Let $X$ be a finite set and $\sigma$ a permutation of $X$. Let $s:\langle\sigma\rangle\backslash X\rightarrow X$ be a section of the canonical surjection of $X$ onto $\langle\sigma\rangle\backslash X$.
What can be said about the relation between the sets $\{x\in X\ |\ \sigma(x)\ne x\}$ and $\{Y\in\langle\sigma\rangle\backslash X\ |\ Y\ne\{s(Y)\}\}$?
I can obviously use the non-singleton orbits to uniquely decompose $\sigma$ into cycles. But is there a way to characterize the lengths and the number of these cycles in terms of the set $\{x\in X\ |\ \sigma(x)\ne x\}$?
For example, what does the cardinality of $\{x\in X\ |\ \sigma(x)\ne x\}$ tell me about the decomposition of $\sigma$ using elements of $\{Y\in\langle\sigma\rangle\backslash X\ |\ Y\ne\{s(Y)\}\}$?