Say I have an set of integers $\{1,2,3,4\}$. The cycles that I mentioned in the title are specimens of cycling notation from group theory. I want to find a formula, that would give me all possible arrangements of cycles that’d cover the set and be at least two in length. For example, with above mentioned set these configurations are possible: $$(1234)$$ $$(1243)$$ $$(1324)$$ $$(1342)$$ $$(1423)$$ $$(1432)$$ $$(12)(34)$$ $$(13)(24)$$ $$(14)(23)$$ Therefore, the expected output would be $6\cdot1+3\cdot2=12$ cycles.
Perhaps this problem can be simplified into simpler term, which is also welcome
I think you're asking for "Number of cycles in all derangement permutations of $\{1,2,\dots,n\}$," tabulated at https://oeis.org/A162973
No closed form formula is given there (which is some evidence that no closed form formula is known), but there is the asymptotic formula, $a(n) \sim e^{-1}n! (\log(n) + \gamma - 1)$, where $\gamma$ is Euler's gamma constant. You'll also find programs for computing it, and an exponential generating function.