[-,+] => 0
[+,-] => 1
[-,-] => 0
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 0
[+,-,+] => 1
[+,+,-] => 2
[-,-,+] => 0
[-,+,-] => 1
[+,-,-] => 2
[-,-,-] => 0
[+,3,2] => 1
[-,3,2] => 0
[2,1,+] => 0
[2,1,-] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 1
[3,-,1] => 1
[+,+,+,+] => 0
[-,+,+,+] => 0
[+,-,+,+] => 1
[+,+,-,+] => 2
[+,+,+,-] => 3
[-,-,+,+] => 0
[-,+,-,+] => 1
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 3
[+,+,-,-] => 4
[-,-,-,+] => 0
[-,-,+,-] => 1
[-,+,-,-] => 2
[+,-,-,-] => 3
[-,-,-,-] => 0
[+,+,4,3] => 2
[-,+,4,3] => 1
[+,-,4,3] => 2
[-,-,4,3] => 0
[+,3,2,+] => 1
[-,3,2,+] => 0
[+,3,2,-] => 3
[-,3,2,-] => 1
[+,3,4,2] => 2
[-,3,4,2] => 0
[+,4,2,3] => 1
[-,4,2,3] => 0
[+,4,+,2] => 2
[-,4,+,2] => 1
[+,4,-,2] => 3
[-,4,-,2] => 1
[2,1,+,+] => 0
[2,1,-,+] => 1
[2,1,+,-] => 2
[2,1,-,-] => 2
[2,1,4,3] => 1
[2,3,1,+] => 0
[2,3,1,-] => 1
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,+,1] => 1
[2,4,-,1] => 1
[3,1,2,+] => 0
[3,1,2,-] => 2
[3,1,4,2] => 1
[3,+,1,+] => 1
[3,-,1,+] => 1
[3,+,1,-] => 3
[3,-,1,-] => 2
[3,+,4,1] => 2
[3,-,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,+,2] => 1
[4,1,-,2] => 2
[4,+,1,3] => 1
[4,-,1,3] => 1
[4,+,+,1] => 2
[4,-,+,1] => 2
[4,+,-,1] => 3
[4,-,-,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
[+,+,+,+,+] => 0
[-,+,+,+,+] => 0
[+,-,+,+,+] => 1
[+,+,-,+,+] => 2
[+,+,+,-,+] => 3
[+,+,+,+,-] => 4
[-,-,+,+,+] => 0
[-,+,-,+,+] => 1
[-,+,+,-,+] => 2
[-,+,+,+,-] => 3
[+,-,-,+,+] => 2
[+,-,+,-,+] => 3
[+,-,+,+,-] => 4
[+,+,-,-,+] => 4
[+,+,-,+,-] => 5
>>> Load all 412 entries. <<<
[+,+,+,-,-] => 6
[-,-,-,+,+] => 0
[-,-,+,-,+] => 1
[-,-,+,+,-] => 2
[-,+,-,-,+] => 2
[-,+,-,+,-] => 3
[-,+,+,-,-] => 4
[+,-,-,-,+] => 3
[+,-,-,+,-] => 4
[+,-,+,-,-] => 5
[+,+,-,-,-] => 6
[-,-,-,-,+] => 0
[-,-,-,+,-] => 1
[-,-,+,-,-] => 2
[-,+,-,-,-] => 3
[+,-,-,-,-] => 4
[-,-,-,-,-] => 0
[+,+,+,5,4] => 3
[-,+,+,5,4] => 2
[+,-,+,5,4] => 3
[+,+,-,5,4] => 4
[-,-,+,5,4] => 1
[-,+,-,5,4] => 2
[+,-,-,5,4] => 3
[-,-,-,5,4] => 0
[+,+,4,3,+] => 2
[-,+,4,3,+] => 1
[+,-,4,3,+] => 2
[+,+,4,3,-] => 5
[-,-,4,3,+] => 0
[-,+,4,3,-] => 3
[+,-,4,3,-] => 4
[-,-,4,3,-] => 1
[+,+,4,5,3] => 4
[-,+,4,5,3] => 2
[+,-,4,5,3] => 3
[-,-,4,5,3] => 0
[+,+,5,3,4] => 2
[-,+,5,3,4] => 1
[+,-,5,3,4] => 2
[-,-,5,3,4] => 0
[+,+,5,+,3] => 3
[-,+,5,+,3] => 2
[+,-,5,+,3] => 3
[+,+,5,-,3] => 5
[-,-,5,+,3] => 1
[-,+,5,-,3] => 3
[+,-,5,-,3] => 4
[-,-,5,-,3] => 1
[+,3,2,+,+] => 1
[-,3,2,+,+] => 0
[+,3,2,-,+] => 3
[+,3,2,+,-] => 4
[-,3,2,-,+] => 1
[-,3,2,+,-] => 2
[+,3,2,-,-] => 5
[-,3,2,-,-] => 2
[+,3,2,5,4] => 3
[-,3,2,5,4] => 1
[+,3,4,2,+] => 2
[-,3,4,2,+] => 0
[+,3,4,2,-] => 4
[-,3,4,2,-] => 1
[+,3,4,5,2] => 3
[-,3,4,5,2] => 0
[+,3,5,2,4] => 2
[-,3,5,2,4] => 0
[+,3,5,+,2] => 3
[-,3,5,+,2] => 1
[+,3,5,-,2] => 4
[-,3,5,-,2] => 1
[+,4,2,3,+] => 1
[-,4,2,3,+] => 0
[+,4,2,3,-] => 4
[-,4,2,3,-] => 2
[+,4,2,5,3] => 3
[-,4,2,5,3] => 1
[+,4,+,2,+] => 2
[-,4,+,2,+] => 1
[+,4,-,2,+] => 3
[+,4,+,2,-] => 5
[-,4,-,2,+] => 1
[-,4,+,2,-] => 3
[+,4,-,2,-] => 5
[-,4,-,2,-] => 2
[+,4,+,5,2] => 4
[-,4,+,5,2] => 2
[+,4,-,5,2] => 4
[-,4,-,5,2] => 1
[+,4,5,2,3] => 2
[-,4,5,2,3] => 0
[+,4,5,3,2] => 3
[-,4,5,3,2] => 1
[+,5,2,3,4] => 1
[-,5,2,3,4] => 0
[+,5,2,+,3] => 2
[-,5,2,+,3] => 1
[+,5,2,-,3] => 4
[-,5,2,-,3] => 2
[+,5,+,2,4] => 2
[-,5,+,2,4] => 1
[+,5,-,2,4] => 3
[-,5,-,2,4] => 1
[+,5,+,+,2] => 3
[-,5,+,+,2] => 2
[+,5,-,+,2] => 4
[+,5,+,-,2] => 5
[-,5,-,+,2] => 2
[-,5,+,-,2] => 3
[+,5,-,-,2] => 5
[-,5,-,-,2] => 2
[+,5,4,2,3] => 3
[-,5,4,2,3] => 1
[+,5,4,3,2] => 4
[-,5,4,3,2] => 2
[2,1,+,+,+] => 0
[2,1,-,+,+] => 1
[2,1,+,-,+] => 2
[2,1,+,+,-] => 3
[2,1,-,-,+] => 2
[2,1,-,+,-] => 3
[2,1,+,-,-] => 4
[2,1,-,-,-] => 3
[2,1,+,5,4] => 2
[2,1,-,5,4] => 2
[2,1,4,3,+] => 1
[2,1,4,3,-] => 3
[2,1,4,5,3] => 2
[2,1,5,3,4] => 1
[2,1,5,+,3] => 2
[2,1,5,-,3] => 3
[2,3,1,+,+] => 0
[2,3,1,-,+] => 1
[2,3,1,+,-] => 2
[2,3,1,-,-] => 2
[2,3,1,5,4] => 1
[2,3,4,1,+] => 0
[2,3,4,1,-] => 1
[2,3,4,5,1] => 0
[2,3,5,1,4] => 0
[2,3,5,+,1] => 1
[2,3,5,-,1] => 1
[2,4,1,3,+] => 0
[2,4,1,3,-] => 2
[2,4,1,5,3] => 1
[2,4,+,1,+] => 1
[2,4,-,1,+] => 1
[2,4,+,1,-] => 3
[2,4,-,1,-] => 2
[2,4,+,5,1] => 2
[2,4,-,5,1] => 1
[2,4,5,1,3] => 0
[2,4,5,3,1] => 1
[2,5,1,3,4] => 0
[2,5,1,+,3] => 1
[2,5,1,-,3] => 2
[2,5,+,1,4] => 1
[2,5,-,1,4] => 1
[2,5,+,+,1] => 2
[2,5,-,+,1] => 2
[2,5,+,-,1] => 3
[2,5,-,-,1] => 2
[2,5,4,1,3] => 1
[2,5,4,3,1] => 2
[3,1,2,+,+] => 0
[3,1,2,-,+] => 2
[3,1,2,+,-] => 3
[3,1,2,-,-] => 4
[3,1,2,5,4] => 2
[3,1,4,2,+] => 1
[3,1,4,2,-] => 3
[3,1,4,5,2] => 2
[3,1,5,2,4] => 1
[3,1,5,+,2] => 2
[3,1,5,-,2] => 3
[3,+,1,+,+] => 1
[3,-,1,+,+] => 1
[3,+,1,-,+] => 3
[3,+,1,+,-] => 4
[3,-,1,-,+] => 2
[3,-,1,+,-] => 3
[3,+,1,-,-] => 5
[3,-,1,-,-] => 3
[3,+,1,5,4] => 3
[3,-,1,5,4] => 2
[3,+,4,1,+] => 2
[3,-,4,1,+] => 1
[3,+,4,1,-] => 4
[3,-,4,1,-] => 2
[3,+,4,5,1] => 3
[3,-,4,5,1] => 1
[3,+,5,1,4] => 2
[3,-,5,1,4] => 1
[3,+,5,+,1] => 3
[3,-,5,+,1] => 2
[3,+,5,-,1] => 4
[3,-,5,-,1] => 2
[3,4,1,2,+] => 0
[3,4,1,2,-] => 2
[3,4,1,5,2] => 1
[3,4,2,1,+] => 1
[3,4,2,1,-] => 3
[3,4,2,5,1] => 2
[3,4,5,1,2] => 0
[3,4,5,2,1] => 1
[3,5,1,2,4] => 0
[3,5,1,+,2] => 1
[3,5,1,-,2] => 2
[3,5,2,1,4] => 1
[3,5,2,+,1] => 2
[3,5,2,-,1] => 3
[3,5,4,1,2] => 1
[3,5,4,2,1] => 2
[4,1,2,3,+] => 0
[4,1,2,3,-] => 3
[4,1,2,5,3] => 2
[4,1,+,2,+] => 1
[4,1,-,2,+] => 2
[4,1,+,2,-] => 4
[4,1,-,2,-] => 4
[4,1,+,5,2] => 3
[4,1,-,5,2] => 3
[4,1,5,2,3] => 1
[4,1,5,3,2] => 2
[4,+,1,3,+] => 1
[4,-,1,3,+] => 1
[4,+,1,3,-] => 4
[4,-,1,3,-] => 3
[4,+,1,5,3] => 3
[4,-,1,5,3] => 2
[4,+,+,1,+] => 2
[4,-,+,1,+] => 2
[4,+,-,1,+] => 3
[4,+,+,1,-] => 5
[4,-,-,1,+] => 2
[4,-,+,1,-] => 4
[4,+,-,1,-] => 5
[4,-,-,1,-] => 3
[4,+,+,5,1] => 4
[4,-,+,5,1] => 3
[4,+,-,5,1] => 4
[4,-,-,5,1] => 2
[4,+,5,1,3] => 2
[4,-,5,1,3] => 1
[4,+,5,3,1] => 3
[4,-,5,3,1] => 2
[4,3,1,2,+] => 1
[4,3,1,2,-] => 3
[4,3,1,5,2] => 2
[4,3,2,1,+] => 2
[4,3,2,1,-] => 4
[4,3,2,5,1] => 3
[4,3,5,1,2] => 1
[4,3,5,2,1] => 2
[4,5,1,2,3] => 0
[4,5,1,3,2] => 1
[4,5,2,1,3] => 1
[4,5,2,3,1] => 2
[4,5,+,1,2] => 2
[4,5,-,1,2] => 2
[4,5,+,2,1] => 3
[4,5,-,2,1] => 3
[5,1,2,3,4] => 0
[5,1,2,+,3] => 1
[5,1,2,-,3] => 3
[5,1,+,2,4] => 1
[5,1,-,2,4] => 2
[5,1,+,+,2] => 2
[5,1,-,+,2] => 3
[5,1,+,-,2] => 4
[5,1,-,-,2] => 4
[5,1,4,2,3] => 2
[5,1,4,3,2] => 3
[5,+,1,3,4] => 1
[5,-,1,3,4] => 1
[5,+,1,+,3] => 2
[5,-,1,+,3] => 2
[5,+,1,-,3] => 4
[5,-,1,-,3] => 3
[5,+,+,1,4] => 2
[5,-,+,1,4] => 2
[5,+,-,1,4] => 3
[5,-,-,1,4] => 2
[5,+,+,+,1] => 3
[5,-,+,+,1] => 3
[5,+,-,+,1] => 4
[5,+,+,-,1] => 5
[5,-,-,+,1] => 3
[5,-,+,-,1] => 4
[5,+,-,-,1] => 5
[5,-,-,-,1] => 3
[5,+,4,1,3] => 3
[5,-,4,1,3] => 2
[5,+,4,3,1] => 4
[5,-,4,3,1] => 3
[5,3,1,2,4] => 1
[5,3,1,+,2] => 2
[5,3,1,-,2] => 3
[5,3,2,1,4] => 2
[5,3,2,+,1] => 3
[5,3,2,-,1] => 4
[5,3,4,1,2] => 2
[5,3,4,2,1] => 3
[5,4,1,2,3] => 1
[5,4,1,3,2] => 2
[5,4,2,1,3] => 2
[5,4,2,3,1] => 3
[5,4,+,1,2] => 3
[5,4,-,1,2] => 3
[5,4,+,2,1] => 4
[5,4,-,2,1] => 4
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Description
The inversion number of the associated bounded affine permutation.
The inversion number is given by $\ell(f) = |\{(i, j) \in [n] \times \mathbb Z \; | \; i < j \text{ and } f(i) > f(j)\}|$
References
[1] Lam, T. Totally nonnegative Grassmannian and Grassmann polytopes MathSciNet:3468251 arXiv:1506.00603
Code
def dectobap(pi):
    bap=[]
    tau = list(pi)
    for j in range(0,len(tau)):
        if tau[j]==(j+1):
            bap.append(j+1+len(tau))
        elif tau[j]<0:
            bap.append(j+1)
        else:
            if tau[j]<(j+1):
                add_n=tau[j]+len(tau)
                bap.append(add_n)
            else:
                bap.append(tau[j])
    return bap

def statistic(pi):
    count=0
    tau = dectobap(pi)
    for i in range(0,len(tau)-1):
        for j in range(i+1,len(tau)):
            if tau[i]>tau[j]:
                count += 1
    return count
Created
May 12, 2020 at 22:38 by Danny Luecke
Updated
May 13, 2020 at 15:04 by Danny Luecke