Signed permutation

A signed permutation on $n$ labels is a permutation of the symbols $\{\pm 1,\dots ,\pm n\}$ satisfying $w(-i)=-w(i)$. It is specified by the list $w(1),\dots ,w(n)$, whose absolute values form a permutation of $1,\dots ,n$. For example, $(-2,3,-1)$ specifies $w(1)=-2$, $w(2)=3$, $w(3)=-1$. There are $2^{n}n!$ choices: select an ordinary permutation and an independent sign for each entry. Under composition they form the hyperoctahedral group, acting on a hypercube by permuting coordinate axes and reversing their signs.