Identifier
Values
=>
[+,+]=>0
[-,+]=>1
[+,-]=>1
[-,-]=>2
[2,1]=>1
[+,+,+]=>0
[-,+,+]=>1
[+,-,+]=>1
[+,+,-]=>1
[-,-,+]=>2
[-,+,-]=>2
[+,-,-]=>2
[-,-,-]=>3
[+,3,2]=>1
[-,3,2]=>2
[2,1,+]=>1
[2,1,-]=>2
[2,3,1]=>2
[3,1,2]=>1
[3,+,1]=>1
[3,-,1]=>2
[+,+,+,+]=>0
[-,+,+,+]=>1
[+,-,+,+]=>1
[+,+,-,+]=>1
[+,+,+,-]=>1
[-,-,+,+]=>2
[-,+,-,+]=>2
[-,+,+,-]=>2
[+,-,-,+]=>2
[+,-,+,-]=>2
[+,+,-,-]=>2
[-,-,-,+]=>3
[-,-,+,-]=>3
[-,+,-,-]=>3
[+,-,-,-]=>3
[-,-,-,-]=>4
[+,+,4,3]=>1
[-,+,4,3]=>2
[+,-,4,3]=>2
[-,-,4,3]=>3
[+,3,2,+]=>1
[-,3,2,+]=>2
[+,3,2,-]=>2
[-,3,2,-]=>3
[+,3,4,2]=>2
[-,3,4,2]=>3
[+,4,2,3]=>1
[-,4,2,3]=>2
[+,4,+,2]=>1
[-,4,+,2]=>2
[+,4,-,2]=>2
[-,4,-,2]=>3
[2,1,+,+]=>1
[2,1,-,+]=>2
[2,1,+,-]=>2
[2,1,-,-]=>3
[2,1,4,3]=>2
[2,3,1,+]=>2
[2,3,1,-]=>3
[2,3,4,1]=>3
[2,4,1,3]=>2
[2,4,+,1]=>2
[2,4,-,1]=>3
[3,1,2,+]=>1
[3,1,2,-]=>2
[3,1,4,2]=>2
[3,+,1,+]=>1
[3,-,1,+]=>2
[3,+,1,-]=>2
[3,-,1,-]=>3
[3,+,4,1]=>2
[3,-,4,1]=>3
[3,4,1,2]=>2
[3,4,2,1]=>2
[4,1,2,3]=>1
[4,1,+,2]=>1
[4,1,-,2]=>2
[4,+,1,3]=>1
[4,-,1,3]=>2
[4,+,+,1]=>1
[4,-,+,1]=>2
[4,+,-,1]=>2
[4,-,-,1]=>3
[4,3,1,2]=>2
[4,3,2,1]=>2
[+,+,+,+,+]=>0
[-,+,+,+,+]=>1
[+,-,+,+,+]=>1
[+,+,-,+,+]=>1
[+,+,+,-,+]=>1
[+,+,+,+,-]=>1
[-,-,+,+,+]=>2
[-,+,-,+,+]=>2
[-,+,+,-,+]=>2
[-,+,+,+,-]=>2
[+,-,-,+,+]=>2
[+,-,+,-,+]=>2
[+,-,+,+,-]=>2
[+,+,-,-,+]=>2
[+,+,-,+,-]=>2
[+,+,+,-,-]=>2
[-,-,-,+,+]=>3
[-,-,+,-,+]=>3
[-,-,+,+,-]=>3
[-,+,-,-,+]=>3
[-,+,-,+,-]=>3
[-,+,+,-,-]=>3
[+,-,-,-,+]=>3
[+,-,-,+,-]=>3
[+,-,+,-,-]=>3
[+,+,-,-,-]=>3
[-,-,-,-,+]=>4
[-,-,-,+,-]=>4
[-,-,+,-,-]=>4
[-,+,-,-,-]=>4
[+,-,-,-,-]=>4
[-,-,-,-,-]=>5
[+,+,+,5,4]=>1
[-,+,+,5,4]=>2
[+,-,+,5,4]=>2
[+,+,-,5,4]=>2
[-,-,+,5,4]=>3
[-,+,-,5,4]=>3
[+,-,-,5,4]=>3
[-,-,-,5,4]=>4
[+,+,4,3,+]=>1
[-,+,4,3,+]=>2
[+,-,4,3,+]=>2
[+,+,4,3,-]=>2
[-,-,4,3,+]=>3
[-,+,4,3,-]=>3
[+,-,4,3,-]=>3
[-,-,4,3,-]=>4
[+,+,4,5,3]=>2
[-,+,4,5,3]=>3
[+,-,4,5,3]=>3
[-,-,4,5,3]=>4
[+,+,5,3,4]=>1
[-,+,5,3,4]=>2
[+,-,5,3,4]=>2
[-,-,5,3,4]=>3
[+,+,5,+,3]=>1
[-,+,5,+,3]=>2
[+,-,5,+,3]=>2
[+,+,5,-,3]=>2
[-,-,5,+,3]=>3
[-,+,5,-,3]=>3
[+,-,5,-,3]=>3
[-,-,5,-,3]=>4
[+,3,2,+,+]=>1
[-,3,2,+,+]=>2
[+,3,2,-,+]=>2
[+,3,2,+,-]=>2
[-,3,2,-,+]=>3
[-,3,2,+,-]=>3
[+,3,2,-,-]=>3
[-,3,2,-,-]=>4
[+,3,2,5,4]=>2
[-,3,2,5,4]=>3
[+,3,4,2,+]=>2
[-,3,4,2,+]=>3
[+,3,4,2,-]=>3
[-,3,4,2,-]=>4
[+,3,4,5,2]=>3
[-,3,4,5,2]=>4
[+,3,5,2,4]=>2
[-,3,5,2,4]=>3
[+,3,5,+,2]=>2
[-,3,5,+,2]=>3
[+,3,5,-,2]=>3
[-,3,5,-,2]=>4
[+,4,2,3,+]=>1
[-,4,2,3,+]=>2
[+,4,2,3,-]=>2
[-,4,2,3,-]=>3
[+,4,2,5,3]=>2
[-,4,2,5,3]=>3
[+,4,+,2,+]=>1
[-,4,+,2,+]=>2
[+,4,-,2,+]=>2
[+,4,+,2,-]=>2
[-,4,-,2,+]=>3
[-,4,+,2,-]=>3
[+,4,-,2,-]=>3
[-,4,-,2,-]=>4
[+,4,+,5,2]=>2
[-,4,+,5,2]=>3
[+,4,-,5,2]=>3
[-,4,-,5,2]=>4
[+,4,5,2,3]=>2
[-,4,5,2,3]=>3
[+,4,5,3,2]=>2
[-,4,5,3,2]=>3
[+,5,2,3,4]=>1
[-,5,2,3,4]=>2
[+,5,2,+,3]=>1
[-,5,2,+,3]=>2
[+,5,2,-,3]=>2
[-,5,2,-,3]=>3
[+,5,+,2,4]=>1
[-,5,+,2,4]=>2
[+,5,-,2,4]=>2
[-,5,-,2,4]=>3
[+,5,+,+,2]=>1
[-,5,+,+,2]=>2
[+,5,-,+,2]=>2
[+,5,+,-,2]=>2
[-,5,-,+,2]=>3
[-,5,+,-,2]=>3
[+,5,-,-,2]=>3
[-,5,-,-,2]=>4
[+,5,4,2,3]=>2
[-,5,4,2,3]=>3
[+,5,4,3,2]=>2
[-,5,4,3,2]=>3
[2,1,+,+,+]=>1
[2,1,-,+,+]=>2
[2,1,+,-,+]=>2
[2,1,+,+,-]=>2
[2,1,-,-,+]=>3
[2,1,-,+,-]=>3
[2,1,+,-,-]=>3
[2,1,-,-,-]=>4
[2,1,+,5,4]=>2
[2,1,-,5,4]=>3
[2,1,4,3,+]=>2
[2,1,4,3,-]=>3
[2,1,4,5,3]=>3
[2,1,5,3,4]=>2
[2,1,5,+,3]=>2
[2,1,5,-,3]=>3
[2,3,1,+,+]=>2
[2,3,1,-,+]=>3
[2,3,1,+,-]=>3
[2,3,1,-,-]=>4
[2,3,1,5,4]=>3
[2,3,4,1,+]=>3
[2,3,4,1,-]=>4
[2,3,4,5,1]=>4
[2,3,5,1,4]=>3
[2,3,5,+,1]=>3
[2,3,5,-,1]=>4
[2,4,1,3,+]=>2
[2,4,1,3,-]=>3
[2,4,1,5,3]=>3
[2,4,+,1,+]=>2
[2,4,-,1,+]=>3
[2,4,+,1,-]=>3
[2,4,-,1,-]=>4
[2,4,+,5,1]=>3
[2,4,-,5,1]=>4
[2,4,5,1,3]=>3
[2,4,5,3,1]=>3
[2,5,1,3,4]=>2
[2,5,1,+,3]=>2
[2,5,1,-,3]=>3
[2,5,+,1,4]=>2
[2,5,-,1,4]=>3
[2,5,+,+,1]=>2
[2,5,-,+,1]=>3
[2,5,+,-,1]=>3
[2,5,-,-,1]=>4
[2,5,4,1,3]=>3
[2,5,4,3,1]=>3
[3,1,2,+,+]=>1
[3,1,2,-,+]=>2
[3,1,2,+,-]=>2
[3,1,2,-,-]=>3
[3,1,2,5,4]=>2
[3,1,4,2,+]=>2
[3,1,4,2,-]=>3
[3,1,4,5,2]=>3
[3,1,5,2,4]=>2
[3,1,5,+,2]=>2
[3,1,5,-,2]=>3
[3,+,1,+,+]=>1
[3,-,1,+,+]=>2
[3,+,1,-,+]=>2
[3,+,1,+,-]=>2
[3,-,1,-,+]=>3
[3,-,1,+,-]=>3
[3,+,1,-,-]=>3
[3,-,1,-,-]=>4
[3,+,1,5,4]=>2
[3,-,1,5,4]=>3
[3,+,4,1,+]=>2
[3,-,4,1,+]=>3
[3,+,4,1,-]=>3
[3,-,4,1,-]=>4
[3,+,4,5,1]=>3
[3,-,4,5,1]=>4
[3,+,5,1,4]=>2
[3,-,5,1,4]=>3
[3,+,5,+,1]=>2
[3,-,5,+,1]=>3
[3,+,5,-,1]=>3
[3,-,5,-,1]=>4
[3,4,1,2,+]=>2
[3,4,1,2,-]=>3
[3,4,1,5,2]=>3
[3,4,2,1,+]=>2
[3,4,2,1,-]=>3
[3,4,2,5,1]=>3
[3,4,5,1,2]=>3
[3,4,5,2,1]=>3
[3,5,1,2,4]=>2
[3,5,1,+,2]=>2
[3,5,1,-,2]=>3
[3,5,2,1,4]=>2
[3,5,2,+,1]=>2
[3,5,2,-,1]=>3
[3,5,4,1,2]=>3
[3,5,4,2,1]=>3
[4,1,2,3,+]=>1
[4,1,2,3,-]=>2
[4,1,2,5,3]=>2
[4,1,+,2,+]=>1
[4,1,-,2,+]=>2
[4,1,+,2,-]=>2
[4,1,-,2,-]=>3
[4,1,+,5,2]=>2
[4,1,-,5,2]=>3
[4,1,5,2,3]=>2
[4,1,5,3,2]=>2
[4,+,1,3,+]=>1
[4,-,1,3,+]=>2
[4,+,1,3,-]=>2
[4,-,1,3,-]=>3
[4,+,1,5,3]=>2
[4,-,1,5,3]=>3
[4,+,+,1,+]=>1
[4,-,+,1,+]=>2
[4,+,-,1,+]=>2
[4,+,+,1,-]=>2
[4,-,-,1,+]=>3
[4,-,+,1,-]=>3
[4,+,-,1,-]=>3
[4,-,-,1,-]=>4
[4,+,+,5,1]=>2
[4,-,+,5,1]=>3
[4,+,-,5,1]=>3
[4,-,-,5,1]=>4
[4,+,5,1,3]=>2
[4,-,5,1,3]=>3
[4,+,5,3,1]=>2
[4,-,5,3,1]=>3
[4,3,1,2,+]=>2
[4,3,1,2,-]=>3
[4,3,1,5,2]=>3
[4,3,2,1,+]=>2
[4,3,2,1,-]=>3
[4,3,2,5,1]=>3
[4,3,5,1,2]=>3
[4,3,5,2,1]=>3
[4,5,1,2,3]=>2
[4,5,1,3,2]=>2
[4,5,2,1,3]=>2
[4,5,2,3,1]=>2
[4,5,+,1,2]=>2
[4,5,-,1,2]=>3
[4,5,+,2,1]=>2
[4,5,-,2,1]=>3
[5,1,2,3,4]=>1
[5,1,2,+,3]=>1
[5,1,2,-,3]=>2
[5,1,+,2,4]=>1
[5,1,-,2,4]=>2
[5,1,+,+,2]=>1
[5,1,-,+,2]=>2
[5,1,+,-,2]=>2
[5,1,-,-,2]=>3
[5,1,4,2,3]=>2
[5,1,4,3,2]=>2
[5,+,1,3,4]=>1
[5,-,1,3,4]=>2
[5,+,1,+,3]=>1
[5,-,1,+,3]=>2
[5,+,1,-,3]=>2
[5,-,1,-,3]=>3
[5,+,+,1,4]=>1
[5,-,+,1,4]=>2
[5,+,-,1,4]=>2
[5,-,-,1,4]=>3
[5,+,+,+,1]=>1
[5,-,+,+,1]=>2
[5,+,-,+,1]=>2
[5,+,+,-,1]=>2
[5,-,-,+,1]=>3
[5,-,+,-,1]=>3
[5,+,-,-,1]=>3
[5,-,-,-,1]=>4
[5,+,4,1,3]=>2
[5,-,4,1,3]=>3
[5,+,4,3,1]=>2
[5,-,4,3,1]=>3
[5,3,1,2,4]=>2
[5,3,1,+,2]=>2
[5,3,1,-,2]=>3
[5,3,2,1,4]=>2
[5,3,2,+,1]=>2
[5,3,2,-,1]=>3
[5,3,4,1,2]=>3
[5,3,4,2,1]=>3
[5,4,1,2,3]=>2
[5,4,1,3,2]=>2
[5,4,2,1,3]=>2
[5,4,2,3,1]=>2
[5,4,+,1,2]=>2
[5,4,-,1,2]=>3
[5,4,+,2,1]=>2
[5,4,-,2,1]=>3
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Description
The dimension of the subspace of the complex vector space for the associated Grassmannian.
Given an affine permutation, this is $$\frac{1}{n} \sum^n_{i=1} (f(i)-i)$$
This value is seen as $k$ in the notation Gr($k,n$), ($k,n$)-bounded affine permutations, ($k,n$)-Grassmann necklaces, and ($k,n$)-Le diagrams.
Given an affine permutation, this is $$\frac{1}{n} \sum^n_{i=1} (f(i)-i)$$
This value is seen as $k$ in the notation Gr($k,n$), ($k,n$)-bounded affine permutations, ($k,n$)-Grassmann necklaces, and ($k,n$)-Le diagrams.
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):
tau = dectobap(pi)
sum=0
indices=0
for i in range(0,len(tau)):
sum = sum+tau[i]
indices = indices+i+1
k = (sum-indices)/len(tau)
return len(tau)-k #to align with k-value from Postnikov's Grassmann necklace and Le-diagram
Created
May 12, 2020 at 22:30 by Danny Luecke
Updated
May 15, 2020 at 08:26 by Danny Luecke
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