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

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

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

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

A cycle $(a_0{a}_1\dots a_{k-1})$ has length $k$ and support $\{a_0,\dots ,a_{k-1}\}$. 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 $\mathrm{\mid }X\mathrm{\mid }=n$, the cycle type of $\sigma$ is the list of natural numbers $c_1(\sigma ),\dots ,c_n(\sigma )$, 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.