Identifier
Values
[+] => 0
[-] => 0
[+,+] => 0
[-,+] => 0
[+,-] => 0
[-,-] => 0
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 0
[+,-,+] => 0
[+,+,-] => 0
[-,-,+] => 0
[-,+,-] => 0
[+,-,-] => 0
[-,-,-] => 0
[+,3,2] => 0
[-,3,2] => 0
[2,1,+] => 0
[2,1,-] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 0
[3,-,1] => 0
[+,+,+,+] => 0
[-,+,+,+] => 0
[+,-,+,+] => 0
[+,+,-,+] => 0
[+,+,+,-] => 0
[-,-,+,+] => 0
[-,+,-,+] => 0
[-,+,+,-] => 0
[+,-,-,+] => 0
[+,-,+,-] => 0
[+,+,-,-] => 0
[-,-,-,+] => 0
[-,-,+,-] => 0
[-,+,-,-] => 0
[+,-,-,-] => 0
[-,-,-,-] => 0
[+,+,4,3] => 0
[-,+,4,3] => 0
[+,-,4,3] => 0
[-,-,4,3] => 0
[+,3,2,+] => 0
[-,3,2,+] => 0
[+,3,2,-] => 0
[-,3,2,-] => 0
[+,3,4,2] => 0
[-,3,4,2] => 0
[+,4,2,3] => 0
[-,4,2,3] => 0
[+,4,+,2] => 0
[-,4,+,2] => 0
[+,4,-,2] => 0
[-,4,-,2] => 0
[2,1,+,+] => 0
[2,1,-,+] => 0
[2,1,+,-] => 0
[2,1,-,-] => 0
[2,1,4,3] => 0
[2,3,1,+] => 0
[2,3,1,-] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,+,1] => 0
[2,4,-,1] => 0
[3,1,2,+] => 0
[3,1,2,-] => 0
[3,1,4,2] => 1
[3,+,1,+] => 0
[3,-,1,+] => 0
[3,+,1,-] => 0
[3,-,1,-] => 0
[3,+,4,1] => 0
[3,-,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,+,2] => 0
[4,1,-,2] => 0
[4,+,1,3] => 0
[4,-,1,3] => 0
[4,+,+,1] => 0
[4,-,+,1] => 0
[4,+,-,1] => 0
[4,-,-,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[+,+,+,+,+] => 0
[-,+,+,+,+] => 0
[+,-,+,+,+] => 0
[+,+,-,+,+] => 0
[+,+,+,-,+] => 0
[+,+,+,+,-] => 0
[-,-,+,+,+] => 0
[-,+,-,+,+] => 0
[-,+,+,-,+] => 0
[-,+,+,+,-] => 0
[+,-,-,+,+] => 0
[+,-,+,-,+] => 0
[+,-,+,+,-] => 0
>>> Load all 1200 entries. <<<
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Description
The number of crossed alignments in a decorated permutation.
Let $w$ be a decorated permutation.
An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment.
Let $w$ be a decorated permutation.
An alignment $(i \mapsto w(i), j \mapsto w(j))$ is crossed if there exists a third arc $(h \mapsto w(h))$ which forms a crossing with both arcs in the alignment.
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 cyclic_interval(a, b, n):
if a <= b:
return list(range(a, b+1))
return list(range(a, n+1)) + list(range(1, b+1))
def alignments(x):
"""
sage: x = DecoratedPermutation([8,9,5,-4,7,6,1,3,2])
sage: (3, 5, 6, 6) in alignments(x)
True
sage: len(alignments(x))
13
sage: x = DecoratedPermutation([8,9,5,4,7,-6,1,3,2])
sage: (3, 5, 6, 6) in alignments(x)
False
"""
pi = x.to_signed_permutation().permutation()
n = len(pi)
al = []
for i, e in enumerate(pi, 1):
for j, f in enumerate(pi, 1):
if i == j:
continue
a, b = sorted([i, e])
c, d = sorted([j, f])
if a <= c <= b <= d or c <= a <= d <= b:
continue
if (e not in cyclic_interval(i, f-1, n)
or f not in cyclic_interval(e+1, j, n)):
continue
if j == f and x[j-1] < 0:
continue
if i == e and x[i-1] > 0:
continue
al.append((i, e, j, f))
return al
def statistic(x):
A = alignments(x)
pi = x.to_signed_permutation().permutation()
n = len(pi)
cal = []
for i, e, j, f in A:
a, b = sorted([i, e])
c, d = sorted([j, f])
for k, g in enumerate(pi, 1):
r, s = sorted([k, g])
if ((a <= r <= b <= s or r <= a <= s <= b) and
(c <= r <= d <= s or r <= c <= s <= d)):
cal.append((i, e, j, f, r, k))
break
return len(cal)
Created
Jul 22, 2022 at 12:07 by Martin Rubey
Updated
Jul 22, 2022 at 12:07 by Martin Rubey
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