**1 Cyclic decomposition**

Let us recall that a cycle *σ* = (*a*_{1}^{ … }*a*_{r}) in *S*_{n} is a permutation such that *σ*(*a*_{i}) = *a*_{i + 1} for i = 1,…, r – 1, *σ*(*a*_{r}) = *a*_{1}, and *σ*(*x*) = *x* for every other *x* in **n**. Two cycles (*a*_{1},…,*a*_{r}) (*b*_{1,} … *b*_{s}) in *S*_{n} are *disjoint* permutations if and only if the sets {*a*_{1},…,* a*_{r}} and {*b*_{1},…,*b*_{s}} are disjoint. Note that a cycle of length *r* can be written in *r* ways, namely, as (*a*_{1}* … a*_{r}) and (*a*_{i}*a*_{i+1} … *a*_{r}*a*_{1} … *a*_{i– 1}), *i =* 2,…, r. A cycle of length *r* is also called an *r-cycle.*

**1.1 Theorem.** *Any permutation* * is a product of pairwise disjoint cycles. This cyclic factorization is unique except for the order in which the cycles ...*