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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
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-----------------------------------------------------------------------------
Statistic identifier: St001818
-----------------------------------------------------------------------------
Collection: Decorated permutations
-----------------------------------------------------------------------------
Description: The number of alignments of a decorated permutation.
An alignment in a decorated permutation $w$ is a pair of directed edges $(i\mapsto w(i), j\mapsto w(j))$ which can be drawn as distinct noncrossing arcs oriented in the same direction in the circular chord diagram.
According to [1], the number of alignments of a decorated permutation on $n$ letters with $k$ anit-exceedances equals and associated Grassmann interval $[u, v]$ equals $k(n - k) - [\ell(v) − \ell(u)]$.
-----------------------------------------------------------------------------
References: [1] A. Postnikov, ''Total positivity, Grassmannians, and networks.'' 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks [[arXiv:math/0609764]]
[2] Billey, S. C., Weaver, J. E. Criteria for smoothness of Positroid varieties via pattern avoidance, Johnson graphs, and spirographs [[arXiv:2207.06508]]
-----------------------------------------------------------------------------
Code:
# please simplify
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):
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):
return len(alignments(x))
-----------------------------------------------------------------------------
Statistic values:
[+] => 0
[-] => 0
[+,+] => 0
[-,+] => 1
[+,-] => 1
[-,-] => 0
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 2
[+,-,+] => 2
[+,+,-] => 2
[-,-,+] => 2
[-,+,-] => 2
[+,-,-] => 2
[-,-,-] => 0
[+,3,2] => 1
[-,3,2] => 1
[2,1,+] => 1
[2,1,-] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 1
[3,-,1] => 1
[+,+,+,+] => 0
[-,+,+,+] => 3
[+,-,+,+] => 3
[+,+,-,+] => 3
[+,+,+,-] => 3
[-,-,+,+] => 4
[-,+,-,+] => 4
[-,+,+,-] => 4
[+,-,-,+] => 4
[+,-,+,-] => 4
[+,+,-,-] => 4
[-,-,-,+] => 3
[-,-,+,-] => 3
[-,+,-,-] => 3
[+,-,-,-] => 3
[-,-,-,-] => 0
[+,+,4,3] => 2
[-,+,4,3] => 3
[+,-,4,3] => 3
[-,-,4,3] => 2
[+,3,2,+] => 2
[-,3,2,+] => 3
[+,3,2,-] => 3
[-,3,2,-] => 2
[+,3,4,2] => 2
[-,3,4,2] => 1
[+,4,2,3] => 1
[-,4,2,3] => 2
[+,4,+,2] => 2
[-,4,+,2] => 3
[+,4,-,2] => 3
[-,4,-,2] => 2
[2,1,+,+] => 2
[2,1,-,+] => 3
[2,1,+,-] => 3
[2,1,-,-] => 2
[2,1,4,3] => 2
[2,3,1,+] => 2
[2,3,1,-] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,+,1] => 2
[2,4,-,1] => 1
[3,1,2,+] => 1
[3,1,2,-] => 2
[3,1,4,2] => 1
[3,+,1,+] => 2
[3,-,1,+] => 3
[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] => 2
[4,+,+,1] => 2
[4,-,+,1] => 3
[4,+,-,1] => 3
[4,-,-,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 2
[+,+,+,+,+] => 0
[-,+,+,+,+] => 4
[+,-,+,+,+] => 4
[+,+,-,+,+] => 4
[+,+,+,-,+] => 4
[+,+,+,+,-] => 4
[-,-,+,+,+] => 6
[-,+,-,+,+] => 6
[-,+,+,-,+] => 6
[-,+,+,+,-] => 6
[+,-,-,+,+] => 6
[+,-,+,-,+] => 6
[+,-,+,+,-] => 6
[+,+,-,-,+] => 6
[+,+,-,+,-] => 6
[+,+,+,-,-] => 6
[-,-,-,+,+] => 6
[-,-,+,-,+] => 6
[-,-,+,+,-] => 6
[-,+,-,-,+] => 6
[-,+,-,+,-] => 6
[-,+,+,-,-] => 6
[+,-,-,-,+] => 6
[+,-,-,+,-] => 6
[+,-,+,-,-] => 6
[+,+,-,-,-] => 6
[-,-,-,-,+] => 4
[-,-,-,+,-] => 4
[-,-,+,-,-] => 4
[-,+,-,-,-] => 4
[+,-,-,-,-] => 4
[-,-,-,-,-] => 0
[+,+,+,5,4] => 3
[-,+,+,5,4] => 5
[+,-,+,5,4] => 5
[+,+,-,5,4] => 5
[-,-,+,5,4] => 5
[-,+,-,5,4] => 5
[+,-,-,5,4] => 5
[-,-,-,5,4] => 3
[+,+,4,3,+] => 3
[-,+,4,3,+] => 5
[+,-,4,3,+] => 5
[+,+,4,3,-] => 5
[-,-,4,3,+] => 5
[-,+,4,3,-] => 5
[+,-,4,3,-] => 5
[-,-,4,3,-] => 3
[+,+,4,5,3] => 4
[-,+,4,5,3] => 4
[+,-,4,5,3] => 4
[-,-,4,5,3] => 2
[+,+,5,3,4] => 2
[-,+,5,3,4] => 4
[+,-,5,3,4] => 4
[-,-,5,3,4] => 4
[+,+,5,+,3] => 3
[-,+,5,+,3] => 5
[+,-,5,+,3] => 5
[+,+,5,-,3] => 5
[-,-,5,+,3] => 5
[-,+,5,-,3] => 5
[+,-,5,-,3] => 5
[-,-,5,-,3] => 3
[+,3,2,+,+] => 3
[-,3,2,+,+] => 5
[+,3,2,-,+] => 5
[+,3,2,+,-] => 5
[-,3,2,-,+] => 5
[-,3,2,+,-] => 5
[+,3,2,-,-] => 5
[-,3,2,-,-] => 3
[+,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,-] => 2
[+,3,4,5,2] => 3
[-,3,4,5,2] => 1
[+,3,5,2,4] => 3
[-,3,5,2,4] => 3
[+,3,5,+,2] => 4
[-,3,5,+,2] => 4
[+,3,5,-,2] => 4
[-,3,5,-,2] => 2
[+,4,2,3,+] => 2
[-,4,2,3,+] => 4
[+,4,2,3,-] => 4
[-,4,2,3,-] => 4
[+,4,2,5,3] => 3
[-,4,2,5,3] => 3
[+,4,+,2,+] => 3
[-,4,+,2,+] => 5
[+,4,-,2,+] => 5
[+,4,+,2,-] => 5
[-,4,-,2,+] => 5
[-,4,+,2,-] => 5
[+,4,-,2,-] => 5
[-,4,-,2,-] => 3
[+,4,+,5,2] => 4
[-,4,+,5,2] => 4
[+,4,-,5,2] => 4
[-,4,-,5,2] => 2
[+,4,5,2,3] => 2
[-,4,5,2,3] => 2
[+,4,5,3,2] => 3
[-,4,5,3,2] => 3
[+,5,2,3,4] => 1
[-,5,2,3,4] => 3
[+,5,2,+,3] => 2
[-,5,2,+,3] => 4
[+,5,2,-,3] => 4
[-,5,2,-,3] => 4
[+,5,+,2,4] => 2
[-,5,+,2,4] => 4
[+,5,-,2,4] => 4
[-,5,-,2,4] => 4
[+,5,+,+,2] => 3
[-,5,+,+,2] => 5
[+,5,-,+,2] => 5
[+,5,+,-,2] => 5
[-,5,-,+,2] => 5
[-,5,+,-,2] => 5
[+,5,-,-,2] => 5
[-,5,-,-,2] => 3
[+,5,4,2,3] => 3
[-,5,4,2,3] => 3
[+,5,4,3,2] => 4
[-,5,4,3,2] => 4
[2,1,+,+,+] => 3
[2,1,-,+,+] => 5
[2,1,+,-,+] => 5
[2,1,+,+,-] => 5
[2,1,-,-,+] => 5
[2,1,-,+,-] => 5
[2,1,+,-,-] => 5
[2,1,-,-,-] => 3
[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] => 3
[2,1,5,3,4] => 3
[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,-,-] => 2
[2,3,1,5,4] => 3
[2,3,4,1,+] => 3
[2,3,4,1,-] => 1
[2,3,4,5,1] => 0
[2,3,5,1,4] => 2
[2,3,5,+,1] => 3
[2,3,5,-,1] => 1
[2,4,1,3,+] => 3
[2,4,1,3,-] => 3
[2,4,1,5,3] => 2
[2,4,+,1,+] => 4
[2,4,-,1,+] => 4
[2,4,+,1,-] => 4
[2,4,-,1,-] => 2
[2,4,+,5,1] => 3
[2,4,-,5,1] => 1
[2,4,5,1,3] => 1
[2,4,5,3,1] => 2
[2,5,1,3,4] => 2
[2,5,1,+,3] => 3
[2,5,1,-,3] => 3
[2,5,+,1,4] => 3
[2,5,-,1,4] => 3
[2,5,+,+,1] => 4
[2,5,-,+,1] => 4
[2,5,+,-,1] => 4
[2,5,-,-,1] => 2
[2,5,4,1,3] => 2
[2,5,4,3,1] => 3
[3,1,2,+,+] => 2
[3,1,2,-,+] => 4
[3,1,2,+,-] => 4
[3,1,2,-,-] => 4
[3,1,2,5,4] => 3
[3,1,4,2,+] => 3
[3,1,4,2,-] => 3
[3,1,4,5,2] => 2
[3,1,5,2,4] => 2
[3,1,5,+,2] => 3
[3,1,5,-,2] => 3
[3,+,1,+,+] => 3
[3,-,1,+,+] => 5
[3,+,1,-,+] => 5
[3,+,1,+,-] => 5
[3,-,1,-,+] => 5
[3,-,1,+,-] => 5
[3,+,1,-,-] => 5
[3,-,1,-,-] => 3
[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,-] => 2
[3,+,4,5,1] => 3
[3,-,4,5,1] => 1
[3,+,5,1,4] => 3
[3,-,5,1,4] => 3
[3,+,5,+,1] => 4
[3,-,5,+,1] => 4
[3,+,5,-,1] => 4
[3,-,5,-,1] => 2
[3,4,1,2,+] => 2
[3,4,1,2,-] => 2
[3,4,1,5,2] => 1
[3,4,2,1,+] => 3
[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] => 1
[3,5,1,+,2] => 2
[3,5,1,-,2] => 2
[3,5,2,1,4] => 2
[3,5,2,+,1] => 3
[3,5,2,-,1] => 3
[3,5,4,1,2] => 1
[3,5,4,2,1] => 2
[4,1,2,3,+] => 1
[4,1,2,3,-] => 3
[4,1,2,5,3] => 2
[4,1,+,2,+] => 2
[4,1,-,2,+] => 4
[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,+] => 2
[4,-,1,3,+] => 4
[4,+,1,3,-] => 4
[4,-,1,3,-] => 4
[4,+,1,5,3] => 3
[4,-,1,5,3] => 3
[4,+,+,1,+] => 3
[4,-,+,1,+] => 5
[4,+,-,1,+] => 5
[4,+,+,1,-] => 5
[4,-,-,1,+] => 5
[4,-,+,1,-] => 5
[4,+,-,1,-] => 5
[4,-,-,1,-] => 3
[4,+,+,5,1] => 4
[4,-,+,5,1] => 4
[4,+,-,5,1] => 4
[4,-,-,5,1] => 2
[4,+,5,1,3] => 2
[4,-,5,1,3] => 2
[4,+,5,3,1] => 3
[4,-,5,3,1] => 3
[4,3,1,2,+] => 3
[4,3,1,2,-] => 3
[4,3,1,5,2] => 2
[4,3,2,1,+] => 4
[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] => 3
[5,1,+,+,2] => 2
[5,1,-,+,2] => 4
[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] => 3
[5,+,1,+,3] => 2
[5,-,1,+,3] => 4
[5,+,1,-,3] => 4
[5,-,1,-,3] => 4
[5,+,+,1,4] => 2
[5,-,+,1,4] => 4
[5,+,-,1,4] => 4
[5,-,-,1,4] => 4
[5,+,+,+,1] => 3
[5,-,+,+,1] => 5
[5,+,-,+,1] => 5
[5,+,+,-,1] => 5
[5,-,-,+,1] => 5
[5,-,+,-,1] => 5
[5,+,-,-,1] => 5
[5,-,-,-,1] => 3
[5,+,4,1,3] => 3
[5,-,4,1,3] => 3
[5,+,4,3,1] => 4
[5,-,4,3,1] => 4
[5,3,1,2,4] => 2
[5,3,1,+,2] => 3
[5,3,1,-,2] => 3
[5,3,2,1,4] => 3
[5,3,2,+,1] => 4
[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
[+,+,+,+,+,+] => 0
[-,+,+,+,+,+] => 5
[+,-,+,+,+,+] => 5
[+,+,-,+,+,+] => 5
[+,+,+,-,+,+] => 5
[+,+,+,+,-,+] => 5
[+,+,+,+,+,-] => 5
[-,-,+,+,+,+] => 8
[-,+,-,+,+,+] => 8
[-,+,+,-,+,+] => 8
[-,+,+,+,-,+] => 8
[-,+,+,+,+,-] => 8
[+,-,-,+,+,+] => 8
[+,-,+,-,+,+] => 8
[+,-,+,+,-,+] => 8
[+,-,+,+,+,-] => 8
[+,+,-,-,+,+] => 8
[+,+,-,+,-,+] => 8
[+,+,-,+,+,-] => 8
[+,+,+,-,-,+] => 8
[+,+,+,-,+,-] => 8
[+,+,+,+,-,-] => 8
[-,-,-,+,+,+] => 9
[-,-,+,-,+,+] => 9
[-,-,+,+,-,+] => 9
[-,-,+,+,+,-] => 9
[-,+,-,-,+,+] => 9
[-,+,-,+,-,+] => 9
[-,+,-,+,+,-] => 9
[-,+,+,-,-,+] => 9
[-,+,+,-,+,-] => 9
[-,+,+,+,-,-] => 9
[+,-,-,-,+,+] => 9
[+,-,-,+,-,+] => 9
[+,-,-,+,+,-] => 9
[+,-,+,-,-,+] => 9
[+,-,+,-,+,-] => 9
[+,-,+,+,-,-] => 9
[+,+,-,-,-,+] => 9
[+,+,-,-,+,-] => 9
[+,+,-,+,-,-] => 9
[+,+,+,-,-,-] => 9
[-,-,-,-,+,+] => 8
[-,-,-,+,-,+] => 8
[-,-,-,+,+,-] => 8
[-,-,+,-,-,+] => 8
[-,-,+,-,+,-] => 8
[-,-,+,+,-,-] => 8
[-,+,-,-,-,+] => 8
[-,+,-,-,+,-] => 8
[-,+,-,+,-,-] => 8
[-,+,+,-,-,-] => 8
[+,-,-,-,-,+] => 8
[+,-,-,-,+,-] => 8
[+,-,-,+,-,-] => 8
[+,-,+,-,-,-] => 8
[+,+,-,-,-,-] => 8
[-,-,-,-,-,+] => 5
[-,-,-,-,+,-] => 5
[-,-,-,+,-,-] => 5
[-,-,+,-,-,-] => 5
[-,+,-,-,-,-] => 5
[+,-,-,-,-,-] => 5
[-,-,-,-,-,-] => 0
[+,+,+,+,6,5] => 4
[-,+,+,+,6,5] => 7
[+,-,+,+,6,5] => 7
[+,+,-,+,6,5] => 7
[+,+,+,-,6,5] => 7
[-,-,+,+,6,5] => 8
[-,+,-,+,6,5] => 8
[-,+,+,-,6,5] => 8
[+,-,-,+,6,5] => 8
[+,-,+,-,6,5] => 8
[+,+,-,-,6,5] => 8
[-,-,-,+,6,5] => 7
[-,-,+,-,6,5] => 7
[-,+,-,-,6,5] => 7
[+,-,-,-,6,5] => 7
[-,-,-,-,6,5] => 4
[+,+,+,5,4,+] => 4
[-,+,+,5,4,+] => 7
[+,-,+,5,4,+] => 7
[+,+,-,5,4,+] => 7
[+,+,+,5,4,-] => 7
[-,-,+,5,4,+] => 8
[-,+,-,5,4,+] => 8
[-,+,+,5,4,-] => 8
[+,-,-,5,4,+] => 8
[+,-,+,5,4,-] => 8
[+,+,-,5,4,-] => 8
[-,-,-,5,4,+] => 7
[-,-,+,5,4,-] => 7
[-,+,-,5,4,-] => 7
[+,-,-,5,4,-] => 7
[-,-,-,5,4,-] => 4
[+,+,+,5,6,4] => 6
[-,+,+,5,6,4] => 7
[+,-,+,5,6,4] => 7
[+,+,-,5,6,4] => 7
[-,-,+,5,6,4] => 6
[-,+,-,5,6,4] => 6
[+,-,-,5,6,4] => 6
[-,-,-,5,6,4] => 3
[+,+,+,6,4,5] => 3
[-,+,+,6,4,5] => 6
[+,-,+,6,4,5] => 6
[+,+,-,6,4,5] => 6
[-,-,+,6,4,5] => 7
[-,+,-,6,4,5] => 7
[+,-,-,6,4,5] => 7
[-,-,-,6,4,5] => 6
[+,+,+,6,+,4] => 4
[-,+,+,6,+,4] => 7
[+,-,+,6,+,4] => 7
[+,+,-,6,+,4] => 7
[+,+,+,6,-,4] => 7
[-,-,+,6,+,4] => 8
[-,+,-,6,+,4] => 8
[-,+,+,6,-,4] => 8
[+,-,-,6,+,4] => 8
[+,-,+,6,-,4] => 8
[+,+,-,6,-,4] => 8
[-,-,-,6,+,4] => 7
[-,-,+,6,-,4] => 7
[-,+,-,6,-,4] => 7
[+,-,-,6,-,4] => 7
[-,-,-,6,-,4] => 4
[+,+,4,3,+,+] => 4
[-,+,4,3,+,+] => 7
[+,-,4,3,+,+] => 7
[+,+,4,3,-,+] => 7
[+,+,4,3,+,-] => 7
[-,-,4,3,+,+] => 8
[-,+,4,3,-,+] => 8
[-,+,4,3,+,-] => 8
[+,-,4,3,-,+] => 8
[+,-,4,3,+,-] => 8
[+,+,4,3,-,-] => 8
[-,-,4,3,-,+] => 7
[-,-,4,3,+,-] => 7
[-,+,4,3,-,-] => 7
[+,-,4,3,-,-] => 7
[-,-,4,3,-,-] => 4
[+,+,4,3,6,5] => 6
[-,+,4,3,6,5] => 7
[+,-,4,3,6,5] => 7
[-,-,4,3,6,5] => 6
[+,+,4,5,3,+] => 6
[-,+,4,5,3,+] => 7
[+,-,4,5,3,+] => 7
[+,+,4,5,3,-] => 7
[-,-,4,5,3,+] => 6
[-,+,4,5,3,-] => 6
[+,-,4,5,3,-] => 6
[-,-,4,5,3,-] => 3
[+,+,4,5,6,3] => 6
[-,+,4,5,6,3] => 5
[+,-,4,5,6,3] => 5
[-,-,4,5,6,3] => 2
[+,+,4,6,3,5] => 5
[-,+,4,6,3,5] => 6
[+,-,4,6,3,5] => 6
[-,-,4,6,3,5] => 5
[+,+,4,6,+,3] => 6
[-,+,4,6,+,3] => 7
[+,-,4,6,+,3] => 7
[+,+,4,6,-,3] => 7
[-,-,4,6,+,3] => 6
[-,+,4,6,-,3] => 6
[+,-,4,6,-,3] => 6
[-,-,4,6,-,3] => 3
[+,+,5,3,4,+] => 3
[-,+,5,3,4,+] => 6
[+,-,5,3,4,+] => 6
[+,+,5,3,4,-] => 6
[-,-,5,3,4,+] => 7
[-,+,5,3,4,-] => 7
[+,-,5,3,4,-] => 7
[-,-,5,3,4,-] => 6
[+,+,5,3,6,4] => 5
[-,+,5,3,6,4] => 6
[+,-,5,3,6,4] => 6
[-,-,5,3,6,4] => 5
[+,+,5,+,3,+] => 4
[-,+,5,+,3,+] => 7
[+,-,5,+,3,+] => 7
[+,+,5,-,3,+] => 7
[+,+,5,+,3,-] => 7
[-,-,5,+,3,+] => 8
[-,+,5,-,3,+] => 8
[-,+,5,+,3,-] => 8
[+,-,5,-,3,+] => 8
[+,-,5,+,3,-] => 8
[+,+,5,-,3,-] => 8
[-,-,5,-,3,+] => 7
[-,-,5,+,3,-] => 7
[-,+,5,-,3,-] => 7
[+,-,5,-,3,-] => 7
[-,-,5,-,3,-] => 4
[+,+,5,+,6,3] => 6
[-,+,5,+,6,3] => 7
[+,-,5,+,6,3] => 7
[+,+,5,-,6,3] => 7
[-,-,5,+,6,3] => 6
[-,+,5,-,6,3] => 6
[+,-,5,-,6,3] => 6
[-,-,5,-,6,3] => 3
[+,+,5,6,3,4] => 4
[-,+,5,6,3,4] => 5
[+,-,5,6,3,4] => 5
[-,-,5,6,3,4] => 4
[+,+,5,6,4,3] => 5
[-,+,5,6,4,3] => 6
[+,-,5,6,4,3] => 6
[-,-,5,6,4,3] => 5
[+,+,6,3,4,5] => 2
[-,+,6,3,4,5] => 5
[+,-,6,3,4,5] => 5
[-,-,6,3,4,5] => 6
[+,+,6,3,+,4] => 3
[-,+,6,3,+,4] => 6
[+,-,6,3,+,4] => 6
[+,+,6,3,-,4] => 6
[-,-,6,3,+,4] => 7
[-,+,6,3,-,4] => 7
[+,-,6,3,-,4] => 7
[-,-,6,3,-,4] => 6
[+,+,6,+,3,5] => 3
[-,+,6,+,3,5] => 6
[+,-,6,+,3,5] => 6
[+,+,6,-,3,5] => 6
[-,-,6,+,3,5] => 7
[-,+,6,-,3,5] => 7
[+,-,6,-,3,5] => 7
[-,-,6,-,3,5] => 6
[+,+,6,+,+,3] => 4
[-,+,6,+,+,3] => 7
[+,-,6,+,+,3] => 7
[+,+,6,-,+,3] => 7
[+,+,6,+,-,3] => 7
[-,-,6,+,+,3] => 8
[-,+,6,-,+,3] => 8
[-,+,6,+,-,3] => 8
[+,-,6,-,+,3] => 8
[+,-,6,+,-,3] => 8
[+,+,6,-,-,3] => 8
[-,-,6,-,+,3] => 7
[-,-,6,+,-,3] => 7
[-,+,6,-,-,3] => 7
[+,-,6,-,-,3] => 7
[-,-,6,-,-,3] => 4
[+,+,6,5,3,4] => 5
[-,+,6,5,3,4] => 6
[+,-,6,5,3,4] => 6
[-,-,6,5,3,4] => 5
[+,+,6,5,4,3] => 6
[-,+,6,5,4,3] => 7
[+,-,6,5,4,3] => 7
[-,-,6,5,4,3] => 6
[+,3,2,+,+,+] => 4
[-,3,2,+,+,+] => 7
[+,3,2,-,+,+] => 7
[+,3,2,+,-,+] => 7
[+,3,2,+,+,-] => 7
[-,3,2,-,+,+] => 8
[-,3,2,+,-,+] => 8
[-,3,2,+,+,-] => 8
[+,3,2,-,-,+] => 8
[+,3,2,-,+,-] => 8
[+,3,2,+,-,-] => 8
[-,3,2,-,-,+] => 7
[-,3,2,-,+,-] => 7
[-,3,2,+,-,-] => 7
[+,3,2,-,-,-] => 7
[-,3,2,-,-,-] => 4
[+,3,2,+,6,5] => 6
[-,3,2,+,6,5] => 7
[+,3,2,-,6,5] => 7
[-,3,2,-,6,5] => 6
[+,3,2,5,4,+] => 6
[-,3,2,5,4,+] => 7
[+,3,2,5,4,-] => 7
[-,3,2,5,4,-] => 6
[+,3,2,5,6,4] => 6
[-,3,2,5,6,4] => 5
[+,3,2,6,4,5] => 5
[-,3,2,6,4,5] => 6
[+,3,2,6,+,4] => 6
[-,3,2,6,+,4] => 7
[+,3,2,6,-,4] => 7
[-,3,2,6,-,4] => 6
[+,3,4,2,+,+] => 6
[-,3,4,2,+,+] => 7
[+,3,4,2,-,+] => 7
[+,3,4,2,+,-] => 7
[-,3,4,2,-,+] => 6
[-,3,4,2,+,-] => 6
[+,3,4,2,-,-] => 6
[-,3,4,2,-,-] => 3
[+,3,4,2,6,5] => 6
[-,3,4,2,6,5] => 5
[+,3,4,5,2,+] => 6
[-,3,4,5,2,+] => 5
[+,3,4,5,2,-] => 5
[-,3,4,5,2,-] => 2
[+,3,4,5,6,2] => 4
[-,3,4,5,6,2] => 1
[+,3,4,6,2,5] => 5
[-,3,4,6,2,5] => 4
[+,3,4,6,+,2] => 6
[-,3,4,6,+,2] => 5
[+,3,4,6,-,2] => 5
[-,3,4,6,-,2] => 2
[+,3,5,2,4,+] => 5
[-,3,5,2,4,+] => 6
[+,3,5,2,4,-] => 6
[-,3,5,2,4,-] => 5
[+,3,5,2,6,4] => 5
[-,3,5,2,6,4] => 4
[+,3,5,+,2,+] => 6
[-,3,5,+,2,+] => 7
[+,3,5,-,2,+] => 7
[+,3,5,+,2,-] => 7
[-,3,5,-,2,+] => 6
[-,3,5,+,2,-] => 6
[+,3,5,-,2,-] => 6
[-,3,5,-,2,-] => 3
[+,3,5,+,6,2] => 6
[-,3,5,+,6,2] => 5
[+,3,5,-,6,2] => 5
[-,3,5,-,6,2] => 2
[+,3,5,6,2,4] => 4
[-,3,5,6,2,4] => 3
[+,3,5,6,4,2] => 5
[-,3,5,6,4,2] => 4
[+,3,6,2,4,5] => 4
[-,3,6,2,4,5] => 5
[+,3,6,2,+,4] => 5
[-,3,6,2,+,4] => 6
[+,3,6,2,-,4] => 6
[-,3,6,2,-,4] => 5
[+,3,6,+,2,5] => 5
[-,3,6,+,2,5] => 6
[+,3,6,-,2,5] => 6
[-,3,6,-,2,5] => 5
[+,3,6,+,+,2] => 6
[-,3,6,+,+,2] => 7
[+,3,6,-,+,2] => 7
[+,3,6,+,-,2] => 7
[-,3,6,-,+,2] => 6
[-,3,6,+,-,2] => 6
[+,3,6,-,-,2] => 6
[-,3,6,-,-,2] => 3
[+,3,6,5,2,4] => 5
[-,3,6,5,2,4] => 4
[+,3,6,5,4,2] => 6
[-,3,6,5,4,2] => 5
[+,4,2,3,+,+] => 3
[-,4,2,3,+,+] => 6
[+,4,2,3,-,+] => 6
[+,4,2,3,+,-] => 6
[-,4,2,3,-,+] => 7
[-,4,2,3,+,-] => 7
[+,4,2,3,-,-] => 7
[-,4,2,3,-,-] => 6
[+,4,2,3,6,5] => 5
[-,4,2,3,6,5] => 6
[+,4,2,5,3,+] => 5
[-,4,2,5,3,+] => 6
[+,4,2,5,3,-] => 6
[-,4,2,5,3,-] => 5
[+,4,2,5,6,3] => 5
[-,4,2,5,6,3] => 4
[+,4,2,6,3,5] => 4
[-,4,2,6,3,5] => 5
[+,4,2,6,+,3] => 5
[-,4,2,6,+,3] => 6
[+,4,2,6,-,3] => 6
[-,4,2,6,-,3] => 5
[+,4,+,2,+,+] => 4
[-,4,+,2,+,+] => 7
[+,4,-,2,+,+] => 7
[+,4,+,2,-,+] => 7
[+,4,+,2,+,-] => 7
[-,4,-,2,+,+] => 8
[-,4,+,2,-,+] => 8
[-,4,+,2,+,-] => 8
[+,4,-,2,-,+] => 8
[+,4,-,2,+,-] => 8
[+,4,+,2,-,-] => 8
[-,4,-,2,-,+] => 7
[-,4,-,2,+,-] => 7
[-,4,+,2,-,-] => 7
[+,4,-,2,-,-] => 7
[-,4,-,2,-,-] => 4
[+,4,+,2,6,5] => 6
[-,4,+,2,6,5] => 7
[+,4,-,2,6,5] => 7
[-,4,-,2,6,5] => 6
[+,4,+,5,2,+] => 6
[-,4,+,5,2,+] => 7
[+,4,-,5,2,+] => 7
[+,4,+,5,2,-] => 7
[-,4,-,5,2,+] => 6
[-,4,+,5,2,-] => 6
[+,4,-,5,2,-] => 6
[-,4,-,5,2,-] => 3
[+,4,+,5,6,2] => 6
[-,4,+,5,6,2] => 5
[+,4,-,5,6,2] => 5
[-,4,-,5,6,2] => 2
[+,4,+,6,2,5] => 5
[-,4,+,6,2,5] => 6
[+,4,-,6,2,5] => 6
[-,4,-,6,2,5] => 5
[+,4,+,6,+,2] => 6
[-,4,+,6,+,2] => 7
[+,4,-,6,+,2] => 7
[+,4,+,6,-,2] => 7
[-,4,-,6,+,2] => 6
[-,4,+,6,-,2] => 6
[+,4,-,6,-,2] => 6
[-,4,-,6,-,2] => 3
[+,4,5,2,3,+] => 4
[-,4,5,2,3,+] => 5
[+,4,5,2,3,-] => 5
[-,4,5,2,3,-] => 4
[+,4,5,2,6,3] => 4
[-,4,5,2,6,3] => 3
[+,4,5,3,2,+] => 5
[-,4,5,3,2,+] => 6
[+,4,5,3,2,-] => 6
[-,4,5,3,2,-] => 5
[+,4,5,3,6,2] => 5
[-,4,5,3,6,2] => 4
[+,4,5,6,2,3] => 3
[-,4,5,6,2,3] => 2
[+,4,5,6,3,2] => 4
[-,4,5,6,3,2] => 3
[+,4,6,2,3,5] => 3
[-,4,6,2,3,5] => 4
[+,4,6,2,+,3] => 4
[-,4,6,2,+,3] => 5
[+,4,6,2,-,3] => 5
[-,4,6,2,-,3] => 4
[+,4,6,3,2,5] => 4
[-,4,6,3,2,5] => 5
[+,4,6,3,+,2] => 5
[-,4,6,3,+,2] => 6
[+,4,6,3,-,2] => 6
[-,4,6,3,-,2] => 5
[+,4,6,5,2,3] => 4
[-,4,6,5,2,3] => 3
[+,4,6,5,3,2] => 5
[-,4,6,5,3,2] => 4
[+,5,2,3,4,+] => 2
[-,5,2,3,4,+] => 5
[+,5,2,3,4,-] => 5
[-,5,2,3,4,-] => 6
[+,5,2,3,6,4] => 4
[-,5,2,3,6,4] => 5
[+,5,2,+,3,+] => 3
[-,5,2,+,3,+] => 6
[+,5,2,-,3,+] => 6
[+,5,2,+,3,-] => 6
[-,5,2,-,3,+] => 7
[-,5,2,+,3,-] => 7
[+,5,2,-,3,-] => 7
[-,5,2,-,3,-] => 6
[+,5,2,+,6,3] => 5
[-,5,2,+,6,3] => 6
[+,5,2,-,6,3] => 6
[-,5,2,-,6,3] => 5
[+,5,2,6,3,4] => 3
[-,5,2,6,3,4] => 4
[+,5,2,6,4,3] => 4
[-,5,2,6,4,3] => 5
[+,5,+,2,4,+] => 3
[-,5,+,2,4,+] => 6
[+,5,-,2,4,+] => 6
[+,5,+,2,4,-] => 6
[-,5,-,2,4,+] => 7
[-,5,+,2,4,-] => 7
[+,5,-,2,4,-] => 7
[-,5,-,2,4,-] => 6
[+,5,+,2,6,4] => 5
[-,5,+,2,6,4] => 6
[+,5,-,2,6,4] => 6
[-,5,-,2,6,4] => 5
[+,5,+,+,2,+] => 4
[-,5,+,+,2,+] => 7
[+,5,-,+,2,+] => 7
[+,5,+,-,2,+] => 7
[+,5,+,+,2,-] => 7
[-,5,-,+,2,+] => 8
[-,5,+,-,2,+] => 8
[-,5,+,+,2,-] => 8
[+,5,-,-,2,+] => 8
[+,5,-,+,2,-] => 8
[+,5,+,-,2,-] => 8
[-,5,-,-,2,+] => 7
[-,5,-,+,2,-] => 7
[-,5,+,-,2,-] => 7
[+,5,-,-,2,-] => 7
[-,5,-,-,2,-] => 4
[+,5,+,+,6,2] => 6
[-,5,+,+,6,2] => 7
[+,5,-,+,6,2] => 7
[+,5,+,-,6,2] => 7
[-,5,-,+,6,2] => 6
[-,5,+,-,6,2] => 6
[+,5,-,-,6,2] => 6
[-,5,-,-,6,2] => 3
[+,5,+,6,2,4] => 4
[-,5,+,6,2,4] => 5
[+,5,-,6,2,4] => 5
[-,5,-,6,2,4] => 4
[+,5,+,6,4,2] => 5
[-,5,+,6,4,2] => 6
[+,5,-,6,4,2] => 6
[-,5,-,6,4,2] => 5
[+,5,4,2,3,+] => 5
[-,5,4,2,3,+] => 6
[+,5,4,2,3,-] => 6
[-,5,4,2,3,-] => 5
[+,5,4,2,6,3] => 5
[-,5,4,2,6,3] => 4
[+,5,4,3,2,+] => 6
[-,5,4,3,2,+] => 7
[+,5,4,3,2,-] => 7
[-,5,4,3,2,-] => 6
[+,5,4,3,6,2] => 6
[-,5,4,3,6,2] => 5
[+,5,4,6,2,3] => 4
[-,5,4,6,2,3] => 3
[+,5,4,6,3,2] => 5
[-,5,4,6,3,2] => 4
[+,5,6,2,3,4] => 2
[-,5,6,2,3,4] => 3
[+,5,6,2,4,3] => 3
[-,5,6,2,4,3] => 4
[+,5,6,3,2,4] => 3
[-,5,6,3,2,4] => 4
[+,5,6,3,4,2] => 4
[-,5,6,3,4,2] => 5
[+,5,6,+,2,3] => 4
[-,5,6,+,2,3] => 5
[+,5,6,-,2,3] => 5
[-,5,6,-,2,3] => 4
[+,5,6,+,3,2] => 5
[-,5,6,+,3,2] => 6
[+,5,6,-,3,2] => 6
[-,5,6,-,3,2] => 5
[+,6,2,3,4,5] => 1
[-,6,2,3,4,5] => 4
[+,6,2,3,+,4] => 2
[-,6,2,3,+,4] => 5
[+,6,2,3,-,4] => 5
[-,6,2,3,-,4] => 6
[+,6,2,+,3,5] => 2
[-,6,2,+,3,5] => 5
[+,6,2,-,3,5] => 5
[-,6,2,-,3,5] => 6
[+,6,2,+,+,3] => 3
[-,6,2,+,+,3] => 6
[+,6,2,-,+,3] => 6
[+,6,2,+,-,3] => 6
[-,6,2,-,+,3] => 7
[-,6,2,+,-,3] => 7
[+,6,2,-,-,3] => 7
[-,6,2,-,-,3] => 6
[+,6,2,5,3,4] => 4
[-,6,2,5,3,4] => 5
[+,6,2,5,4,3] => 5
[-,6,2,5,4,3] => 6
[+,6,+,2,4,5] => 2
[-,6,+,2,4,5] => 5
[+,6,-,2,4,5] => 5
[-,6,-,2,4,5] => 6
[+,6,+,2,+,4] => 3
[-,6,+,2,+,4] => 6
[+,6,-,2,+,4] => 6
[+,6,+,2,-,4] => 6
[-,6,-,2,+,4] => 7
[-,6,+,2,-,4] => 7
[+,6,-,2,-,4] => 7
[-,6,-,2,-,4] => 6
[+,6,+,+,2,5] => 3
[-,6,+,+,2,5] => 6
[+,6,-,+,2,5] => 6
[+,6,+,-,2,5] => 6
[-,6,-,+,2,5] => 7
[-,6,+,-,2,5] => 7
[+,6,-,-,2,5] => 7
[-,6,-,-,2,5] => 6
[+,6,+,+,+,2] => 4
[-,6,+,+,+,2] => 7
[+,6,-,+,+,2] => 7
[+,6,+,-,+,2] => 7
[+,6,+,+,-,2] => 7
[-,6,-,+,+,2] => 8
[-,6,+,-,+,2] => 8
[-,6,+,+,-,2] => 8
[+,6,-,-,+,2] => 8
[+,6,-,+,-,2] => 8
[+,6,+,-,-,2] => 8
[-,6,-,-,+,2] => 7
[-,6,-,+,-,2] => 7
[-,6,+,-,-,2] => 7
[+,6,-,-,-,2] => 7
[-,6,-,-,-,2] => 4
[+,6,+,5,2,4] => 5
[-,6,+,5,2,4] => 6
[+,6,-,5,2,4] => 6
[-,6,-,5,2,4] => 5
[+,6,+,5,4,2] => 6
[-,6,+,5,4,2] => 7
[+,6,-,5,4,2] => 7
[-,6,-,5,4,2] => 6
[+,6,4,2,3,5] => 4
[-,6,4,2,3,5] => 5
[+,6,4,2,+,3] => 5
[-,6,4,2,+,3] => 6
[+,6,4,2,-,3] => 6
[-,6,4,2,-,3] => 5
[+,6,4,3,2,5] => 5
[-,6,4,3,2,5] => 6
[+,6,4,3,+,2] => 6
[-,6,4,3,+,2] => 7
[+,6,4,3,-,2] => 7
[-,6,4,3,-,2] => 6
[+,6,4,5,2,3] => 5
[-,6,4,5,2,3] => 4
[+,6,4,5,3,2] => 6
[-,6,4,5,3,2] => 5
[+,6,5,2,3,4] => 3
[-,6,5,2,3,4] => 4
[+,6,5,2,4,3] => 4
[-,6,5,2,4,3] => 5
[+,6,5,3,2,4] => 4
[-,6,5,3,2,4] => 5
[+,6,5,3,4,2] => 5
[-,6,5,3,4,2] => 6
[+,6,5,+,2,3] => 5
[-,6,5,+,2,3] => 6
[+,6,5,-,2,3] => 6
[-,6,5,-,2,3] => 5
[+,6,5,+,3,2] => 6
[-,6,5,+,3,2] => 7
[+,6,5,-,3,2] => 7
[-,6,5,-,3,2] => 6
[2,1,+,+,+,+] => 4
[2,1,-,+,+,+] => 7
[2,1,+,-,+,+] => 7
[2,1,+,+,-,+] => 7
[2,1,+,+,+,-] => 7
[2,1,-,-,+,+] => 8
[2,1,-,+,-,+] => 8
[2,1,-,+,+,-] => 8
[2,1,+,-,-,+] => 8
[2,1,+,-,+,-] => 8
[2,1,+,+,-,-] => 8
[2,1,-,-,-,+] => 7
[2,1,-,-,+,-] => 7
[2,1,-,+,-,-] => 7
[2,1,+,-,-,-] => 7
[2,1,-,-,-,-] => 4
[2,1,+,+,6,5] => 6
[2,1,-,+,6,5] => 7
[2,1,+,-,6,5] => 7
[2,1,-,-,6,5] => 6
[2,1,+,5,4,+] => 6
[2,1,-,5,4,+] => 7
[2,1,+,5,4,-] => 7
[2,1,-,5,4,-] => 6
[2,1,+,5,6,4] => 6
[2,1,-,5,6,4] => 5
[2,1,+,6,4,5] => 5
[2,1,-,6,4,5] => 6
[2,1,+,6,+,4] => 6
[2,1,-,6,+,4] => 7
[2,1,+,6,-,4] => 7
[2,1,-,6,-,4] => 6
[2,1,4,3,+,+] => 6
[2,1,4,3,-,+] => 7
[2,1,4,3,+,-] => 7
[2,1,4,3,-,-] => 6
[2,1,4,3,6,5] => 6
[2,1,4,5,3,+] => 6
[2,1,4,5,3,-] => 5
[2,1,4,5,6,3] => 4
[2,1,4,6,3,5] => 5
[2,1,4,6,+,3] => 6
[2,1,4,6,-,3] => 5
[2,1,5,3,4,+] => 5
[2,1,5,3,4,-] => 6
[2,1,5,3,6,4] => 5
[2,1,5,+,3,+] => 6
[2,1,5,-,3,+] => 7
[2,1,5,+,3,-] => 7
[2,1,5,-,3,-] => 6
[2,1,5,+,6,3] => 6
[2,1,5,-,6,3] => 5
[2,1,5,6,3,4] => 4
[2,1,5,6,4,3] => 5
[2,1,6,3,4,5] => 4
[2,1,6,3,+,4] => 5
[2,1,6,3,-,4] => 6
[2,1,6,+,3,5] => 5
[2,1,6,-,3,5] => 6
[2,1,6,+,+,3] => 6
[2,1,6,-,+,3] => 7
[2,1,6,+,-,3] => 7
[2,1,6,-,-,3] => 6
[2,1,6,5,3,4] => 5
[2,1,6,5,4,3] => 6
[2,3,1,+,+,+] => 6
[2,3,1,-,+,+] => 7
[2,3,1,+,-,+] => 7
[2,3,1,+,+,-] => 7
[2,3,1,-,-,+] => 6
[2,3,1,-,+,-] => 6
[2,3,1,+,-,-] => 6
[2,3,1,-,-,-] => 3
[2,3,1,+,6,5] => 6
[2,3,1,-,6,5] => 5
[2,3,1,5,4,+] => 6
[2,3,1,5,4,-] => 5
[2,3,1,5,6,4] => 4
[2,3,1,6,4,5] => 5
[2,3,1,6,+,4] => 6
[2,3,1,6,-,4] => 5
[2,3,4,1,+,+] => 6
[2,3,4,1,-,+] => 5
[2,3,4,1,+,-] => 5
[2,3,4,1,-,-] => 2
[2,3,4,1,6,5] => 4
[2,3,4,5,1,+] => 4
[2,3,4,5,1,-] => 1
[2,3,4,5,6,1] => 0
[2,3,4,6,1,5] => 3
[2,3,4,6,+,1] => 4
[2,3,4,6,-,1] => 1
[2,3,5,1,4,+] => 5
[2,3,5,1,4,-] => 4
[2,3,5,1,6,4] => 3
[2,3,5,+,1,+] => 6
[2,3,5,-,1,+] => 5
[2,3,5,+,1,-] => 5
[2,3,5,-,1,-] => 2
[2,3,5,+,6,1] => 4
[2,3,5,-,6,1] => 1
[2,3,5,6,1,4] => 2
[2,3,5,6,4,1] => 3
[2,3,6,1,4,5] => 4
[2,3,6,1,+,4] => 5
[2,3,6,1,-,4] => 4
[2,3,6,+,1,5] => 5
[2,3,6,-,1,5] => 4
[2,3,6,+,+,1] => 6
[2,3,6,-,+,1] => 5
[2,3,6,+,-,1] => 5
[2,3,6,-,-,1] => 2
[2,3,6,5,1,4] => 3
[2,3,6,5,4,1] => 4
[2,4,1,3,+,+] => 5
[2,4,1,3,-,+] => 6
[2,4,1,3,+,-] => 6
[2,4,1,3,-,-] => 5
[2,4,1,3,6,5] => 5
[2,4,1,5,3,+] => 5
[2,4,1,5,3,-] => 4
[2,4,1,5,6,3] => 3
[2,4,1,6,3,5] => 4
[2,4,1,6,+,3] => 5
[2,4,1,6,-,3] => 4
[2,4,+,1,+,+] => 6
[2,4,-,1,+,+] => 7
[2,4,+,1,-,+] => 7
[2,4,+,1,+,-] => 7
[2,4,-,1,-,+] => 6
[2,4,-,1,+,-] => 6
[2,4,+,1,-,-] => 6
[2,4,-,1,-,-] => 3
[2,4,+,1,6,5] => 6
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Created: Jul 15, 2022 at 18:19 by Martin Rubey
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Last Updated: Jul 15, 2022 at 18:19 by Martin Rubey