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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 $\sigma(-i) = -\sigma(i)$.
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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
$$\pi(1)=5,\pi(2)=-4,\pi(3)=2,\pi(4)=-3,\pi(5)=-1.$$
| 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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