It might be a well known fact, but I wanted to clarify if it is correct to state that the number
of all possible n-permutation cycles (unsigned Stirling numbers of first kind) is factorial of n?
$$ \sum_{k=1}^{n} c(n,k) = n! $$
If correct, I assume that one could prove the statement by looking at canonical cycle notation of each permutation and argue that there exists a unique way to put parentheses to form cycles.*.
*by the lemma given in my textbook (Bona, A Walk Through Combinatorics, p130):
Lemma 6.15 (Transition Lemma). Let p : [n] → [n] be a permutation
written in canonical cycle notation. Let g(p) be the permutation obtained
from p by removing the parentheses and reading the entries as a permutation
in the one-line notation. Then g is a bijection from the set Sn of all
permutations on [n] onto Sn.
I apologize if it is an obvious statement, I just wanted to clarify that it is correct.
Best Answer
Here is the proof, from Bolker and Gleason, Counting Permuations.