Identifier
Values
=>
[+]=>1
[-]=>1
[+,+]=>1
[-,+]=>1
[+,-]=>1
[-,-]=>1
[2,1]=>2
[+,+,+]=>1
[-,+,+]=>1
[+,-,+]=>1
[+,+,-]=>1
[-,-,+]=>1
[-,+,-]=>1
[+,-,-]=>1
[-,-,-]=>1
[+,3,2]=>2
[-,3,2]=>2
[2,1,+]=>2
[2,1,-]=>2
[2,3,1]=>4
[3,1,2]=>4
[3,+,1]=>2
[3,-,1]=>2
[+,+,+,+]=>1
[-,+,+,+]=>1
[+,-,+,+]=>1
[+,+,-,+]=>1
[+,+,+,-]=>1
[-,-,+,+]=>1
[-,+,-,+]=>1
[-,+,+,-]=>1
[+,-,-,+]=>1
[+,-,+,-]=>1
[+,+,-,-]=>1
[-,-,-,+]=>1
[-,-,+,-]=>1
[-,+,-,-]=>1
[+,-,-,-]=>1
[-,-,-,-]=>1
[+,+,4,3]=>2
[-,+,4,3]=>2
[+,-,4,3]=>2
[-,-,4,3]=>2
[+,3,2,+]=>2
[-,3,2,+]=>2
[+,3,2,-]=>2
[-,3,2,-]=>2
[+,3,4,2]=>4
[-,3,4,2]=>4
[+,4,2,3]=>4
[-,4,2,3]=>4
[+,4,+,2]=>2
[-,4,+,2]=>2
[+,4,-,2]=>2
[-,4,-,2]=>2
[2,1,+,+]=>2
[2,1,-,+]=>2
[2,1,+,-]=>2
[2,1,-,-]=>2
[2,1,4,3]=>4
[2,3,1,+]=>4
[2,3,1,-]=>4
[2,3,4,1]=>8
[2,4,1,3]=>10
[2,4,+,1]=>4
[2,4,-,1]=>4
[3,1,2,+]=>4
[3,1,2,-]=>4
[3,1,4,2]=>8
[3,+,1,+]=>2
[3,-,1,+]=>2
[3,+,1,-]=>2
[3,-,1,-]=>2
[3,+,4,1]=>4
[3,-,4,1]=>4
[3,4,1,2]=>14
[3,4,2,1]=>8
[4,1,2,3]=>8
[4,1,+,2]=>4
[4,1,-,2]=>4
[4,+,1,3]=>4
[4,-,1,3]=>4
[4,+,+,1]=>2
[4,-,+,1]=>2
[4,+,-,1]=>2
[4,-,-,1]=>2
[4,3,1,2]=>8
[4,3,2,1]=>4
[+,+,+,+,+]=>1
[-,+,+,+,+]=>1
[+,-,+,+,+]=>1
[+,+,-,+,+]=>1
[+,+,+,-,+]=>1
[+,+,+,+,-]=>1
[-,-,+,+,+]=>1
[-,+,-,+,+]=>1
[-,+,+,-,+]=>1
[-,+,+,+,-]=>1
[+,-,-,+,+]=>1
[+,-,+,-,+]=>1
[+,-,+,+,-]=>1
[+,+,-,-,+]=>1
[+,+,-,+,-]=>1
[+,+,+,-,-]=>1
[-,-,-,+,+]=>1
[-,-,+,-,+]=>1
[-,-,+,+,-]=>1
[-,+,-,-,+]=>1
[-,+,-,+,-]=>1
[-,+,+,-,-]=>1
[+,-,-,-,+]=>1
[+,-,-,+,-]=>1
[+,-,+,-,-]=>1
[+,+,-,-,-]=>1
[-,-,-,-,+]=>1
[-,-,-,+,-]=>1
[-,-,+,-,-]=>1
[-,+,-,-,-]=>1
[+,-,-,-,-]=>1
[-,-,-,-,-]=>1
[+,+,+,5,4]=>2
[-,+,+,5,4]=>2
[+,-,+,5,4]=>2
[+,+,-,5,4]=>2
[-,-,+,5,4]=>2
[-,+,-,5,4]=>2
[+,-,-,5,4]=>2
[-,-,-,5,4]=>2
[+,+,4,3,+]=>2
[-,+,4,3,+]=>2
[+,-,4,3,+]=>2
[+,+,4,3,-]=>2
[-,-,4,3,+]=>2
[-,+,4,3,-]=>2
[+,-,4,3,-]=>2
[-,-,4,3,-]=>2
[+,+,4,5,3]=>4
[-,+,4,5,3]=>4
[+,-,4,5,3]=>4
[-,-,4,5,3]=>4
[+,+,5,3,4]=>4
[-,+,5,3,4]=>4
[+,-,5,3,4]=>4
[-,-,5,3,4]=>4
[+,+,5,+,3]=>2
[-,+,5,+,3]=>2
[+,-,5,+,3]=>2
[+,+,5,-,3]=>2
[-,-,5,+,3]=>2
[-,+,5,-,3]=>2
[+,-,5,-,3]=>2
[-,-,5,-,3]=>2
[+,3,2,+,+]=>2
[-,3,2,+,+]=>2
[+,3,2,-,+]=>2
[+,3,2,+,-]=>2
[-,3,2,-,+]=>2
[-,3,2,+,-]=>2
[+,3,2,-,-]=>2
[-,3,2,-,-]=>2
[+,3,2,5,4]=>4
[-,3,2,5,4]=>4
[+,3,4,2,+]=>4
[-,3,4,2,+]=>4
[+,3,4,2,-]=>4
[-,3,4,2,-]=>4
[+,3,4,5,2]=>8
[-,3,4,5,2]=>8
[+,3,5,2,4]=>10
[-,3,5,2,4]=>10
[+,3,5,+,2]=>4
[-,3,5,+,2]=>4
[+,3,5,-,2]=>4
[-,3,5,-,2]=>4
[+,4,2,3,+]=>4
[-,4,2,3,+]=>4
[+,4,2,3,-]=>4
[-,4,2,3,-]=>4
[+,4,2,5,3]=>8
[-,4,2,5,3]=>8
[+,4,+,2,+]=>2
[-,4,+,2,+]=>2
[+,4,-,2,+]=>2
[+,4,+,2,-]=>2
[-,4,-,2,+]=>2
[-,4,+,2,-]=>2
[+,4,-,2,-]=>2
[-,4,-,2,-]=>2
[+,4,+,5,2]=>4
[-,4,+,5,2]=>4
[+,4,-,5,2]=>4
[-,4,-,5,2]=>4
[+,4,5,2,3]=>14
[-,4,5,2,3]=>14
[+,4,5,3,2]=>8
[-,4,5,3,2]=>8
[+,5,2,3,4]=>8
[-,5,2,3,4]=>8
[+,5,2,+,3]=>4
[-,5,2,+,3]=>4
[+,5,2,-,3]=>4
[-,5,2,-,3]=>4
[+,5,+,2,4]=>4
[-,5,+,2,4]=>4
[+,5,-,2,4]=>4
[-,5,-,2,4]=>4
[+,5,+,+,2]=>2
[-,5,+,+,2]=>2
[+,5,-,+,2]=>2
[+,5,+,-,2]=>2
[-,5,-,+,2]=>2
[-,5,+,-,2]=>2
[+,5,-,-,2]=>2
[-,5,-,-,2]=>2
[+,5,4,2,3]=>8
[-,5,4,2,3]=>8
[+,5,4,3,2]=>4
[-,5,4,3,2]=>4
[2,1,+,+,+]=>2
[2,1,-,+,+]=>2
[2,1,+,-,+]=>2
[2,1,+,+,-]=>2
[2,1,-,-,+]=>2
[2,1,-,+,-]=>2
[2,1,+,-,-]=>2
[2,1,-,-,-]=>2
[2,1,+,5,4]=>4
[2,1,-,5,4]=>4
[2,1,4,3,+]=>4
[2,1,4,3,-]=>4
[2,1,4,5,3]=>8
[2,1,5,3,4]=>8
[2,1,5,+,3]=>4
[2,1,5,-,3]=>4
[2,3,1,+,+]=>4
[2,3,1,-,+]=>4
[2,3,1,+,-]=>4
[2,3,1,-,-]=>4
[2,3,1,5,4]=>8
[2,3,4,1,+]=>8
[2,3,4,1,-]=>8
[2,3,4,5,1]=>16
[2,3,5,1,4]=>22
[2,3,5,+,1]=>8
[2,3,5,-,1]=>8
[2,4,1,3,+]=>10
[2,4,1,3,-]=>10
[2,4,1,5,3]=>20
[2,4,+,1,+]=>4
[2,4,-,1,+]=>4
[2,4,+,1,-]=>4
[2,4,-,1,-]=>4
[2,4,+,5,1]=>8
[2,4,-,5,1]=>8
[2,4,5,1,3]=>34
[2,4,5,3,1]=>16
[2,5,1,3,4]=>22
[2,5,1,+,3]=>10
[2,5,1,-,3]=>10
[2,5,+,1,4]=>10
[2,5,-,1,4]=>10
[2,5,+,+,1]=>4
[2,5,-,+,1]=>4
[2,5,+,-,1]=>4
[2,5,-,-,1]=>4
[2,5,4,1,3]=>20
[2,5,4,3,1]=>8
[3,1,2,+,+]=>4
[3,1,2,-,+]=>4
[3,1,2,+,-]=>4
[3,1,2,-,-]=>4
[3,1,2,5,4]=>8
[3,1,4,2,+]=>8
[3,1,4,2,-]=>8
[3,1,4,5,2]=>16
[3,1,5,2,4]=>20
[3,1,5,+,2]=>8
[3,1,5,-,2]=>8
[3,+,1,+,+]=>2
[3,-,1,+,+]=>2
[3,+,1,-,+]=>2
[3,+,1,+,-]=>2
[3,-,1,-,+]=>2
[3,-,1,+,-]=>2
[3,+,1,-,-]=>2
[3,-,1,-,-]=>2
[3,+,1,5,4]=>4
[3,-,1,5,4]=>4
[3,+,4,1,+]=>4
[3,-,4,1,+]=>4
[3,+,4,1,-]=>4
[3,-,4,1,-]=>4
[3,+,4,5,1]=>8
[3,-,4,5,1]=>8
[3,+,5,1,4]=>10
[3,-,5,1,4]=>10
[3,+,5,+,1]=>4
[3,-,5,+,1]=>4
[3,+,5,-,1]=>4
[3,-,5,-,1]=>4
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Description
The size of the Grassmannian interval associated with a decorated permutation.
References
[1] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs arXiv:2207.06508
Code
def anti_exceedance_positions(x):
"""
sage: x = DecoratedPermutation([5,4,1,2,7,6,9,-8,3])
sage: anti_exceedance_positions(x)
[3, 4, 9, 6]
"""
pi = x.to_signed_permutation().permutation()
pi_inv = pi.inverse()
n = len(pi)
anti = []
for i in range(1, n+1):
if i < pi_inv(i) or x[i-1] == i:
anti.append(pi_inv(i))
return anti
def to_grassmann_interval(x):
"""
sage: x = DecoratedPermutation([5,4,1,2,7,6,9,-8,3])
sage: to_grassmann_interval(x)
([1, 2, 6, 3, 5, 4, 7, 9, 8], [3, 4, 6, 9, 1, 2, 5, 7, 8])
"""
n = len(x)
I = anti_exceedance_positions(x)
u1 = []
u2 = []
v1 = []
v2 = []
for i in range(1, n+1):
if i in I:
v1.append(i)
u1.append(abs(x[i-1]))
else:
v2.append(i)
u2.append(abs(x[i-1]))
return Permutation(u1+u2), Permutation(v1+v2)
def interval_size(x):
u, v = to_grassmann_interval(x)
return sum(1 for pi in u.bruhat_greater() if pi.bruhat_lequal(v))
Created
Jul 23, 2022 at 18:16 by Martin Rubey
Updated
Jul 23, 2022 at 18:16 by Martin Rubey
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