Definition:Disjoint Permutations

Definition
Let $$S_n$$ denote the symmetric group on $n$ letters.

Let $$\pi, \rho \in S_n$$ both be permutations on $$S_n$$.

Then $$\pi$$ and $$\rho$$ are disjoint iff:
 * 1) $$i \notin \operatorname{Fix} \left({\pi}\right) \implies i \in \operatorname{Fix} \left({\rho}\right)$$;
 * 2) $$i \notin \operatorname{Fix} \left({\rho}\right) \implies i \in \operatorname{Fix} \left({\pi}\right)$$.

That is, each number moved by $\pi$ is fixed by $\rho$ and (equivalently) each number moved by $$\rho$$ is fixed by $$\pi$$.

We may say that:


 * $$\pi$$ is disjoint from $$\rho$$;
 * $$\rho$$ is disjoint from $$\pi$$;
 * $$\pi$$ and $$\rho$$ are (mutually) disjoint.

Note of course that it is perfectly possible for $$i \in \operatorname{Fix} \left({\pi}\right)$$ and also $$i \in \operatorname{Fix} \left({\rho}\right)$$, that is, there may well be elements fixed by more than one disjoint permutation.