<> 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 *one-line notation*. This is given by $\pi = [\pi(1),\ldots,\pi(n)]$. E.g., $\pi = [5,-4,2,-3,-1]$ says that $$\pi(1)=5,\pi(2)=-4,\pi(3)=2,\pi(4)=-3,\pi(5)=-1.$$ <> - 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 ====================== - 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 ========== TBA Remarks ======= TBA References ========== - [](https://en.wikipedia.org/wiki/Hyperoctahedral_group) <> Sage examples ============= {{{#!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 ======================================== - Signed permutations are graded by size. - The database contains all signed permutations of size at most 5.