Wikipedia states that Stirling numbers of the first kind can be obtained from the following representation of the set of all permutations: $$\mathcal{P}=\text{SET}(\text{CYC}(\mathcal{Z}))$$
where $\mathcal{Z}$ is the singleton class. While I understand the general idea that permutations are a set of cycles, I do not understand why $\mathcal{Z}$ is used. Cycles of a singleton set are in my understanding just sequences of the same object - why not use the $\text{SEQ}$ operator here?
Moreover, I do not understand how Stirling numbers can be obtained from this expression. If $\text{CYC}(\mathcal{Z})$ (restricted to $n$) produces cycles on a set $A$, where $|A|=n$, then $\text{SET}(\text{CYC}(\mathcal{Z}))$ (restricted to $k$) does indeed produce permutations with $k$ cycles on $A$, and the size of this set is the desired Strinling number. But how does $\text{CYC}(\mathcal{Z})$ produce cycles on $A$?
There is a distinction between unlabelled and labelled structures. I believe your confusion is that you are thinking of the unlabelled case (in which case $\text{CYC}(\mathcal{Z})$ is uninteresting), but the discussion you quote is in the context of labelled structures. This is the difference between Chapters I and II of Flajolet's book on analytic combinatorics.
In the labeled setting, $\text{CYC}(\mathcal{Z})$ contains all cycles of $\{1, \ldots, n\}$, for any $n \ge 0$. Then applying $\text{SET}(\cdot)$ to this produces all sets of disjoint cycles, which are in bijection with permutations. I am skimming over a lot of details (the notion of a labelled product to correctly relabel things when one takes a $\text{SET}(\cdot)$ of multiple cycles); you should read Chapter II of the Flajolet book for details.