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Definition & Example
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- A **signed permutation** of **size** $n$ is an bijection $\sigma$ of $\{\pm 1,\ldots,\pm n\}$ such that $\sigma(-i) = -\sigma(i)$.
- We usually denote a signed permutation in *one-line notation*. This is given by $\pi = [\pi(1),\ldots,\pi(n)]$. E.g., $\pi = [5,-4,2,-3,-1]$ says that
$$\pi(1)=5,\pi(2)=-4,\pi(3)=2,\pi(4)=-3,\pi(5)=-1.$$
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- There are $2^n\cdot n! = 2^n \cdot 1 \cdot 2 \cdot 3 \cdots n$ signed permutations of size $n$, see [A000165](https://oeis.org/A000165).
Additional information
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- The group of signed permutations of size $n$is the [Coxeter group](https://en.wikipedia.org/wiki/Coxeter_group) of type $B_n$. It is the group of symmetries of a regular [hypercube](https://en.wikipedia.org/wiki/Hypercube) and also known under the name *hyperoctahedral group*.
Properties
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TBA
Remarks
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TBA
References
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- [](https://en.wikipedia.org/wiki/Hyperoctahedral_group)
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Sage examples
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{{{#!sagecell
def statistic(pi):
return sum(1 for a in pi if a < 0)
for n in [1..3]:
for pi in SignedPermutations(n):
print pi,"=>",statistic(pi)
}}}
Technical information for database usage
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- Signed permutations are graded by size.
- The database contains all signed permutations of size at most 5.