I have a very basic doubt here.
How actually is
$$\displaystyle \sum_{k=0}^n{S_1(n,k)} = n!$$?
For example, if we have a string 'ABCDEF', its permutation be 'BACDEF' and which cycle is this representing. Since if we are assuming it to be [AB][CDEF] then it is not counted differently in a cycle. I don't understand how to visualize this sum.
The Stirling number of the first kind $s(n,k)$ counts the number of permutations in $S_n$ with $k$ cycles. So $\sum_k s(n,k)$ counts all permutations in $S_n$; of course there are $n!$ of them.