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-----------------------------------------------------------------------------
Statistic identifier: St001895
-----------------------------------------------------------------------------
Collection: Signed permutations
-----------------------------------------------------------------------------
Description: The oddness of a signed permutation.
The direct sum of two signed permutations $\sigma\in\mathfrak H_k$ and $\tau\in\mathfrak H_m$ is the signed permutation in $\mathfrak H_{k+m}$ obtained by concatenating $\sigma$ with the result of increasing the absolute value of every entry in $\tau$ by $k$.
This statistic records the number of blocks with an odd number of signs in the direct sum decomposition of a signed permutation.
-----------------------------------------------------------------------------
References: [1] Bagno, E., Biagioli, R., Novick, M., Woo, A. Depth in classical Coexter groups [[arXiv:1507.01180]]
-----------------------------------------------------------------------------
Code:
def statistic(sigma):
return sum(1 for b in decomposition(sigma)
if is_odd(b.colors().count(-1)))
def decomposition(sigma):
"""
sage: sigma = SignedPermutation([2,-4,3,-1,7,-5,6])
sage: decomposition(sigma)
[[2, -4, 3, -1], [3, -1, 2]]
sage: sigma = SignedPermutation([2,-4,3,-6,7,-5,1])
sage: decomposition(sigma)
[[2, -4, 3, -6, 7, -5, 1]]
"""
pi = sigma.permutation()
summands = []
j = 0
for i in range(1, len(pi)):
if max(pi[:i]) <= i:
summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:i]])
summands.append(summand)
j = i
summand = SignedPermutation([e-j if e > 0 else e + j for e in sigma[j:]])
summands.append(summand)
return summands
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 2
[2,1] => 0
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 0
[1,2,3] => 0
[1,2,-3] => 1
[1,-2,3] => 1
[1,-2,-3] => 2
[-1,2,3] => 1
[-1,2,-3] => 2
[-1,-2,3] => 2
[-1,-2,-3] => 3
[1,3,2] => 0
[1,3,-2] => 1
[1,-3,2] => 1
[1,-3,-2] => 0
[-1,3,2] => 1
[-1,3,-2] => 2
[-1,-3,2] => 2
[-1,-3,-2] => 1
[2,1,3] => 0
[2,1,-3] => 1
[2,-1,3] => 1
[2,-1,-3] => 2
[-2,1,3] => 1
[-2,1,-3] => 2
[-2,-1,3] => 0
[-2,-1,-3] => 1
[2,3,1] => 0
[2,3,-1] => 1
[2,-3,1] => 1
[2,-3,-1] => 0
[-2,3,1] => 1
[-2,3,-1] => 0
[-2,-3,1] => 0
[-2,-3,-1] => 1
[3,1,2] => 0
[3,1,-2] => 1
[3,-1,2] => 1
[3,-1,-2] => 0
[-3,1,2] => 1
[-3,1,-2] => 0
[-3,-1,2] => 0
[-3,-1,-2] => 1
[3,2,1] => 0
[3,2,-1] => 1
[3,-2,1] => 1
[3,-2,-1] => 0
[-3,2,1] => 1
[-3,2,-1] => 0
[-3,-2,1] => 0
[-3,-2,-1] => 1
[1,2,3,4] => 0
[1,2,3,-4] => 1
[1,2,-3,4] => 1
[1,2,-3,-4] => 2
[1,-2,3,4] => 1
[1,-2,3,-4] => 2
[1,-2,-3,4] => 2
[1,-2,-3,-4] => 3
[-1,2,3,4] => 1
[-1,2,3,-4] => 2
[-1,2,-3,4] => 2
[-1,2,-3,-4] => 3
[-1,-2,3,4] => 2
[-1,-2,3,-4] => 3
[-1,-2,-3,4] => 3
[-1,-2,-3,-4] => 4
[1,2,4,3] => 0
[1,2,4,-3] => 1
[1,2,-4,3] => 1
[1,2,-4,-3] => 0
[1,-2,4,3] => 1
[1,-2,4,-3] => 2
[1,-2,-4,3] => 2
[1,-2,-4,-3] => 1
[-1,2,4,3] => 1
[-1,2,4,-3] => 2
[-1,2,-4,3] => 2
[-1,2,-4,-3] => 1
[-1,-2,4,3] => 2
[-1,-2,4,-3] => 3
[-1,-2,-4,3] => 3
[-1,-2,-4,-3] => 2
[1,3,2,4] => 0
[1,3,2,-4] => 1
[1,3,-2,4] => 1
[1,3,-2,-4] => 2
[1,-3,2,4] => 1
[1,-3,2,-4] => 2
[1,-3,-2,4] => 0
[1,-3,-2,-4] => 1
[-1,3,2,4] => 1
[-1,3,2,-4] => 2
[-1,3,-2,4] => 2
[-1,3,-2,-4] => 3
[-1,-3,2,4] => 2
[-1,-3,2,-4] => 3
[-1,-3,-2,4] => 1
[-1,-3,-2,-4] => 2
[1,3,4,2] => 0
[1,3,4,-2] => 1
[1,3,-4,2] => 1
[1,3,-4,-2] => 0
[1,-3,4,2] => 1
[1,-3,4,-2] => 0
[1,-3,-4,2] => 0
[1,-3,-4,-2] => 1
[-1,3,4,2] => 1
[-1,3,4,-2] => 2
[-1,3,-4,2] => 2
[-1,3,-4,-2] => 1
[-1,-3,4,2] => 2
[-1,-3,4,-2] => 1
[-1,-3,-4,2] => 1
[-1,-3,-4,-2] => 2
[1,4,2,3] => 0
[1,4,2,-3] => 1
[1,4,-2,3] => 1
[1,4,-2,-3] => 0
[1,-4,2,3] => 1
[1,-4,2,-3] => 0
[1,-4,-2,3] => 0
[1,-4,-2,-3] => 1
[-1,4,2,3] => 1
[-1,4,2,-3] => 2
[-1,4,-2,3] => 2
[-1,4,-2,-3] => 1
[-1,-4,2,3] => 2
[-1,-4,2,-3] => 1
[-1,-4,-2,3] => 1
[-1,-4,-2,-3] => 2
[1,4,3,2] => 0
[1,4,3,-2] => 1
[1,4,-3,2] => 1
[1,4,-3,-2] => 0
[1,-4,3,2] => 1
[1,-4,3,-2] => 0
[1,-4,-3,2] => 0
[1,-4,-3,-2] => 1
[-1,4,3,2] => 1
[-1,4,3,-2] => 2
[-1,4,-3,2] => 2
[-1,4,-3,-2] => 1
[-1,-4,3,2] => 2
[-1,-4,3,-2] => 1
[-1,-4,-3,2] => 1
[-1,-4,-3,-2] => 2
[2,1,3,4] => 0
[2,1,3,-4] => 1
[2,1,-3,4] => 1
[2,1,-3,-4] => 2
[2,-1,3,4] => 1
[2,-1,3,-4] => 2
[2,-1,-3,4] => 2
[2,-1,-3,-4] => 3
[-2,1,3,4] => 1
[-2,1,3,-4] => 2
[-2,1,-3,4] => 2
[-2,1,-3,-4] => 3
[-2,-1,3,4] => 0
[-2,-1,3,-4] => 1
[-2,-1,-3,4] => 1
[-2,-1,-3,-4] => 2
[2,1,4,3] => 0
[2,1,4,-3] => 1
[2,1,-4,3] => 1
[2,1,-4,-3] => 0
[2,-1,4,3] => 1
[2,-1,4,-3] => 2
[2,-1,-4,3] => 2
[2,-1,-4,-3] => 1
[-2,1,4,3] => 1
[-2,1,4,-3] => 2
[-2,1,-4,3] => 2
[-2,1,-4,-3] => 1
[-2,-1,4,3] => 0
[-2,-1,4,-3] => 1
[-2,-1,-4,3] => 1
[-2,-1,-4,-3] => 0
[2,3,1,4] => 0
[2,3,1,-4] => 1
[2,3,-1,4] => 1
[2,3,-1,-4] => 2
[2,-3,1,4] => 1
[2,-3,1,-4] => 2
[2,-3,-1,4] => 0
[2,-3,-1,-4] => 1
[-2,3,1,4] => 1
[-2,3,1,-4] => 2
[-2,3,-1,4] => 0
[-2,3,-1,-4] => 1
[-2,-3,1,4] => 0
[-2,-3,1,-4] => 1
[-2,-3,-1,4] => 1
[-2,-3,-1,-4] => 2
[2,3,4,1] => 0
[2,3,4,-1] => 1
[2,3,-4,1] => 1
[2,3,-4,-1] => 0
[2,-3,4,1] => 1
[2,-3,4,-1] => 0
[2,-3,-4,1] => 0
[2,-3,-4,-1] => 1
[-2,3,4,1] => 1
[-2,3,4,-1] => 0
[-2,3,-4,1] => 0
[-2,3,-4,-1] => 1
[-2,-3,4,1] => 0
[-2,-3,4,-1] => 1
[-2,-3,-4,1] => 1
[-2,-3,-4,-1] => 0
[2,4,1,3] => 0
[2,4,1,-3] => 1
[2,4,-1,3] => 1
[2,4,-1,-3] => 0
[2,-4,1,3] => 1
[2,-4,1,-3] => 0
[2,-4,-1,3] => 0
[2,-4,-1,-3] => 1
[-2,4,1,3] => 1
[-2,4,1,-3] => 0
[-2,4,-1,3] => 0
[-2,4,-1,-3] => 1
[-2,-4,1,3] => 0
[-2,-4,1,-3] => 1
[-2,-4,-1,3] => 1
[-2,-4,-1,-3] => 0
[2,4,3,1] => 0
[2,4,3,-1] => 1
[2,4,-3,1] => 1
[2,4,-3,-1] => 0
[2,-4,3,1] => 1
[2,-4,3,-1] => 0
[2,-4,-3,1] => 0
[2,-4,-3,-1] => 1
[-2,4,3,1] => 1
[-2,4,3,-1] => 0
[-2,4,-3,1] => 0
[-2,4,-3,-1] => 1
[-2,-4,3,1] => 0
[-2,-4,3,-1] => 1
[-2,-4,-3,1] => 1
[-2,-4,-3,-1] => 0
[3,1,2,4] => 0
[3,1,2,-4] => 1
[3,1,-2,4] => 1
[3,1,-2,-4] => 2
[3,-1,2,4] => 1
[3,-1,2,-4] => 2
[3,-1,-2,4] => 0
[3,-1,-2,-4] => 1
[-3,1,2,4] => 1
[-3,1,2,-4] => 2
[-3,1,-2,4] => 0
[-3,1,-2,-4] => 1
[-3,-1,2,4] => 0
[-3,-1,2,-4] => 1
[-3,-1,-2,4] => 1
[-3,-1,-2,-4] => 2
[3,1,4,2] => 0
[3,1,4,-2] => 1
[3,1,-4,2] => 1
[3,1,-4,-2] => 0
[3,-1,4,2] => 1
[3,-1,4,-2] => 0
[3,-1,-4,2] => 0
[3,-1,-4,-2] => 1
[-3,1,4,2] => 1
[-3,1,4,-2] => 0
[-3,1,-4,2] => 0
[-3,1,-4,-2] => 1
[-3,-1,4,2] => 0
[-3,-1,4,-2] => 1
[-3,-1,-4,2] => 1
[-3,-1,-4,-2] => 0
[3,2,1,4] => 0
[3,2,1,-4] => 1
[3,2,-1,4] => 1
[3,2,-1,-4] => 2
[3,-2,1,4] => 1
[3,-2,1,-4] => 2
[3,-2,-1,4] => 0
[3,-2,-1,-4] => 1
[-3,2,1,4] => 1
[-3,2,1,-4] => 2
[-3,2,-1,4] => 0
[-3,2,-1,-4] => 1
[-3,-2,1,4] => 0
[-3,-2,1,-4] => 1
[-3,-2,-1,4] => 1
[-3,-2,-1,-4] => 2
[3,2,4,1] => 0
[3,2,4,-1] => 1
[3,2,-4,1] => 1
[3,2,-4,-1] => 0
[3,-2,4,1] => 1
[3,-2,4,-1] => 0
[3,-2,-4,1] => 0
[3,-2,-4,-1] => 1
[-3,2,4,1] => 1
[-3,2,4,-1] => 0
[-3,2,-4,1] => 0
[-3,2,-4,-1] => 1
[-3,-2,4,1] => 0
[-3,-2,4,-1] => 1
[-3,-2,-4,1] => 1
[-3,-2,-4,-1] => 0
[3,4,1,2] => 0
[3,4,1,-2] => 1
[3,4,-1,2] => 1
[3,4,-1,-2] => 0
[3,-4,1,2] => 1
[3,-4,1,-2] => 0
[3,-4,-1,2] => 0
[3,-4,-1,-2] => 1
[-3,4,1,2] => 1
[-3,4,1,-2] => 0
[-3,4,-1,2] => 0
[-3,4,-1,-2] => 1
[-3,-4,1,2] => 0
[-3,-4,1,-2] => 1
[-3,-4,-1,2] => 1
[-3,-4,-1,-2] => 0
[3,4,2,1] => 0
[3,4,2,-1] => 1
[3,4,-2,1] => 1
[3,4,-2,-1] => 0
[3,-4,2,1] => 1
[3,-4,2,-1] => 0
[3,-4,-2,1] => 0
[3,-4,-2,-1] => 1
[-3,4,2,1] => 1
[-3,4,2,-1] => 0
[-3,4,-2,1] => 0
[-3,4,-2,-1] => 1
[-3,-4,2,1] => 0
[-3,-4,2,-1] => 1
[-3,-4,-2,1] => 1
[-3,-4,-2,-1] => 0
[4,1,2,3] => 0
[4,1,2,-3] => 1
[4,1,-2,3] => 1
[4,1,-2,-3] => 0
[4,-1,2,3] => 1
[4,-1,2,-3] => 0
[4,-1,-2,3] => 0
[4,-1,-2,-3] => 1
[-4,1,2,3] => 1
[-4,1,2,-3] => 0
[-4,1,-2,3] => 0
[-4,1,-2,-3] => 1
[-4,-1,2,3] => 0
[-4,-1,2,-3] => 1
[-4,-1,-2,3] => 1
[-4,-1,-2,-3] => 0
[4,1,3,2] => 0
[4,1,3,-2] => 1
[4,1,-3,2] => 1
[4,1,-3,-2] => 0
[4,-1,3,2] => 1
[4,-1,3,-2] => 0
[4,-1,-3,2] => 0
[4,-1,-3,-2] => 1
[-4,1,3,2] => 1
[-4,1,3,-2] => 0
[-4,1,-3,2] => 0
[-4,1,-3,-2] => 1
[-4,-1,3,2] => 0
[-4,-1,3,-2] => 1
[-4,-1,-3,2] => 1
[-4,-1,-3,-2] => 0
[4,2,1,3] => 0
[4,2,1,-3] => 1
[4,2,-1,3] => 1
[4,2,-1,-3] => 0
[4,-2,1,3] => 1
[4,-2,1,-3] => 0
[4,-2,-1,3] => 0
[4,-2,-1,-3] => 1
[-4,2,1,3] => 1
[-4,2,1,-3] => 0
[-4,2,-1,3] => 0
[-4,2,-1,-3] => 1
[-4,-2,1,3] => 0
[-4,-2,1,-3] => 1
[-4,-2,-1,3] => 1
[-4,-2,-1,-3] => 0
[4,2,3,1] => 0
[4,2,3,-1] => 1
[4,2,-3,1] => 1
[4,2,-3,-1] => 0
[4,-2,3,1] => 1
[4,-2,3,-1] => 0
[4,-2,-3,1] => 0
[4,-2,-3,-1] => 1
[-4,2,3,1] => 1
[-4,2,3,-1] => 0
[-4,2,-3,1] => 0
[-4,2,-3,-1] => 1
[-4,-2,3,1] => 0
[-4,-2,3,-1] => 1
[-4,-2,-3,1] => 1
[-4,-2,-3,-1] => 0
[4,3,1,2] => 0
[4,3,1,-2] => 1
[4,3,-1,2] => 1
[4,3,-1,-2] => 0
[4,-3,1,2] => 1
[4,-3,1,-2] => 0
[4,-3,-1,2] => 0
[4,-3,-1,-2] => 1
[-4,3,1,2] => 1
[-4,3,1,-2] => 0
[-4,3,-1,2] => 0
[-4,3,-1,-2] => 1
[-4,-3,1,2] => 0
[-4,-3,1,-2] => 1
[-4,-3,-1,2] => 1
[-4,-3,-1,-2] => 0
[4,3,2,1] => 0
[4,3,2,-1] => 1
[4,3,-2,1] => 1
[4,3,-2,-1] => 0
[4,-3,2,1] => 1
[4,-3,2,-1] => 0
[4,-3,-2,1] => 0
[4,-3,-2,-1] => 1
[-4,3,2,1] => 1
[-4,3,2,-1] => 0
[-4,3,-2,1] => 0
[-4,3,-2,-1] => 1
[-4,-3,2,1] => 0
[-4,-3,2,-1] => 1
[-4,-3,-2,1] => 1
[-4,-3,-2,-1] => 0
[1,2,3,4,5] => 0
[1,2,3,4,-5] => 1
[1,2,3,-4,5] => 1
[1,2,3,-4,-5] => 2
[1,2,-3,4,5] => 1
[1,2,-3,4,-5] => 2
[1,2,-3,-4,5] => 2
[1,2,-3,-4,-5] => 3
[1,-2,3,4,5] => 1
[1,-2,3,4,-5] => 2
[1,-2,3,-4,5] => 2
[1,-2,3,-4,-5] => 3
[1,-2,-3,4,5] => 2
[1,-2,-3,4,-5] => 3
[1,-2,-3,-4,5] => 3
[1,-2,-3,-4,-5] => 4
[-1,2,3,4,5] => 1
[-1,2,3,4,-5] => 2
[-1,2,3,-4,5] => 2
[-1,2,3,-4,-5] => 3
[-1,2,-3,4,5] => 2
[-1,2,-3,4,-5] => 3
[-1,2,-3,-4,5] => 3
[-1,2,-3,-4,-5] => 4
[-1,-2,3,4,5] => 2
[-1,-2,3,4,-5] => 3
[-1,-2,3,-4,5] => 3
[-1,-2,3,-4,-5] => 4
[-1,-2,-3,4,5] => 3
[-1,-2,-3,4,-5] => 4
[-1,-2,-3,-4,5] => 4
[-1,-2,-3,-4,-5] => 5
[1,2,3,5,4] => 0
[1,2,3,5,-4] => 1
[1,2,3,-5,4] => 1
[1,2,3,-5,-4] => 0
[1,2,-3,5,4] => 1
[1,2,-3,5,-4] => 2
[1,2,-3,-5,4] => 2
[1,2,-3,-5,-4] => 1
[1,-2,3,5,4] => 1
[1,-2,3,5,-4] => 2
[1,-2,3,-5,4] => 2
[1,-2,3,-5,-4] => 1
[1,-2,-3,5,4] => 2
[1,-2,-3,5,-4] => 3
[1,-2,-3,-5,4] => 3
[1,-2,-3,-5,-4] => 2
[-1,2,3,5,4] => 1
[-1,2,3,5,-4] => 2
[-1,2,3,-5,4] => 2
[-1,2,3,-5,-4] => 1
[-1,2,-3,5,4] => 2
[-1,2,-3,5,-4] => 3
[-1,2,-3,-5,4] => 3
[-1,2,-3,-5,-4] => 2
[-1,-2,3,5,4] => 2
[-1,-2,3,5,-4] => 3
[-1,-2,3,-5,4] => 3
[-1,-2,3,-5,-4] => 2
[-1,-2,-3,5,4] => 3
[-1,-2,-3,5,-4] => 4
[-1,-2,-3,-5,4] => 4
[-1,-2,-3,-5,-4] => 3
[1,2,4,3,5] => 0
[1,2,4,3,-5] => 1
[1,2,4,-3,5] => 1
[1,2,4,-3,-5] => 2
[1,2,-4,3,5] => 1
[1,2,-4,3,-5] => 2
[1,2,-4,-3,5] => 0
[1,2,-4,-3,-5] => 1
[1,-2,4,3,5] => 1
[1,-2,4,3,-5] => 2
[1,-2,4,-3,5] => 2
[1,-2,4,-3,-5] => 3
[1,-2,-4,3,5] => 2
[1,-2,-4,3,-5] => 3
[1,-2,-4,-3,5] => 1
[1,-2,-4,-3,-5] => 2
[-1,2,4,3,5] => 1
[-1,2,4,3,-5] => 2
[-1,2,4,-3,5] => 2
[-1,2,4,-3,-5] => 3
[-1,2,-4,3,5] => 2
[-1,2,-4,3,-5] => 3
[-1,2,-4,-3,5] => 1
[-1,2,-4,-3,-5] => 2
[-1,-2,4,3,5] => 2
[-1,-2,4,3,-5] => 3
[-1,-2,4,-3,5] => 3
[-1,-2,4,-3,-5] => 4
[-1,-2,-4,3,5] => 3
[-1,-2,-4,3,-5] => 4
[-1,-2,-4,-3,5] => 2
[-1,-2,-4,-3,-5] => 3
[1,2,4,5,3] => 0
[1,2,4,5,-3] => 1
[1,2,4,-5,3] => 1
[1,2,4,-5,-3] => 0
[1,2,-4,5,3] => 1
[1,2,-4,5,-3] => 0
[1,2,-4,-5,3] => 0
[1,2,-4,-5,-3] => 1
[1,-2,4,5,3] => 1
[1,-2,4,5,-3] => 2
[1,-2,4,-5,3] => 2
[1,-2,4,-5,-3] => 1
[1,-2,-4,5,3] => 2
[1,-2,-4,5,-3] => 1
[1,-2,-4,-5,3] => 1
[1,-2,-4,-5,-3] => 2
[-1,2,4,5,3] => 1
[-1,2,4,5,-3] => 2
[-1,2,4,-5,3] => 2
[-1,2,4,-5,-3] => 1
[-1,2,-4,5,3] => 2
[-1,2,-4,5,-3] => 1
[-1,2,-4,-5,3] => 1
[-1,2,-4,-5,-3] => 2
[-1,-2,4,5,3] => 2
[-1,-2,4,5,-3] => 3
[-1,-2,4,-5,3] => 3
[-1,-2,4,-5,-3] => 2
[-1,-2,-4,5,3] => 3
[-1,-2,-4,5,-3] => 2
[-1,-2,-4,-5,3] => 2
[-1,-2,-4,-5,-3] => 3
[1,2,5,3,4] => 0
[1,2,5,3,-4] => 1
[1,2,5,-3,4] => 1
[1,2,5,-3,-4] => 0
[1,2,-5,3,4] => 1
[1,2,-5,3,-4] => 0
[1,2,-5,-3,4] => 0
[1,2,-5,-3,-4] => 1
[1,-2,5,3,4] => 1
[1,-2,5,3,-4] => 2
[1,-2,5,-3,4] => 2
[1,-2,5,-3,-4] => 1
[1,-2,-5,3,4] => 2
[1,-2,-5,3,-4] => 1
[1,-2,-5,-3,4] => 1
[1,-2,-5,-3,-4] => 2
[-1,2,5,3,4] => 1
[-1,2,5,3,-4] => 2
[-1,2,5,-3,4] => 2
[-1,2,5,-3,-4] => 1
[-1,2,-5,3,4] => 2
[-1,2,-5,3,-4] => 1
[-1,2,-5,-3,4] => 1
[-1,2,-5,-3,-4] => 2
[-1,-2,5,3,4] => 2
[-1,-2,5,3,-4] => 3
[-1,-2,5,-3,4] => 3
[-1,-2,5,-3,-4] => 2
[-1,-2,-5,3,4] => 3
[-1,-2,-5,3,-4] => 2
[-1,-2,-5,-3,4] => 2
[-1,-2,-5,-3,-4] => 3
[1,2,5,4,3] => 0
[1,2,5,4,-3] => 1
[1,2,5,-4,3] => 1
[1,2,5,-4,-3] => 0
[1,2,-5,4,3] => 1
[1,2,-5,4,-3] => 0
[1,2,-5,-4,3] => 0
[1,2,-5,-4,-3] => 1
[1,-2,5,4,3] => 1
[1,-2,5,4,-3] => 2
[1,-2,5,-4,3] => 2
[1,-2,5,-4,-3] => 1
[1,-2,-5,4,3] => 2
[1,-2,-5,4,-3] => 1
[1,-2,-5,-4,3] => 1
[1,-2,-5,-4,-3] => 2
[-1,2,5,4,3] => 1
[-1,2,5,4,-3] => 2
[-1,2,5,-4,3] => 2
[-1,2,5,-4,-3] => 1
[-1,2,-5,4,3] => 2
[-1,2,-5,4,-3] => 1
[-1,2,-5,-4,3] => 1
[-1,2,-5,-4,-3] => 2
[-1,-2,5,4,3] => 2
[-1,-2,5,4,-3] => 3
[-1,-2,5,-4,3] => 3
[-1,-2,5,-4,-3] => 2
[-1,-2,-5,4,3] => 3
[-1,-2,-5,4,-3] => 2
[-1,-2,-5,-4,3] => 2
[-1,-2,-5,-4,-3] => 3
[1,3,2,4,5] => 0
[1,3,2,4,-5] => 1
[1,3,2,-4,5] => 1
[1,3,2,-4,-5] => 2
[1,3,-2,4,5] => 1
[1,3,-2,4,-5] => 2
[1,3,-2,-4,5] => 2
[1,3,-2,-4,-5] => 3
[1,-3,2,4,5] => 1
[1,-3,2,4,-5] => 2
[1,-3,2,-4,5] => 2
[1,-3,2,-4,-5] => 3
[1,-3,-2,4,5] => 0
[1,-3,-2,4,-5] => 1
[1,-3,-2,-4,5] => 1
[1,-3,-2,-4,-5] => 2
[-1,3,2,4,5] => 1
[-1,3,2,4,-5] => 2
[-1,3,2,-4,5] => 2
[-1,3,2,-4,-5] => 3
[-1,3,-2,4,5] => 2
[-1,3,-2,4,-5] => 3
[-1,3,-2,-4,5] => 3
[-1,3,-2,-4,-5] => 4
[-1,-3,2,4,5] => 2
[-1,-3,2,4,-5] => 3
[-1,-3,2,-4,5] => 3
[-1,-3,2,-4,-5] => 4
[-1,-3,-2,4,5] => 1
[-1,-3,-2,4,-5] => 2
[-1,-3,-2,-4,5] => 2
[-1,-3,-2,-4,-5] => 3
[1,3,2,5,4] => 0
[1,3,2,5,-4] => 1
[1,3,2,-5,4] => 1
[1,3,2,-5,-4] => 0
[1,3,-2,5,4] => 1
[1,3,-2,5,-4] => 2
[1,3,-2,-5,4] => 2
[1,3,-2,-5,-4] => 1
[1,-3,2,5,4] => 1
[1,-3,2,5,-4] => 2
[1,-3,2,-5,4] => 2
[1,-3,2,-5,-4] => 1
[1,-3,-2,5,4] => 0
[1,-3,-2,5,-4] => 1
[1,-3,-2,-5,4] => 1
[1,-3,-2,-5,-4] => 0
[-1,3,2,5,4] => 1
[-1,3,2,5,-4] => 2
[-1,3,2,-5,4] => 2
[-1,3,2,-5,-4] => 1
[-1,3,-2,5,4] => 2
[-1,3,-2,5,-4] => 3
[-1,3,-2,-5,4] => 3
[-1,3,-2,-5,-4] => 2
[-1,-3,2,5,4] => 2
[-1,-3,2,5,-4] => 3
[-1,-3,2,-5,4] => 3
[-1,-3,2,-5,-4] => 2
[-1,-3,-2,5,4] => 1
[-1,-3,-2,5,-4] => 2
[-1,-3,-2,-5,4] => 2
[-1,-3,-2,-5,-4] => 1
[1,3,4,2,5] => 0
[1,3,4,2,-5] => 1
[1,3,4,-2,5] => 1
[1,3,4,-2,-5] => 2
[1,3,-4,2,5] => 1
[1,3,-4,2,-5] => 2
[1,3,-4,-2,5] => 0
[1,3,-4,-2,-5] => 1
[1,-3,4,2,5] => 1
[1,-3,4,2,-5] => 2
[1,-3,4,-2,5] => 0
[1,-3,4,-2,-5] => 1
[1,-3,-4,2,5] => 0
[1,-3,-4,2,-5] => 1
[1,-3,-4,-2,5] => 1
[1,-3,-4,-2,-5] => 2
[-1,3,4,2,5] => 1
[-1,3,4,2,-5] => 2
[-1,3,4,-2,5] => 2
[-1,3,4,-2,-5] => 3
[-1,3,-4,2,5] => 2
[-1,3,-4,2,-5] => 3
[-1,3,-4,-2,5] => 1
[-1,3,-4,-2,-5] => 2
[-1,-3,4,2,5] => 2
[-1,-3,4,2,-5] => 3
[-1,-3,4,-2,5] => 1
[-1,-3,4,-2,-5] => 2
[-1,-3,-4,2,5] => 1
[-1,-3,-4,2,-5] => 2
[-1,-3,-4,-2,5] => 2
[-1,-3,-4,-2,-5] => 3
[1,3,4,5,2] => 0
[1,3,4,5,-2] => 1
[1,3,4,-5,2] => 1
[1,3,4,-5,-2] => 0
[1,3,-4,5,2] => 1
[1,3,-4,5,-2] => 0
[1,3,-4,-5,2] => 0
[1,3,-4,-5,-2] => 1
[1,-3,4,5,2] => 1
[1,-3,4,5,-2] => 0
[1,-3,4,-5,2] => 0
[1,-3,4,-5,-2] => 1
[1,-3,-4,5,2] => 0
[1,-3,-4,5,-2] => 1
[1,-3,-4,-5,2] => 1
[1,-3,-4,-5,-2] => 0
[-1,3,4,5,2] => 1
[-1,3,4,5,-2] => 2
[-1,3,4,-5,2] => 2
[-1,3,4,-5,-2] => 1
[-1,3,-4,5,2] => 2
[-1,3,-4,5,-2] => 1
[-1,3,-4,-5,2] => 1
[-1,3,-4,-5,-2] => 2
[-1,-3,4,5,2] => 2
[-1,-3,4,5,-2] => 1
[-1,-3,4,-5,2] => 1
[-1,-3,4,-5,-2] => 2
[-1,-3,-4,5,2] => 1
[-1,-3,-4,5,-2] => 2
[-1,-3,-4,-5,2] => 2
[-1,-3,-4,-5,-2] => 1
[1,3,5,2,4] => 0
[1,3,5,2,-4] => 1
[1,3,5,-2,4] => 1
[1,3,5,-2,-4] => 0
[1,3,-5,2,4] => 1
[1,3,-5,2,-4] => 0
[1,3,-5,-2,4] => 0
[1,3,-5,-2,-4] => 1
[1,-3,5,2,4] => 1
[1,-3,5,2,-4] => 0
[1,-3,5,-2,4] => 0
[1,-3,5,-2,-4] => 1
[1,-3,-5,2,4] => 0
[1,-3,-5,2,-4] => 1
[1,-3,-5,-2,4] => 1
[1,-3,-5,-2,-4] => 0
[-1,3,5,2,4] => 1
[-1,3,5,2,-4] => 2
[-1,3,5,-2,4] => 2
[-1,3,5,-2,-4] => 1
[-1,3,-5,2,4] => 2
[-1,3,-5,2,-4] => 1
[-1,3,-5,-2,4] => 1
[-1,3,-5,-2,-4] => 2
[-1,-3,5,2,4] => 2
[-1,-3,5,2,-4] => 1
[-1,-3,5,-2,4] => 1
[-1,-3,5,-2,-4] => 2
[-1,-3,-5,2,4] => 1
[-1,-3,-5,2,-4] => 2
[-1,-3,-5,-2,4] => 2
[-1,-3,-5,-2,-4] => 1
[1,3,5,4,2] => 0
[1,3,5,4,-2] => 1
[1,3,5,-4,2] => 1
[1,3,5,-4,-2] => 0
[1,3,-5,4,2] => 1
[1,3,-5,4,-2] => 0
[1,3,-5,-4,2] => 0
[1,3,-5,-4,-2] => 1
[1,-3,5,4,2] => 1
[1,-3,5,4,-2] => 0
[1,-3,5,-4,2] => 0
[1,-3,5,-4,-2] => 1
[1,-3,-5,4,2] => 0
[1,-3,-5,4,-2] => 1
[1,-3,-5,-4,2] => 1
[1,-3,-5,-4,-2] => 0
[-1,3,5,4,2] => 1
[-1,3,5,4,-2] => 2
[-1,3,5,-4,2] => 2
[-1,3,5,-4,-2] => 1
[-1,3,-5,4,2] => 2
[-1,3,-5,4,-2] => 1
[-1,3,-5,-4,2] => 1
[-1,3,-5,-4,-2] => 2
[-1,-3,5,4,2] => 2
[-1,-3,5,4,-2] => 1
[-1,-3,5,-4,2] => 1
[-1,-3,5,-4,-2] => 2
[-1,-3,-5,4,2] => 1
[-1,-3,-5,4,-2] => 2
[-1,-3,-5,-4,2] => 2
[-1,-3,-5,-4,-2] => 1
[1,4,2,3,5] => 0
[1,4,2,3,-5] => 1
[1,4,2,-3,5] => 1
[1,4,2,-3,-5] => 2
[1,4,-2,3,5] => 1
[1,4,-2,3,-5] => 2
[1,4,-2,-3,5] => 0
[1,4,-2,-3,-5] => 1
[1,-4,2,3,5] => 1
[1,-4,2,3,-5] => 2
[1,-4,2,-3,5] => 0
[1,-4,2,-3,-5] => 1
[1,-4,-2,3,5] => 0
[1,-4,-2,3,-5] => 1
[1,-4,-2,-3,5] => 1
[1,-4,-2,-3,-5] => 2
[-1,4,2,3,5] => 1
[-1,4,2,3,-5] => 2
[-1,4,2,-3,5] => 2
[-1,4,2,-3,-5] => 3
[-1,4,-2,3,5] => 2
[-1,4,-2,3,-5] => 3
[-1,4,-2,-3,5] => 1
[-1,4,-2,-3,-5] => 2
[-1,-4,2,3,5] => 2
[-1,-4,2,3,-5] => 3
[-1,-4,2,-3,5] => 1
[-1,-4,2,-3,-5] => 2
[-1,-4,-2,3,5] => 1
[-1,-4,-2,3,-5] => 2
[-1,-4,-2,-3,5] => 2
[-1,-4,-2,-3,-5] => 3
[1,4,2,5,3] => 0
[1,4,2,5,-3] => 1
[1,4,2,-5,3] => 1
[1,4,2,-5,-3] => 0
[1,4,-2,5,3] => 1
[1,4,-2,5,-3] => 0
[1,4,-2,-5,3] => 0
[1,4,-2,-5,-3] => 1
[1,-4,2,5,3] => 1
[1,-4,2,5,-3] => 0
[1,-4,2,-5,3] => 0
[1,-4,2,-5,-3] => 1
[1,-4,-2,5,3] => 0
[1,-4,-2,5,-3] => 1
[1,-4,-2,-5,3] => 1
[1,-4,-2,-5,-3] => 0
[-1,4,2,5,3] => 1
[-1,4,2,5,-3] => 2
[-1,4,2,-5,3] => 2
[-1,4,2,-5,-3] => 1
[-1,4,-2,5,3] => 2
[-1,4,-2,5,-3] => 1
[-1,4,-2,-5,3] => 1
[-1,4,-2,-5,-3] => 2
[-1,-4,2,5,3] => 2
[-1,-4,2,5,-3] => 1
[-1,-4,2,-5,3] => 1
[-1,-4,2,-5,-3] => 2
[-1,-4,-2,5,3] => 1
[-1,-4,-2,5,-3] => 2
[-1,-4,-2,-5,3] => 2
[-1,-4,-2,-5,-3] => 1
[1,4,3,2,5] => 0
[1,4,3,2,-5] => 1
[1,4,3,-2,5] => 1
[1,4,3,-2,-5] => 2
[1,4,-3,2,5] => 1
[1,4,-3,2,-5] => 2
[1,4,-3,-2,5] => 0
[1,4,-3,-2,-5] => 1
[1,-4,3,2,5] => 1
[1,-4,3,2,-5] => 2
[1,-4,3,-2,5] => 0
[1,-4,3,-2,-5] => 1
[1,-4,-3,2,5] => 0
[1,-4,-3,2,-5] => 1
[1,-4,-3,-2,5] => 1
[1,-4,-3,-2,-5] => 2
[-1,4,3,2,5] => 1
[-1,4,3,2,-5] => 2
[-1,4,3,-2,5] => 2
[-1,4,3,-2,-5] => 3
[-1,4,-3,2,5] => 2
[-1,4,-3,2,-5] => 3
[-1,4,-3,-2,5] => 1
[-1,4,-3,-2,-5] => 2
[-1,-4,3,2,5] => 2
[-1,-4,3,2,-5] => 3
[-1,-4,3,-2,5] => 1
[-1,-4,3,-2,-5] => 2
[-1,-4,-3,2,5] => 1
[-1,-4,-3,2,-5] => 2
[-1,-4,-3,-2,5] => 2
[-1,-4,-3,-2,-5] => 3
[1,4,3,5,2] => 0
[1,4,3,5,-2] => 1
[1,4,3,-5,2] => 1
[1,4,3,-5,-2] => 0
[1,4,-3,5,2] => 1
[1,4,-3,5,-2] => 0
[1,4,-3,-5,2] => 0
[1,4,-3,-5,-2] => 1
[1,-4,3,5,2] => 1
[1,-4,3,5,-2] => 0
[1,-4,3,-5,2] => 0
[1,-4,3,-5,-2] => 1
[1,-4,-3,5,2] => 0
[1,-4,-3,5,-2] => 1
[1,-4,-3,-5,2] => 1
[1,-4,-3,-5,-2] => 0
[-1,4,3,5,2] => 1
[-1,4,3,5,-2] => 2
[-1,4,3,-5,2] => 2
[-1,4,3,-5,-2] => 1
[-1,4,-3,5,2] => 2
[-1,4,-3,5,-2] => 1
[-1,4,-3,-5,2] => 1
[-1,4,-3,-5,-2] => 2
[-1,-4,3,5,2] => 2
[-1,-4,3,5,-2] => 1
[-1,-4,3,-5,2] => 1
[-1,-4,3,-5,-2] => 2
[-1,-4,-3,5,2] => 1
[-1,-4,-3,5,-2] => 2
[-1,-4,-3,-5,2] => 2
[-1,-4,-3,-5,-2] => 1
[1,4,5,2,3] => 0
[1,4,5,2,-3] => 1
[1,4,5,-2,3] => 1
[1,4,5,-2,-3] => 0
[1,4,-5,2,3] => 1
[1,4,-5,2,-3] => 0
[1,4,-5,-2,3] => 0
[1,4,-5,-2,-3] => 1
[1,-4,5,2,3] => 1
[1,-4,5,2,-3] => 0
[1,-4,5,-2,3] => 0
[1,-4,5,-2,-3] => 1
[1,-4,-5,2,3] => 0
[1,-4,-5,2,-3] => 1
[1,-4,-5,-2,3] => 1
[1,-4,-5,-2,-3] => 0
[-1,4,5,2,3] => 1
[-1,4,5,2,-3] => 2
[-1,4,5,-2,3] => 2
[-1,4,5,-2,-3] => 1
[-1,4,-5,2,3] => 2
[-1,4,-5,2,-3] => 1
[-1,4,-5,-2,3] => 1
[-1,4,-5,-2,-3] => 2
[-1,-4,5,2,3] => 2
[-1,-4,5,2,-3] => 1
[-1,-4,5,-2,3] => 1
[-1,-4,5,-2,-3] => 2
[-1,-4,-5,2,3] => 1
[-1,-4,-5,2,-3] => 2
[-1,-4,-5,-2,3] => 2
[-1,-4,-5,-2,-3] => 1
[1,4,5,3,2] => 0
[1,4,5,3,-2] => 1
[1,4,5,-3,2] => 1
[1,4,5,-3,-2] => 0
[1,4,-5,3,2] => 1
[1,4,-5,3,-2] => 0
[1,4,-5,-3,2] => 0
[1,4,-5,-3,-2] => 1
[1,-4,5,3,2] => 1
[1,-4,5,3,-2] => 0
[1,-4,5,-3,2] => 0
[1,-4,5,-3,-2] => 1
[1,-4,-5,3,2] => 0
[1,-4,-5,3,-2] => 1
[1,-4,-5,-3,2] => 1
[1,-4,-5,-3,-2] => 0
[-1,4,5,3,2] => 1
[-1,4,5,3,-2] => 2
[-1,4,5,-3,2] => 2
[-1,4,5,-3,-2] => 1
[-1,4,-5,3,2] => 2
[-1,4,-5,3,-2] => 1
[-1,4,-5,-3,2] => 1
[-1,4,-5,-3,-2] => 2
[-1,-4,5,3,2] => 2
[-1,-4,5,3,-2] => 1
[-1,-4,5,-3,2] => 1
[-1,-4,5,-3,-2] => 2
[-1,-4,-5,3,2] => 1
[-1,-4,-5,3,-2] => 2
[-1,-4,-5,-3,2] => 2
[-1,-4,-5,-3,-2] => 1
[1,5,2,3,4] => 0
[1,5,2,3,-4] => 1
[1,5,2,-3,4] => 1
[1,5,2,-3,-4] => 0
[1,5,-2,3,4] => 1
[1,5,-2,3,-4] => 0
[1,5,-2,-3,4] => 0
[1,5,-2,-3,-4] => 1
[1,-5,2,3,4] => 1
[1,-5,2,3,-4] => 0
[1,-5,2,-3,4] => 0
[1,-5,2,-3,-4] => 1
[1,-5,-2,3,4] => 0
[1,-5,-2,3,-4] => 1
[1,-5,-2,-3,4] => 1
[1,-5,-2,-3,-4] => 0
[-1,5,2,3,4] => 1
[-1,5,2,3,-4] => 2
[-1,5,2,-3,4] => 2
[-1,5,2,-3,-4] => 1
[-1,5,-2,3,4] => 2
[-1,5,-2,3,-4] => 1
[-1,5,-2,-3,4] => 1
[-1,5,-2,-3,-4] => 2
[-1,-5,2,3,4] => 2
[-1,-5,2,3,-4] => 1
[-1,-5,2,-3,4] => 1
[-1,-5,2,-3,-4] => 2
[-1,-5,-2,3,4] => 1
[-1,-5,-2,3,-4] => 2
[-1,-5,-2,-3,4] => 2
[-1,-5,-2,-3,-4] => 1
[1,5,2,4,3] => 0
[1,5,2,4,-3] => 1
[1,5,2,-4,3] => 1
[1,5,2,-4,-3] => 0
[1,5,-2,4,3] => 1
[1,5,-2,4,-3] => 0
[1,5,-2,-4,3] => 0
[1,5,-2,-4,-3] => 1
[1,-5,2,4,3] => 1
[1,-5,2,4,-3] => 0
[1,-5,2,-4,3] => 0
[1,-5,2,-4,-3] => 1
[1,-5,-2,4,3] => 0
[1,-5,-2,4,-3] => 1
[1,-5,-2,-4,3] => 1
[1,-5,-2,-4,-3] => 0
[-1,5,2,4,3] => 1
[-1,5,2,4,-3] => 2
[-1,5,2,-4,3] => 2
[-1,5,2,-4,-3] => 1
[-1,5,-2,4,3] => 2
[-1,5,-2,4,-3] => 1
[-1,5,-2,-4,3] => 1
[-1,5,-2,-4,-3] => 2
[-1,-5,2,4,3] => 2
[-1,-5,2,4,-3] => 1
[-1,-5,2,-4,3] => 1
[-1,-5,2,-4,-3] => 2
[-1,-5,-2,4,3] => 1
[-1,-5,-2,4,-3] => 2
[-1,-5,-2,-4,3] => 2
[-1,-5,-2,-4,-3] => 1
[1,5,3,2,4] => 0
[1,5,3,2,-4] => 1
[1,5,3,-2,4] => 1
[1,5,3,-2,-4] => 0
[1,5,-3,2,4] => 1
[1,5,-3,2,-4] => 0
[1,5,-3,-2,4] => 0
[1,5,-3,-2,-4] => 1
[1,-5,3,2,4] => 1
[1,-5,3,2,-4] => 0
[1,-5,3,-2,4] => 0
[1,-5,3,-2,-4] => 1
[1,-5,-3,2,4] => 0
[1,-5,-3,2,-4] => 1
[1,-5,-3,-2,4] => 1
[1,-5,-3,-2,-4] => 0
[-1,5,3,2,4] => 1
[-1,5,3,2,-4] => 2
[-1,5,3,-2,4] => 2
[-1,5,3,-2,-4] => 1
[-1,5,-3,2,4] => 2
[-1,5,-3,2,-4] => 1
[-1,5,-3,-2,4] => 1
[-1,5,-3,-2,-4] => 2
[-1,-5,3,2,4] => 2
[-1,-5,3,2,-4] => 1
[-1,-5,3,-2,4] => 1
[-1,-5,3,-2,-4] => 2
[-1,-5,-3,2,4] => 1
[-1,-5,-3,2,-4] => 2
[-1,-5,-3,-2,4] => 2
[-1,-5,-3,-2,-4] => 1
[1,5,3,4,2] => 0
[1,5,3,4,-2] => 1
[1,5,3,-4,2] => 1
[1,5,3,-4,-2] => 0
[1,5,-3,4,2] => 1
[1,5,-3,4,-2] => 0
[1,5,-3,-4,2] => 0
[1,5,-3,-4,-2] => 1
[1,-5,3,4,2] => 1
[1,-5,3,4,-2] => 0
[1,-5,3,-4,2] => 0
[1,-5,3,-4,-2] => 1
[1,-5,-3,4,2] => 0
[1,-5,-3,4,-2] => 1
[1,-5,-3,-4,2] => 1
[1,-5,-3,-4,-2] => 0
[-1,5,3,4,2] => 1
[-1,5,3,4,-2] => 2
[-1,5,3,-4,2] => 2
[-1,5,3,-4,-2] => 1
[-1,5,-3,4,2] => 2
[-1,5,-3,4,-2] => 1
[-1,5,-3,-4,2] => 1
[-1,5,-3,-4,-2] => 2
[-1,-5,3,4,2] => 2
[-1,-5,3,4,-2] => 1
[-1,-5,3,-4,2] => 1
[-1,-5,3,-4,-2] => 2
[-1,-5,-3,4,2] => 1
[-1,-5,-3,4,-2] => 2
[-1,-5,-3,-4,2] => 2
[-1,-5,-3,-4,-2] => 1
[1,5,4,2,3] => 0
[1,5,4,2,-3] => 1
[1,5,4,-2,3] => 1
[1,5,4,-2,-3] => 0
[1,5,-4,2,3] => 1
[1,5,-4,2,-3] => 0
[1,5,-4,-2,3] => 0
[1,5,-4,-2,-3] => 1
[1,-5,4,2,3] => 1
[1,-5,4,2,-3] => 0
[1,-5,4,-2,3] => 0
[1,-5,4,-2,-3] => 1
[1,-5,-4,2,3] => 0
[1,-5,-4,2,-3] => 1
[1,-5,-4,-2,3] => 1
[1,-5,-4,-2,-3] => 0
[-1,5,4,2,3] => 1
[-1,5,4,2,-3] => 2
[-1,5,4,-2,3] => 2
[-1,5,4,-2,-3] => 1
[-1,5,-4,2,3] => 2
[-1,5,-4,2,-3] => 1
[-1,5,-4,-2,3] => 1
[-1,5,-4,-2,-3] => 2
[-1,-5,4,2,3] => 2
[-1,-5,4,2,-3] => 1
[-1,-5,4,-2,3] => 1
[-1,-5,4,-2,-3] => 2
[-1,-5,-4,2,3] => 1
[-1,-5,-4,2,-3] => 2
[-1,-5,-4,-2,3] => 2
[-1,-5,-4,-2,-3] => 1
[1,5,4,3,2] => 0
[1,5,4,3,-2] => 1
[1,5,4,-3,2] => 1
[1,5,4,-3,-2] => 0
[1,5,-4,3,2] => 1
[1,5,-4,3,-2] => 0
[1,5,-4,-3,2] => 0
[1,5,-4,-3,-2] => 1
[1,-5,4,3,2] => 1
[1,-5,4,3,-2] => 0
[1,-5,4,-3,2] => 0
[1,-5,4,-3,-2] => 1
[1,-5,-4,3,2] => 0
[1,-5,-4,3,-2] => 1
[1,-5,-4,-3,2] => 1
[1,-5,-4,-3,-2] => 0
[-1,5,4,3,2] => 1
[-1,5,4,3,-2] => 2
[-1,5,4,-3,2] => 2
[-1,5,4,-3,-2] => 1
[-1,5,-4,3,2] => 2
[-1,5,-4,3,-2] => 1
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Created: Mar 31, 2023 at 21:09 by Martin Rubey
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Last Updated: Mar 31, 2023 at 21:09 by Martin Rubey