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Definition & Example

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 oneline notation. This is given by $\pi = [\pi(1),\ldots,\pi(n)]$. E.g., $\pi = [5,4,2,3,1]$ says that
$$\pi(1)=5,\pi(2)=4,\pi(3)=2,\pi(4)=3,\pi(5)=1.$$
the 8 Signed permutations of size 2  
[1,2]  [1,2]  [1,2]  [1,2]  [2,1]  [2,1]  [2,1]  [2,1] 
 There are $2^n\cdot n! = 2^n \cdot 1 \cdot 2 \cdot 3 \cdots n$ signed permutations of size $n$, see A000165.
Additional information
 The group of signed permutations of size $n$is the Coxeter group of type $B_n$. It is the group of symmetries of a regular hypercube and also known under the name hyperoctahedral group.
Properties
TBA
Remarks
TBA
References
Sage examples
Technical information for database usage
 Signed permutations are graded by size.
 The database contains all signed permutations of size at most 5.
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