Support, fixed points, disjoint cycles, cycle length, disjoint-cycle decompositions, and cycle type

Let $X$ be a finite set and let $\sigma\in\operatorname{Sym}(X)$, with cycle notation and composition as in The symmetric group $\operatorname{Sym}(X)$: the bijections of a set $X$ under composition.

The support and fixed-point set of $\sigma$ are

$\operatorname{supp}(\sigma):=\{x\in X:\sigma(x)\ne x\},\qquad \operatorname{Fix}(\sigma):=\{x\in X:\sigma(x)=x\}.$

A cycle $(a_0\,a_1\,\ldots\,a_{k-1})$ has length $k$ and support $\{a_0,\ldots,a_{k-1}\}$. Two cycles are disjoint when their supports are disjoint. A disjoint-cycle decomposition of $\sigma$ is an expression for $\sigma$ as a product of pairwise disjoint cycles of length at least $2$. One-cycles are omitted, and the empty product is the identity permutation.

When $|X|=n$, the cycle type of $\sigma$ is the list of natural numbers $c_1(\sigma),\ldots,c_n(\sigma)$, where $c_k(\sigma)$ is the number of $k$-element orbits of the action generated by $\sigma$. Thus $c_1(\sigma)$ is the number of fixed points. Equivalently, the cycle type records the lengths of all cycles after each fixed point is inserted as a one-cycle.