Identifier
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
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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.
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
Created
Mar 31, 2023 at 21:09 by Martin Rubey
Updated
Mar 31, 2023 at 21:09 by Martin Rubey
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